text:
WP women's team continue their five-year old domestic dominance
What do you need to do to win 9 out of 11 trophies on offer, triumph in 21 from 23 T20-matches and mesmerize the opposition in 51 50-over matches while only losing five times?
Cobus Roodt, coach of the all-conquering Western Province women's team, provided a few insightful answers to the mystery and shared the winning formula behind one of the proudest records in world cricket.
Yet, one had to look beyond the star qualities in the team to find the correct answer.
Western Province was without their superb fast bowler Shabnim Ismail for the last season, while Bernadine Bezuidenhout immigrated to New Zealand, so it was not only the world-class stars that elevated them to champions. It was a very young squad that took care of all-comers.
Western Province won two of the three titles on offer the past season and only came a cropper in the semi-final of the national 50-over tournament in Bloemfontein where they were defeated by North West in the semi-finals, a defeat that Roodt attributes to a lack of clinical focus and perhaps a little bit of complacency.
Roodt said one of the most important qualities in a winning squad is the team culture and the family spirit. It is about team members achieving great feats for one another, playing for the star next to them in the trenches.
The annual team-building weekends away have done much to improve the bonds and friendships in the team.
Alexis le Breton and her vice-captain Robyn Appels have led the team well, made instinctive decisions on the field that aided the team's dominance.
And Roodt has also been a meticulous planner. "I always say if you fail to plan, you plan to fail."
Roodt is proud of the fact that many of the players in the squad are younger than 25, which makes their feats so much more noteworthy and extraordinary.
"Saarah Smith is 18, Laura Wolvaardt is 18, Lara Goodall is 21 and Tala Ross is 20, while Tatum le Roux is 24.
"Alexis is the eldest, but I want us to get away from the culture in the country where women retire in their mid-twenties. I feel that you must stir the passion so that the players stay excited and enthusiastic about playing," he said.
Andrie Steyn was one of the standout players, averaging 57.33 in the 50-over competition, while Laura Wolvaardt averaged 44.4 in the same competition. The spinner Appels captured 15 wickets at an average of 17.60.
Steyn averaged 65 in the T20 campaign at a strike-rate of 102.36 and Goodall boasted an average of 53 while smashing it at a strike-rate of 126.19.
He is proud of the spinner Nadia Mbokotwana, who was the Western Province school player of the year in 2016/2017. He has been nurturing her the past three years and at national level there is a growing awareness that she is a special talent and evolving fast. She did not have the best of seasons after two previous sensational years, but he is not too concerned about that, Roodt said. "Things like that happen."
Saarah Smith is an emerging middle-order batsman and averaged 33.50 the past season. He is excited about her batting and the fact that she bats around the experienced Le Breton is a bonus.
The 16-year-old Leah Jones is a bowling all-rounder and she is developing at the rate of knots.
"What is so amazing about this team is that in five years we have been virtually unbeatable and there are so many players who are not yet 20. The discipline in the team is something special, and we don't enforce team rules. The team as a unit, make the rules, I don't instil it, the team decides on those rules.
"Ultimately, the team deserves enormous credit for what they have achieved over five years of excellence," he said.
Strengthening base pivotal as award-winning UCT aims to reach next level, says elated Paterson
The enjoyment factor, focusing on fortifying and improving the second and third teams and socializing together, were key factors in the University of Cape Town (UCT) winning arguably one of the most popular and elite trophies at the Western Province Cricket Association (WPCA) awards-evening, said Don Paterson, the chairman of the club.
"I think to really have a social aspect to the club was a factor that assisted us. To play in a two-day competition when you are a student can be a challenge so the enjoyment factor is important," said Paterson.
UCT appointed an assistant first-team coach Bulelani Libazi to support Ryan Maron and Hayden Higgs took over the mantle as second-team coach.
The first team finished third in the First Division A League, the second team ended third in Reserve B, the third team won Reserve F and boasted a clean disciplinary record, Paterson said.
The club realized the importance of strengthening the quality of the second and third team, thereby increasing the depth of the squad and raising the bar for the first team. There was no room for complacency as it became more competitive, Paterson remarked.
"Over the last few years, UCT has not done well and we have fallen away a bit, but we have reassessed the structures and also emphasized the importance of enjoying the game and one another's company on the field," he said.
"The players are seriously up-beat about the award, and that our efforts have paid off. Now it is a case of onwards and upwards to the next level," he said
From Township final to semi-pro, Mlumbi's future looks bright
When Neo Mlumbi was eight, he saw the images of cricketers on a breakfast cereal box. It was the moment his intimate relationship with the game started.
Remarkably, within 365 days of playing in a township final at PPC Newlands on 12th April 2016, the 21-year-old Mlumbi, born and bred in Mandalay, was picked as one of the contracted semi-professional players of the Cape Cobras for the 2017/2018-season.
Arguably the two turning points of his career came at that township final at Newlands and in a match for the Western Province Academy against the Western Province semi-professional team.
Dropped on a duck in the final between Mandalay and Langa, Mlumbi steadied himself, struck a half-century and powered Mandalay to a win in a nerve-wrecking match that went down to the wire.
"I was trying to make an impression. There was a lot of tension at the start before I decided I needed to bat until the 20th over to ensure a win," he said.
The Western Province amateur manager, Clinton du Preez, contacted his coach and manager Mfundo Liwani and requested that he join the Western Province Academy.
Things started off a bit shakily, and in his first match he was removed for a duck.
But subsequently, his improvement was steady, and at times, spectacular.
Mlumbi struck a few half-centuries. In a clash against the Western Province semi-professional team, he managed a ton.
"I think that is when they realized that I had some talent," he said.
"I was always dreaming at school about at least playing semi-professional cricket, professional cricket and even for South Africa.
"I always liked Michael Clarke, the way he played, his personality and his conduct off the field. When he retired, Temba Bavuma became my role model. He became my hero after his century at PPC Newlands," he said.
Mlumbi said he has ironed out technical issues and is now focussing on the basics, like getting into correct positions, executing well and keeping his head still.
"I started the past week to work with Ashwell Prince and my academy coach, Siya Sibiya. It is such a privilege to work with a former South African player like Ashwell and I take everything in that he is saying to me, making sure I take full advantage of his presence," he said.
"Western Province has been fantastic in dealing with me and in aiding my development," he said.
"Last year in the off-season I worked with Laurie Ward of Montague Gardens twice a week, and my skills level improved.
"I know now, after five first-class games, that I must improve my skills level even more," he says.
"I love everything about batting – the drive, the cut, the hook – but perhaps the shot I enjoy the most, is the drive over cover," he says.
Experts like Du Preez, a selector of the South African women's team, has remarked that Mlumbi possesses the traits of a good player. He has time to play his shots, execute them at the last moment and hits the ball crisply.
Mlumbi's breakthrough season was completed when he was awarded the WPCA Academy Player of the Season at the recent WPCA Awards evening.
"My goal for this season is just to perform as well as possible. Ultimately, the 'me' is not as important as contributing well for the team's sake," Mlumbi concluded
Sibiya: Thando Ntini is a prodigious talent
It is not only the tale of the tape – he is broad-shouldered and 1.90 metres tall and his surname is Ntini – that impresses observers and analysts the most about the 16-year-old Thando Ntini with whom the Western Province Academy interim head coach and recently awarded WPCA Coach of the Year, Siya Sibiya, has worked tirelessly to propel him into the South African U19 team.
"Sure, he is Makhaya Ntini's son," says Cricket South Africa's Head of Tertiary and Youth Cricket Niels Momberg, "But what really strikes me, is his natural talent," he says.
Makhaya Ntini captured 390 wickets for South Africa and is a legendary former Proteas' fast bowler.
"Thando hits good areas regularly and nips the ball around, thereby creating pressure," said Momberg, who has been working at CSA for 19 years.
"I think he can bowl at 135 km/h once he grows a bit stronger," he said.
Sibiya thinks the young Ntini can bowl 140 km/h or faster and is a prodigious talent. He swings the new ball a bit, but moves it off the seam as well. And his dedication and determination as well as work ethic has impressed Sibiya enormously.
Western Province have taken Ntini under their wings while he is playing for Wynberg High School.
They gave him a trial run at a Colts one-day match in Durbanville and he impressed immensely.
At the Colts week in Potchefstroom, he bowled superbly and also put in some decent performances with the bat.
Sibiya said when he failed at number four against KwaZulu-Natal and struggled against the spinners, he went into the nets and spent a long time with the Cape Cobras Colts to improve his game.
In his next match, against Northerns, he was undefeated on 40 and played the spinners well.
Two years ago, while still at Selborne College, Ntini was perceived to be an extremely promising left-handed top-order star.
"Now he is seen as a potential phenomenal opening bowler who can become South Africa's first true black all-rounder," said Sibiya.
And the coach said he was enormously impressed with the determination and ambition of Ntini. He said while he was training at the Western province Academy under Sibiya, he ran from the high-performance centre to Wynberg Boys High and back. He did this four times per week.
At boarding school, he gets up six o' clock in the morning and sprints around the track. He has invited some of the other athletes to join him.
The young Ntini is aggressive, is not afraid to slip in a fast bouncer or a yorker.
The next step is representing South Africa at U19 level in July against the West Indies in Pietermaritzburg and Durban. "To a certain agree, he is already following in Makhaya's footsteps, because Makhaya represented the SA U19 team in 1995," Momberg said.
"One other thing I like about him, is his action. He has a very good action," Momberg commented.
"He is a very focused young man and he wants to go the distance, without the aid of his father," said Sibiya
All WPCA Awards Winners
A superb club cricket structure that is religiously supported by a passionate community, excellent progress by the decorated Siya Sibiya, splendid performances by the University of Cape Town club and a tsunami of runs by Pieter Malan were some of the standout-features of Western Province club cricket in 2016/2017, said Clinton du Preez, the WPCA Amateur Cricket Manager.
Western Province Cricket Association (WPCA) celebrated the most outstanding club and individual performances at the annual awards evening on Saturday 20th May.
Du Preez applauded club cricket, saying it was a huge success. The fact that 67 clubs plied their trades, is a testimony to the passion and love for the game which receives remarkable community support. "There is a good culture of community involvement in the game.
"Last season Western Province won the club cricket award on the CSA Incentive Scorecard and the only reason we dropped to number three is that we did not have a black African club in the Premier division.
"Because of geo-political changes, we will have a black African club in the Premier division and one in the first division next season," he added.
"Siya served as hub coach at Primrose, then rose to the position as assistant coach at the Western Province academy and after Salieg Nackerdien took over at Western Province, he was the interim-head coach of the Academy.
"Western Province won the category two CSA award on the incentive scorecard and Siya must be congratulated for the enormous role he played.
"He also sometimes assisted at the Cape Cobras and was instrumental in assisting Thando Ntini to get into the South African U19 team," said Du Preez.
"We must also take notice of the dwindling numbers in our club structures and the water restrictions around some of the facilities, which are sources of concern," said Du Preez.
"Generally though, we possess healthy club structures, and we can only admire the work done by administrators on a voluntary base at clubs to retain its cutting-edge qualities. Without club cricket, which is the heartbeat of cricket in every province, we would be infinitely poorer," Du Preez said.
He congratulated the University of Cape Town, the club of the year.
XstraSpace Bellville Cricket Club won the Premier League in the 2016/2017-season.
Milnerton was the last man standing in the first division A league and Strandfontein was the winner in first division B league.
Malan, who only turned out for the Durbanville club occasionally, slammed 1069 runs for the Western Province semi-professional team and was awarded with a professional contract for the 2017/2018 season.
In the Grand Challenge One-Day Cup, Rondebosch won the spoils, while Durbanville lifted the silverware in the AMA20/20 Premier League.
Malan was the first-class player of the year and Andrie Steyn the women's player of the year.
Strandfontein received the elite award as the most sporting team of the year.
UWC and Bellville shared the trophy in the women's league of 2016/2017.
Jesse Christensen was the U19-youth player of the year and Nadia Mbokotwana was his women counterpart as U19 youth player of the year.
Neo Mlumbi won the Academy Player of the Season Award.
WP Academy crowned joint category winners on CSA Incentive Scorecard.
Western Province underlined its enormous reserve strength and deep reservoir of talent as well as the solidity of its pipeline structures when the CSA Incentive Scorecard announced its winners who exceeded in delivering the basic activities and compliance requirements of the CSA operating and funding objectives on Monday.
In category two, Western Province and Northerns were announced joint winners of the Provincial Academy and Personal Performance Plan.
Salieg Nackerdien, coach of the academy and interim coach of the Western Province semi-professional team, hailed the team work and the excellent support by Graham October (administrator) and Siya Sibiya (assistant coach) in propelling the Academy to the summit in their category of the CSA Incentive Scorecard.
"We as a leadership team brain-stormed, wrestled with concepts, and our robust planning helped to ensure that the team operated optimally.
"Both Siya and I had a personal performance plan, had a one-on-one with each player to discuss our goals for the season. We had a mid-term review and an end-of-season discussion.
"We played comfortably more than 30 games. In our match against the Western Province semi-professional team, we triumphed.
"In fact, our win/loss ratio was very impressive. We won between 80 and 85 percent of all our matches.
"This accolade means the world to the academy, to all the staff and to the players. I believe it is also significant because the foundations for the next step – to the WP semi-professional team – are sound.
"This team won four out of the five one-day games in the Western Province league.
"But this was about more than just winning. It was about strictly sticking to the criteria and to ensure that the team is a living and walking billboard of the quality of the academy program," Nackerdien said.
"This is an award which belongs to the whole support staff and the team not to a single individual. It bodes well for the medium-term future of Western Province and indeed Cape Cobras' cricket."
Nabeal Dien, chief executive officer of the Cape Cobras, said he was proud of the advances made by Western Province in finishing fourth in category one behind Northerns, Gauteng and Free State.
Yet, winning the incentive scorecard in 2018 remains the ultimate goal. Western Province will assess the scorecard and take deliberate steps to improve under-performing categories.
"An idea that is not executed, is an illusion, so our practical implementation plans will be formulated soon and reviewed intermittently to lift our operational standards and achieve platinum status," Dien said.
Flint, Malan and Howley proud flag-bearers of WP and Cape Cobras at CSA awards
Arguably South Africa's most picturesque ground, PPC Newlands, also boasts the best groundsman in the country, Evan Flint, and that is official.
At the Cricket South Africa awards ceremony on Saturday, Flint scooped up the elite award as the country's finest curator, while the Western Province cricketer and new acquisition of the professional squad of the Cape Cobras, Pieter Malan lifted the cup as the Sunfoil Cup Three-Day Cricketer of the Year.
Malan scored 1069 runs at an average of 118.77 and was almost a pencil-in as three-day semi-professional cricketer of the year.
Shamiel Howley of Western Province received the prize as CSA deaf cricketer of the year.
PPC Newlands has seen steadily improving performances in the 50-over domestic showpiece. The team batting second, now enjoys almost a 50/50 chance of clinching it. Prior to Flint's arrival in 2008, the team batting first in white-ball cricket, had a distinct advantage.
With Flint at the helm, some dramatic tests between South Africa and England were on the brinks of results, but the visitors held out for draws. In 2009/2010, South Africa was one wicket away from ecstasy but England defied the attack at PPC Newlands.
In 2015/2016, England blasted 629 for six declared and South Africa replied with 627 for seven, thanks to a double ton from Hashim Amla and a maiden hundred from Temba Bavuma.
Alistair Cook and his troops almost stumbled to defeat at 159 for six in the second innings before steadying the ship. More than 85 000 people flocked to one of the most enjoyable and dramatic tests on South African soil in many years, and Flint's meticulous preparations behind the scenes set up a tantalizing contest after lunch on day five.
Flint said the arrival of a new roller in 2013 possibly made Newlands a tad more batsman-friendly and batsmen could play more freely on the up.
But he says the low totals in the 2011 Test at PPC Newlands had little to do with the pitch. He said Australia scored well in the first innings thanks to Michael Clarke's 151. Bad shot-making and good bowling was the deadly cocktail which resulted in South Africa scoring 96 in their first innings and Australia crumbling to 47 all out after Vernon Philander nipped out 5-17.
In the second innings, South Africa raced to 236 for two. "I am still scratching my head over it, but I think it is a combination of good bowling and batsmen trying to dominate too much," he added.
The same situation occurred in 2012/2013 when Dale Steyn, Philander and Morné Morkel, who were at the peak of their powers, just operated brilliantly to dismiss New Zealand for a lowly 45.
"The award is a triumph for the whole ground staff, not only for me. It doesn't happen that often that a coastal curator gets the award. It might be a tad easier up-country, as situations change a lot, and quicker at coastal grounds," says Flint.
"The Bidvest Wanderers Stadium's curator won it many times consecutively. In terms of confidence, I guess it is good to have it on your CV that you are the curator of the year, but it is more difficult to stay at the top," said Flint.
The past season, there have been some phenomenal scores at PPC Newlands in the Momentum One Day Cup campaign, which included 400 for three by the Multiply Titans and 305 for six by the Warriors.
"I guess the new Newlands might help the top and middle order a bit in the sense that you can play on the up more freely.
"But what I like about the ground is that in the first two hours of the morning session on the opening day, the ball is moving around, and only when the sun starts baking down in the afternoon does it flatten out."
Flint says it cannot be said generally that the team prescribes what the character of the pitch should be. Yet, he is not going to prepare a spinner's paradise if South Africa's strengths are their fast bowling.
It can generally be said that the South African team has become more confident and, with England, are arguably the best performing units when the ball is nipping around, Flint remarked.
"If I can have my way, I try to prepare a pitch which will last for four or five days, but it doesn't always pan out that way, and the atmospheric conditions also play a role," Flint said
Mlumbi continues to make strides at WP
– South African cricket supporters might be hard-pressed to find an answer to the question as to who Neo Mlumbi is, even if they were allowed to phone a friend. But don't despair. The answer will be on the tip of the tongue of most Cape Cobras supporters soon. Yes, his fame can spread fast as he attractively establishes his foot print with the blade as a member of the Western Province cricket team.
Mlumbi (21) has played in five first-class games, averaging 14.50 with his 47 against Easterns his finest contribution. He has produced glimpses of greatness the past two years.
The first time Clinton du Preez, amateur cricket manager of the Western Province Cricket Association, watched Mlumbi, was two years ago in a Township final between Langa and Mandalay at PPC Newlands.
Neo previously represented WP in the Youth "B" age group teams and was a member of the Mandalay CC.
"I liked the way he stroked the ball during that match, and went up to him afterwards and spoke to him," Du Preez, who is an astute talent spotter, said. He was invited to trial at the WP provincial academy. The coaches enjoyed the player and added him to the squad for the 2016/17 season.
"During the pre-season, he represented the Western Province Cricket Academy against the Western Province senior team and struck a century which gave him and the coaches a lot of confidence," reflected Du Preez.
"At the Western Province Academy he showed glimpses of good form and made his debut for the Western Province senior provincial team and made a score of 47.
Recently he participated at the national Colts week and was selected for the South African Colts team that participated in the CSA Future Cup.
"He is a very flexible player, who shows much skill in all areas with the bat around the crease and strikes the ball crisply. The upcoming winter requires that he makes the hard yards with his coaches and this will enhance his preparation for the ensuing season" he said.
"It would be far-fetched for me to say that Neo is the next Temba Bavuma, but I believe he can follow in his foot-steps," Du Preez said when comparing Mlumbi to the South African middle-order batsmen who hammered his maiden test century in front of an ecstatic PPC Newlands crowd in January 2016.
"I discussed his future goals with him when meeting him this past week and he said his vision is to establish himself in the Cape Cobras side, possibly advance and hopefully get into the South Africa A squad," Du Preez said.
If Mlumbi can continue the impressive upward curve that saw him step into the Western Provincial team the past season, the public will soon produce the answer about his identity without a moment's hesitation.
And the boardroom is abuzz with discussions about the 21-year old's talent. That's why he is one of the seven semi-professional players who have just been picked by the Western Province selection committee and the Coaches for next season.
Is he a future Bavuma? Only time will tell, but Mlumbi is determined to lift his game to new heights in 2017/2018 and create his own legacy.


text:
La série EX 600, créée par la maison de disques BAM est une nomenclature d'édition de disque EP 45 au format 17 cm / 4 titres.

Préambule
Elle débute en 1963 avec l'édition de titres de Jacques Douai, fraîchement récompensé par un grand prix du disque de l'académie Charles-Cros.

Attribution des références et faits marquants

Récompenses discographiques

Catalogue des publications phonographiques : numération détaillée

Série EX 600 (Édition des EP 17 cm / 45 mono/ puis stéréo)
Index des titres publiés au format EP en 1963
EX 600 : Jacques Douai  - À Saint-Malo
EX 601 : Jacques Douai - La Chanson de Tessa
EX 602 : Jacques Douai - Il n'aurait fallu
EX 603 : Les hommes bleus : anthologie de la musique populaire marocaine
EX 604 : Cachemire : expédition lotus - Documents sonores recueillis et enregistrés par Jacques Biltgen (cinéaste), Guy Thomas (photographe), Francine Tissot (documentaliste)
EX 605 :
EX 606 : Paul Villaz - L'anthropopithèque
EX 607 : Claire : La poupée qui chante de Jacques Saintout (BO série d'animation TV)
EX 608 : Francesca Soleville - Vingt ans
EX 609 : Lise Médini - Sourires N°2
EX 610 : Christian Borel - Surinam
EX 611 : Jacques Douai - Trois Escargots (Jacques Douai Chante pour les enfants n°4)
EX 612 : Nouvelle-Guinée - Documents sonores recueillis et enregistrés par …
EX 614 : Hélène Martin - Le Condamné à mort
EX 615 :
EX 616 : Christian Borel - Montéhus

Index des titres publiés au format EP en 1964
EX 617 : Francesca Soleville - Paris-Cayenne
EX 618 : Francesca Soleville - Nuit Et Brouillard (Reprise du standard de Jean Ferrat
EX 619 :
EX 620 : Serge Kerval - Chansons poétiques
EX 621 : Claude Fonfrède - La danse de la renaissance
EX 622 : Ping Crawford - Chansons du Far-West N°1
EX 623 :
EX 624 : Jacques Marchais - Chanson pour la nommer
EX 625 : Ping Crawford - Chansons du Far-West N°2
EX 626 : Jacques Douai & Thérèse Palau - Chants et danses de France
EX 627 :
EX 628 : Argentine - Documents folkloriques d'argentine recueillis par Jacques Cornet
EX 629 : Ile de Paques : Musique du silence (6 titres)

Index des titres publiés au format EP en 1965
EX 630 : Claude Fonfrède - Les plus fines dentelles
EX 631 : Francesca Soleville - Les Tuileries
EX 632 :
EX 633 : Marie-Claire Pichaud - Les pigeons de Poitiers
EX 634 : Serge Kerval - Chansons des pays de France
EX 635 : Francesca Soleville - Marie
EX 636 : Serge Kerval - Folklore de France
EX 637 : Marie-Claire Pichaud - Chansons de ce temps-ci
EX 638 : Philipines - Documents sonores recueillis et enregistrés par Gabriel …
EX 639 : Roumanie (4 titres)
EX 640 : Marie-Claire Pichaud - Entre terre et soleil.

Index des titres publiés au format EP en 1966
EX 641 : Francesca Soleville - La petite juiveEX 642 : Catherine Derain - PierrotEX 643 : Claude Fonfrède - Monseigneur L'AtomeEX 644 : Jacques Marchais  - Quand Biron voulut danserEX 645 : Jacques Bertin - CorentinEX 646 : Phily Form et son ensemble Western - Western folk dancesEX 647 : Guy Monfaur - Dans ma petite villeIndex des titres publiés au format EP en 1967
EX 648 : Jean Vasca - GalacteaEX 649 : Gospel Songs du Vieux Sud - Documents sonores recueillis et enregistrés par Alain Saint-Hilaire et Jean-Claude Berrer
EX 650 : Ping Crawford - Western and folksongsEX 651 : Les indiens du PanamaEX 652 : Michel Aubert - Si tu voulaisEX 653 :
EX 654 : Francesca Soleville - Viêt-Nam / Et je t'appelleEX 655 : Jacques Bertin - La BretagneEX 656 : Serge Kerval - Les coiffes noiresIndex des titres publiés au format EP en 1968
EX 657 : Indonésie - Documents sonores recueillis et enregistrés par …
EX 658 : Afghanistan - Documents sonores recueillis et enregistrés par Jacques Cornet
EX 659 : Marc Vincent - Le chameau du FinistèreIndex des titres publiés au format EP en 1969
EX 660 : Francesca Soleville - Paul ÉluardEX 661 : Henri Gougaud - Le gendarme et le voleurEX 662 : Françoise Moreau - Comptine en robe de luneEX 663 :

Index des titres publiés au format EP en 1970
EX 664 : Serge Kerval - M'en vas à la fontaineEX 665 : Jacqueline Pons - La fille de l'îleEX 666 : Francesca Soleville - Je t'aimeEX 667 : Gilles, Jules et Jean - Hugh smackEX 668 : Ensemble Achalay - El Condor el passa (musique des Andes)
EX 669 : Claude Fonfrède - J'aime bien manger, j'aime bien boire
EX 670 : Serge Kerval - Boute au vent

Index des titres publiés au format EP en 1971
EX 671 : Henri Gougaud - Nous dormirons ensemble
EX 672 : Marie-Paule Belle - Le judas
EX 673 :
EX 674 : Jacques Douai - Jacques Douai Chante pour les enfants n°5

Index des titres publiés au format EP en 1972
EX 675 : Mouloudji - Que le temps passe vite

Notes et références

Liens externes
  Article consacré au label BAM du Magazine Jukebox Mag (N° 230 page 28 -  téléchargeable en ).
 Les Disques BAM sur le site rateyourmusic.com
 Les Disques BAM sur le site discogs.com
 Musiques du monde en LP et en CD

Label discographique indépendant
Label discographique disparu

redpajama_set_name: RedPajamaWikipedia


text:
When is the next FPL deadline? Dates & times for picking Premier League fantasy team
Soham Mukherjee
12:13 AM MYT 23/12/2022
Worried about the next FPL deadline? Relax, GOAL has got you covered.
Premier League action resumes after the World Cup break and so does the excitement around Fantasy Premier League (FPL).
Gameweek 17 kicks off on Boxing Day with Brentford facing Tottenham in the first Premier League encounter in six weeks. Naturally, FPL managers are busy making changes to their squad keeping in mind the performance of the players in the World Cup.
You can make unlimited transfers to the team with no price changes but must remain careful that several players are yet to report back to the club after participating in the World Cup with their respective national teams. So, by when should you finish updating your teams? GOAL gives you the details about FPL deadlines.
Fantasy Premier League deadlines for 2022-23
The Brentford game is set to kick off 12:30 pm GMT (7:30 am ET) on December 26. So the deadline to finish updating your team is 11 am GMT (6:00 am ET). The deadline is usually set 90 minutes before the kick-off time in the first match of the Gameweek.
Let us take a look at all the 2022-23 FPL deadlines.
Gameweek
Deadline (GMT/ET)
Gameweek 1 August 5 (6:30pm/1:30pm)
Gameweek 2 August 13 (11am/6am)
Gameweek 5 August 30 (6pm/1pm)
Gameweek 6 September 3 (11am/6am)
Gameweek 7 September 10 (11am/6am)
Gameweek 8 September 16 (6:30pm/1:30p)
Gameweek 9 October 1 (11am/6am)
Gameweek 10 October 8 (1:30pm/8:30am)
Gameweek 11 October 14 (6:30pm/1:30pm)
Gameweek 12 October 18 (6pm/1pm)
Gameweek 13 October 22 (11am/6am)
Gameweek 15 November 5 (2:30pm/9:30am)
Gameweek 16 November 12 (2pm/9am)
Gameweek 17 December 26 (11am/6am)
Gameweek 18 December 30 (6:15pm/1:15pm)
Gameweek 19 January 2 (4pm/11am)
Gameweek 20 January 13 (6:30pm/1:30pm)
Gameweek 21 January 21 (11am/6am)
Gameweek 22 February 3 (6:30pm/1:30pm)
Gameweek 23 February 11 (11am/6am)
Gameweek 25 February 24 (6:30pm/1:30pm)
Gameweek 26 March 4 (1:30pm/8:30am)
Gameweek 27 March 11 (2:30pm/9:30am)
Gameweek 29 April 1 (1:30pm/8:30am)
Gameweek 31 April 15 (1:30pm/8:30am)
Gameweek 33 April 25 (6:15pm/1:15pm)
Gameweek 35 May 6 (1:30pm/8:30am)
Gameweek 36 May 13 (1:30pm/8:30am)
GOAL's ultimate FPL guide can help you with all the tips and tricks you need to navigate the rest of this season.
The Premier League have brought a breath of fresh air for struggling FPL managers with 'FPL Second Chance League', starting next week. It is a fresh competition, that will start on matchweek 17 and run until the end of the 2022-23 Premier League season where managers will be added to a new league and they would have to start from zero.
It is a wonderful opportunity for beginners or managers who have not fared well in the first phase to make an impression in this league.
The prizes on offer are the same as the top five managers at the end of the season will get hospitality tickets to a 2023-24 Premier League match along with travel and accommodation, a Bluetooth speaker, an FPL goody bag, and many more exciting hampers.
Copyright © 2023 Goal (Malaysia) All rights reserved. The information contained in Goal (Malaysia) may not be published, broadcast, rewritten, or redistributed without the prior written authority of Goal (Malaysia)

redpajama_set_name: RedPajamaCommonCrawl


text:
Home 2011: Actaball Game Preview: Cleveland (80-81) vs. Detroit (94-67)
Game Preview: Cleveland (80-81) vs. Detroit (94-67)
2011: Actaball Staff Special September 28, 2011 0 Comment
By Matt Van Wormer
One-hundred sixty-one down, one to go. The Cleveland Indians will wrap up the 2011 campaign trying to hold second place in the division outright and finish at .500 while the Detroit Tigers will be playing as hard as they can to earn first round home field advantage in the playoffs. The Tigers need a win and a Texas Rangers loss to host the Wild Card instead of playing the Yankees in the first round.
After an April and half of May that had the Indians looking like the top dogs in the American League, injuries and bad luck befell the Tribe as the Tigers closed the gap and took hold of the Central Division. I'll tell you what though, if you would have told me at the beginning of the year that the Indians would contend for 90% of the season, I would have taken that in a heartbeat.
So, tonight wraps things up with Zach McAllister (0-1, 8.53, 2.05 WHIP) taking the mound for the Tribe trying to earn his first career victory and get the Tribe to .500 on the year. McAllister pitched well in his last start, going six innings and giving up just two runs on six hits and a walk against the White Sox. He took a no-decision as the Indians lost the game, 5-4.
Rick Porcello (14-9, 4.76, 1.41 WHIP) will try to join Justin Verlander and Max Scherzer in the 15 win club in 2011 as the Tigers wrap up the regular season. The Tigers will be going for their 10th straight win against the Indians after the Tribe took six of the first 10 games played between the two teams.
The Tigers have quite a bit to be playing for so don't expect to see too many players being rested by manager Jim Leyland. The Indians will have to fight with everything they have left to close their season on a winning note.
On a personal note, I'd like to thank the readers who have read my game previews and want to encourage you to let me know if you have any thoughts, critiques or criticisms. Thank you for a wonderful season and I hope next year I get to write game previews into October!
Cleveland Indians Starting Lineup:
1. Kosuke Fukudome, RF
2. Jason Kipnis, 2B
3. Asdrubal Cabrera, SS
4. Travis Hafner, DH
5. Carlos Santana, C
6. Shelley Duncan, LF
7. Jack Hannahan, 3B
8. Matt LaPorta, 1B
9. Ezequiel Carrera, CF
Detroit Tigers Starting Lineup:
1. Austin Jackson, CF
2. Wilson Betemit, 3B
3. Delmon Young, LF
4. Miguel Cabrera, 1B
5. Victor Martinez, DH
6. Alex Avila, C
7. Jhonny Peralta, SS
8. Don Kelly, 3B
9. Ramon Santiago, 2B
Photo: Jason Miller/Getty Images
Game Previews Zach McAllister
Staff Special
Today in Tribe History: May 6, 2003
Bob Toth May 6, 2021 0 Comment
Former Indians pitcher Art Houtteman passes away in Rochester Hills, Michigan, at the age of 75.
Today in Tribe History: February 28, 1988
The Cleveland Indians sign free agent utility man Terry Francona.
Today in Tribe History: January 8, 1991
Bob Toth January 8, 2021 0 Comment
Gaylord Perry, the Cleveland Indians' first Cy Young Award winning pitcher, is elected to the National Baseball Hall of Fame.
Today in Tribe History: November 13, 2008
Bob Toth November 13, 2020 0 Comment
Starting pitcher Cliff Lee takes home the American League Cy Young Award after his 22-3 season for the Indians during the regular season. Cleveland is now home…
Remembering Jobu
Sports have a way of uniting perfect strangers behind blind loyalty, passion, and hometown pride. In our wins and our losses, we find acceptance, hope, and new…


Donald Trump fired Attorney General Jeff Sessions. Sessions 'resigned' at Trump's request and Trump replaced him with Sessions' chief of staff Matthew Whitaker, a loyalist who wrote a column for CNN last year asserting that special counsel Robert Mueller was going too far with his investigation.
The move comes amid chatter that Mueller is planning indictments, and that Donald Trump Jr. is concerned about being indicted.


text:
\section*{1. Introduction}

The subject of this paper is  a Jacobian, introduced by F. Lazzeri
(unpublished),
associated to every compact oriented riemannian manifold  whose
 dimension is twice an  odd number; if $(M,g)$
 is a compact oriented riemannian manifold  of
 dimension $n=2m =2(2k+1)$,
Lazzeri's Jacobian of $(M, g)$ is the following p.p.a.v.:

the torus $H^{m}(M,{\bf R})/(H^{m}(M,{\bf Z})/torsion)$ with the complex
structure given by the operator $\ast$ and the polarization whose imaginary
part is $(\alpha , \beta )=-\int \alpha  \wedge \beta $,  $\alpha ,\beta  \in
H^{m}(M,{\bf Z})$.

 (It is really a p.p.a.v., in fact:  the operator $\ast : H^{m}(M, {\bf R})
\rightarrow H^{m} (M, {\bf R})$
(defined through the isomorphism of $H^{m}(M, {\bf R}) $ with the space of
harmonic $m$-forms)
has square $-1$, so it induces a complex structure
on $ H^{m} (M, {\bf R})$; besides  $ \int \cdot \wedge \cdot =
\int \ast \cdot \wedge \ast \cdot$ and  $ \int \cdot \wedge \ast \cdot $
is positive definite.)

One can easily see that
the following definition is good:

{\bf Definitions 1.1.} {\it
Let  $k \in {\bf N}$ and $m =2k+1$.
 Let ${\cal R} \subset \{(M,g) \, | \;  M$
  oriented compact $C^{\infty}$
manifold of dimension $2m$, $g$ riemannian
metric on $M\} $.
Let $\sim $ be the following equivalence relation on ${\cal R}$:
 $(M_{1},g_{1}) \sim (M_{2},g_{2})$ iff there are an orientation
preserving diffeomorphism $f: M_{1} \rightarrow M_{2}$
and a $ C^{ \infty } $ map $t: M_{1} \rightarrow
{\bf R}^{+} $  such that $(M_{1}, t \, g_{1}) \in {\cal R}$ and
 $f^{\ast}g_{2}=t \, g_{1}$.
Let ${\cal A}_{h}$ be the set of
p.p.a.v.'s of dimension $h$
 up to isomorphisms.
Let
\[ T_{\cal R}:
{\cal R}/\sim \;\; \longrightarrow \; {\cal A}_{1/2 \, b_{m}(M)} \]
be the map  associating to the class of $(M,g)$
Lazzeri'Jacobian of $(M,g)$.

Now  fix  $M$
and  a symplectic basis  of $H^{m}(M,{\bf Z})/torsion$ with respect
$ - \int \cdot \wedge \cdot $ and
let  ${\cal R} \subset \{g\;|\;g\;metric\;on\;M\}$.
Then $ T_{{\cal R}}$ can be lifted   to a map
        \[ \hat{T}_{\cal R}:{\cal R}/ conformal\, equivalence
\;\longrightarrow \; {\cal H}_{ 1/2 \, b_{m}(M)}, \]
where ${\cal H}_{h}$  is the $h$-Siegel upper half space
(two metrics on $M$, $g_{1}$ and $g_{2}$, are said conformal equivalent
iff $g_{1} =t  g_{2}$ where $t$ is a
$ C^{ \infty }$ map: $M \rightarrow {\bf R}^{+}$).

We  often omit the subscripts ${\cal R}$ in $T_{\cal R}$ and
$\hat{T}_{\cal R}$.}

If we consider the metrics of constant curvature equal to $-1$ on
a $C^{\infty}$  compact surface, we have that Lazzeri's
Jacobian is the   usual Jacobian of the correspondent Riemann surface.

The outline of the paper is the following:
Section 2 deals with
 Torelli and Schottky  problem for Lazzeri's Jacobian
 of tori with flat metrics,
in Sect. 3 we study Lazzeri's Jacobian for K\"{a}hler manifolds and its
relationship with other Jacobians and
in Sect. 4  we examine Lazzeri's Jacobian of a bundle.

\vspace{0.1cm}

 To state our results on Lazzeri's
Jacobian we fix here some notation   we will use in all the
paper.

{\bf Notation  1.2.} {\em
$\bullet$ {\boldmath ${\cal A}_{h}$, ${\cal H}_{h}$:}
defined above;
we will often drop the subscripts $h$.

$\bullet$  {\boldmath $G_{{\bf R}}(h,V), G_{{\bf C}}(k,W)$.}
 $G_{{\bf R}}(h,V)$ is the Grassmannian of the real
 $h$-subspaces of the real vector space $V$,
 $G_{{\bf C}}(k,W)$ is the Grassmannian of the complex
 $k$-subspaces of the complex vector space $W$.

$\bullet$ {\boldmath${\cal F}_{n}$.}
We define ${\cal F}_{n}= \{({\bf R}^{n}/\Lambda,g)\; | \; \Lambda \;\;
lattice, \;\; g \;\; flat \;\; metric \;\; on \;\;
 {\bf R}^{n}/\Lambda \}/\sim$,
where $\sim $ is the equivalence relation defined in Def. 1.1; here it
becomes
$({\bf R}^{n}/\Lambda, g) \sim ({\bf R}^{n}/\Lambda', g')$ iff $\exists$
a  orientation preserving map
 $f: {\bf R}^{n}/\Lambda \rightarrow
{\bf R}^{n}/\Lambda'$ induced by a linear map $ {\bf R}^{n} \rightarrow {\bf R}^{n} $
such that $f^{\ast}g'=c g$ for some $c \in {\bf R}^{+}$, where
the orientation is the standard one of ${\bf R}^{n} $, (in fact
  if $({\bf R}^{n}/ \Lambda, g) $ and $
({\bf R}^{n}/ \Lambda', g')$ are equivalent for $\sim $  through
a  map $\varphi$, then $\varphi $ is given by an affine map ${\bf R}^{n}
\rightarrow {\bf R}^{n}$).

$\bullet$ {\boldmath${\cal T}_{n}$.}
Set    ${\cal T}_{n} := \{ T \in M(n \times n,  {\bf R}) \mid  T  \; lower
\; triangular, \; T_{i,i} >0 \; \forall i , \; det \, T = 1  \} $.

$ \bullet $ Let $m \in {\bf N}$ and $n=2m$.
Consider the standard basis of ${\bf R}^{n}$, $\{e_{i}\}_{i=1,..n}$.
We call {\bf lexicographic ordered basis} of
$\wedge^{m} {\bf R}^{n} $ the following ordered basis:
$e_{I}$, $ I= (i_{1},..., i_{m})  \in {{\bf N}}^{m}$  with $ 1 \leq
i_{1} < i_{2} < ... < i_{m} \leq n $,
  with the multindeces ordered by the  lexicographic order.
Let ${\cal I} : = \{ I = (1,  i_{2} ,.., i_{m} ) \in  {\bf N}^{m}
 \,|\; 1< i_{2} < ..< i_{m}
 \leq n \}$. If $I \in {\cal I}$, we  choose and fix forever
one of the  multindeces $J$ such that we obtain
$ (I, J)$ from $(1,....,n)$ with an even number of
transpositions and we
 call it  $\tilde{I}$. Let $ {\cal E} = \{ \tilde{I}\, |\,
I \in {\cal I} \}$.
We want to define another ordered basis of
$\wedge^{m} {\bf R}^{n} $:
  we take first the multindices  in ${\cal I}$  in the lexicographic
order: we call them  $I_{1}, I_{2},...$;
 then we consider  $\tilde{I}_{1}, \tilde{I}_{2},...$;
  we call {\bf the symmetric lexicographic ordered basis}
of $\wedge^{m} {\bf R}^{n} $ the ordered basis
$e_{I_{1} } , e_{I_{2}}, ...,e_{\tilde{I}_{1} } , e_{\tilde{I}_{2}}, ...,$
and we call ``the symmetric lexicographic order'' the order
 $I_{1}, I_{2},...,\tilde{I}_{1}, \tilde{I}_{2},...$ of the multindeces
of ${\cal I} \cup {\cal E}$. Analogously for $({\bf R}^{n}) ^{\vee}$ instead
of ${\bf R}^{n}$.

$\bullet$ Let $M$ be a complex manifold.
 If  $H^{q}(M,{\bf C})= V \oplus W $, $\pi_{V,W}$  will
 denote  the projection onto $V$.
We will often omit the subscript $V,W$ in $\pi_{V,W}$ when it will be clear.

$\bullet$ {\boldmath
$ K(M,g)$, $K'(M,g)$, $ F^{p}_{q}(M)$, $J_{q}(M)$.}
Let  $(M,g)$ be a compact K\"{a}hler
manifold of complex dimension $m$; let $\Omega $ be the
(1,1)-form associated to $g$.

{\it $K= K(M,g):=\{[\eta] \in H^{m}(M,{\bf C})| \,
[ \eta] = \sum_{r \geq 0,\, r  \equiv -m(m+1)/2 \; (mod \,2)}
[\Omega^{r} \eta_{r}] $ with $\eta_{r}$ primitive form  of degree
$m-2r  \}$ and

$K'= K'(M,g):=\{[\eta]
\in H^{m}(M,{\bf C})| \, [\eta] = \sum_{r \geq 0, \,r
\equiv 1-m(m+1)/2 \; (mod \,2)} [\Omega^{r} \eta_{r}] $
 with $\eta_{r}$ primitive form of degree $m-2r \}$}.

$ F^{p}_{q}(M)  :=  \oplus_{a+b=q,\; a \geq p} H^{a,b}(M,{\bf C})$
(we will omit the subscript $q$ when no confusion can arise).

    If $q$ is an odd number, we define
$ J_{q}(M)$ $ = \oplus_{a+b=q,\; a-b \equiv 1 \; (4)} H^{a,b}(M,{\bf C})$.}

\vspace{0.2cm}

First we examine the case of flat metrics on tori.
  Let $n=2m= 2(2k+1)$ and $ N := \frac{1}{2} \dim \wedge^{m}{\bf R}^{n}$;
we  study  $T:{\cal F}_{n} \rightarrow {\cal A}_{N}$
and, choosing a symplectic basis, we study $\hat{T}$ and
in  particular Im $\hat{T}$ (and then Im $T$):

{\bf Theorem A. }
{\it {\bf i)} Choosen any symplectic basis of $H^{m}({\bf R}^{n}/{\bf Z}^{n},{\bf Z})$
with respect to $ -   \int \cdot \wedge \cdot$, the map
$\hat{T}:   \{g\;|\;g\;flat\; metric\;on\;{\bf R}^{n}/{\bf Z}^{n}\}
/ conf.\; equiv . \;\longrightarrow \;{\cal H}_{N} $
is injective.

{\bf ii)} Now
call $\{ dx_{1},..., dx_{n}\}$
 the standard basis of $({\bf R}^{n})^{\vee}$.
If we take  as a symplectic basis of $H^{m}({\bf R}^{n}/{\bf Z}^{n},{\bf Z})$
with respect to $ -   \int \cdot \wedge \cdot$,
the sym. lex. ordered basis
$\{ dx_{I} = dx_{i_{1}} \wedge ... \wedge dx_{i_{m}}
\}_{I \in {\cal I} \cup {\cal E}}$, we have that
$Im  \hat{T}=
\{ X + i Y \in {\cal H}_{N} \mid 1) \; and \; 2) \; hold \}$, where:

1) \(Y= E E^{t}\) where
 $E_{I,J}= det (T)_{I,J} $, $I,J \in {\cal I}$,
${\cal I}$  ordered in  lex. order., for some $ T $
upper triangular matrix $ n \times n $
 with determinant $1$ and $ T_{i,i} \geq 0$
(in  particular $E$ is upper triangular
with positive diagonal elements and the entries of every column
of $E$ are the Pl\"{u}cker coordinates of an element of
$G_{{\bf R}}(2k, {\bf R}^{n-1})$ and the same for the entries of every
 row of $E$).

2) $  X_{IJ}= 0$  if  $I$ and $ J$  have more than
one index in common,
where  $I,J \in {\cal I} $, ${\cal I}$
ordered in  lex. order.

{\bf iii)}   Lazzeri's Jacobian of a generic flat oriented torus has not
nontrivial automorphisms as p.p.a.v..

{\bf iv)} The map $T: {\cal F}_{n} \longrightarrow
{\cal A}_{N}$ is generically locally injective, {\bf v)} but not injective. }

In Section 3 we consider Lazzeri's Jacobian for K\"{a}hler manifolds $(M,g)$
and we
see that it depends only on the complex structure of $M$  and on the
cohomology class of the $(1,1)$-form associated to $g$; using  that, we
examine  the relationship of Lazzeri's Jacobian with Weil's  and
Griffiths' Jacobians:

{\bf Theorem  B.} {\it
Let $(M,g)$ be a compact K\"{a}hler manifold
of complex dimension $m=2k+1$; (in the sequel
$\pi $  is $ \pi_{J_{m}(M), \overline{J_{m}(M)}}$);
we have that, as complex tori,

$\bullet $ $k^{th}$ Weil's Jacobian is
$H^{m}(M,{\bf R})/(H^{m}(M,{\bf Z})/torsion)$ with
the complex structure given by $C$
(or $J_{m}(M)/ \pi(H^{m}(M,{\bf Z}))$ with the complex structure given by
$i$),

$\bullet $ $k^{th}$ Griffiths' Jacobian is
 $H^{m}(M,{\bf R})/(H^{m}(M,{\bf Z})/torsion)$ with
the complex structure given by $C$ on the set $\{\eta + \overline{\eta }\mid\;
\eta \in J_{m}(M) \cap \overline{F^{k+1}(M)}\}$
and $-C $ on $\{\eta + \overline{\eta } \mid \;
\eta \in J_{m}(M) \cap F^{k+1}(M)\}$
(or $J_{m}(M)/ \pi(H^{m}(M,{\bf Z}))$ with the complex structure
$i|_{J_{m}(M) \cap   \overline{F^{k+1}(M)}}
 \oplus -i|_{J_{m}(M)\cap   F^{k+1}(M)}$),

$\bullet $
Lazzeri's Jacobian is $H^{m}(M,{\bf R})/(H^{m}(M,{\bf Z})/torsion)$ with
the complex structure  $C|_{H^{m}(M,{\bf R}) \cap K} \oplus
-C|_{H^{m}(M,{\bf R}) \cap K'}$
(or $J_{m}(M)/ \pi(H^{m}(M,{\bf Z}))$
with the complex structure  $i|_{J_{m}(M) \cap K}
\oplus -i|_{J_{m}(M) \cap K'}$);

$k^{th}$ Weil's Jacobian and Lazzeri's Jacobian
are p.p.a.v.'s (also if the 1-1 form associated to the metric isn't rational)
and  the real part of the polarization is the same (equal to $\int \cdot
\wedge \ast \cdot$ on $H^{m}(M,{\bf R})/(H^{m}(M,{\bf Z})/torsion)$).}

Besides
we consider another class of Ricci-flat metrics (besides the flat ones on tori)
 K\"{a}hler-Einstein metrics on the complex manifolds with trivial
canonical bundle and  we find another  local Torelli
theorem  for Lazzeri's Jacobian ({\bf Corollary C}),  and
one could formulate the following conjecture:

{\bf Conjecture.} {\it $\hat{T}_{\cal R}$ is locally
 injective if ${\cal R} $ is a set of Ricci-flat metrics on a manifold $M$.}

Finally in Section 4
 given a bundle $F  \rightarrow M$, we study the relationship
between Lazzeri's Jacobian of $F$ and Lazzeri's Jacobian of $M$
{\bf (Prop. D)}.

We think
that one can easily find  open problems about Lazzeri's Jacobian,
for instance to go on with the study of Schottky and Torelli type problems,
 to study Prym-Tyurin varieties for Lazzeri's Jacobians and
the relationship between Lazzeri's Jacobian and the theory of degeneration of
abelian varieties or more precisely
to study  a possible ``object'' $T(M,g_{0})$ to
associate to every compact oriented manifold $M$  of dimension
$2(2k+1)$ with a singular metric
$g_{0}$ (i.e.   $(g_{0})_{P}$ is
semipositive definite $ \forall \, P \in M$),
such that, if $g_{t}$  are riemannian
metrics on $M$ for $t>0$
 and $g_{t} \longrightarrow g_{0}$ for $t \longrightarrow 0$,
 then $T(M,g_{t}) \longrightarrow T(M,g_{0})$ in some sense.
Observe that if we consider  a torus $M= {\bf R}^{2(2k+1)}/ \Lambda $
 with a singular flat metric $g_{0}$, one could define $T(M,g_{0})$ in the
 following way:
let $g$ be a flat metric  on $M$ extending  $g_{0}$ (i.e.
  $(g_{0})_{P}= (g)_{P} $
on $ (\ker (g_{0})_{P})^{\perp_{g}}$ $\forall P \in M$,
or equivalently for one point of $M$);
we define  $T(M, g_{0})= {\cal H}^{2k+1}_{g}(M, {\bf R})/
\psi ({\cal H}^{2k+1}_{g}(M, {\bf Z}))$ with the complex structure $\ast_{g}$,
where ${\cal H}^{2k+1}_{g}$ is the set of
  harmonic $(2k+1)$-forms for $g$,
i.e. invariant $(2k+1)$-forms,
$\psi $ is defined pointwise as the projection  onto
$ \wedge^{2k+1}( (\ker (g_{0})_{P})^{\perp_{g}})^{\vee}$ $ \forall  P$
and $\ast_{g}$ the  operator $\ast $ for $g$
(one can easily see that this definition doesn't depend on the metric $g$
extending $g_{0}$, see Appendix.)

\section*{2. The case of tori with flat metrics}

\subsection*{2.a. Some lemmas of linear algebra}

{\bf Lemma 2.1.} {\it
 Let $ \Omega $ be a real upper  triangular matrix
 $n \times n$, where $n=2m$. In the lex. ordered  basis of
$\wedge^{m} {\bf R}^{n} $, the matrix
$\bigwedge ^{m} \Omega $ is upper triangular.
Besides, if  $ \bigwedge ^{m} \Omega $ is equal to
{\footnotesize\( \left( \begin{array}{cc}
                                                  A & C \\
                                                  B & D
\end{array} \right) \)}
in the sym. lex. ordered  basis of $\wedge^{m} {\bf R}^{n} $,
then     $B=0$, $A$ is  upper triangular and $D$ lower triangular.}

{\sc Proof.} Left to the reader. \hfill $\Box$

{\bf Lemma 2.2.} {\it  Let $ \Omega $ be a real upper triangular matrix
 $n \times n$, where $n=2m$.

Let us fix an ordered  basis of $\wedge^{m} {\bf R}^{n} $
defined in the following way:
order in any fixed way the multindeces
 $I  $ in  $  {\cal I}$; call them $K_{1}, K_{2},..  $,
 then, order the multindices in $ {\cal E} $ in the following way:
$\tilde{K}_{1}, \tilde{K}_{2},.. $; consider
$e_{K_{1}} , e_{K_{2} },...,e_{\tilde{K}_{1}} , e_{\tilde{K}_{2}},...$;
(for instance the sym.  lex. ordered basis).

In this ordered basis of $\wedge^{m} {\bf R}^{n} $
 let $ \bigwedge ^{m} \Omega $ be
{\footnotesize\( \left( \begin{array}{cc}
                                                  A & C \\
                                                  B & D
\end{array} \right). \)}
We have:

a)$A^{t}D= (\det\,\Omega ) I$;

b)suppose $\det \Omega \neq 0$; for $I \in {\cal I}$ and $J \in  {\cal E}$}

\[ (A^{-1}C)_{IJ}= \left  \{ \begin{array}{ll}
0                        & if \; \tilde{I}\; and\; J\; have\; more\; than
\;one\;index\; in\; common \\
\pm \frac{\Omega_{1,l}}{\Omega_{1,1}}
& if\; \tilde{I} \;and\; J \;have\; only\; l\;
in\; common
\end{array}
    \right    . \]

{\sc Proof.}
If $P$ and $Q$ are two multindeces, $\Omega_{P,Q} $ will denote
                   the determinant of the minor $P,Q$ of $\Omega$.
We will denote  the $j^{th}$-column of $\Omega$ by $v_{j}$ and
$v_{j}-\Omega_{1,j}e_{1}$ by  $\overline{v}_{j}$.

{\bf  a)}
     Let   $I=(i_{1},...,i_{m}) \in {\cal I}$
and  $J=(j_{1},...,j_{m}) \in {\cal E}$.
 Observe that

$(A^{t}D)_{I,J} e_{1} \wedge ...\wedge e_{n}
=\sum_{S \in {\cal I} }
  \Omega_{S,I} \Omega_{\tilde{S},J} e_{1} \wedge ...\wedge e_{n}=$

$= ( \sum_{S \in {\cal I} }
  \Omega_{S,I}  e_{s_{1}} \wedge ....\wedge e_{s_{m}}) \wedge
(\sum_{T \in  {\cal E}}
 \Omega_{T,J} e_{t_{1}} \wedge ...\wedge e_{t_{m}})= $

$= ( \sum_{S \in {\cal I} \cup {\cal E}}
  \Omega_{S,I}  e_{s_{1}} \wedge ....\wedge e_{s_{m}}) \wedge
(\sum_{T \in {\cal I} \cup {\cal E}}
 \Omega_{T,J} e_{t_{1}} \wedge ...\wedge e_{t_{m}})=$

$=     v_{i_{1}} \wedge ...\wedge v_{i_{m}} \wedge v_{j_{1}} \wedge ...\wedge
v_{j_{m}}   =
\left\{ \begin{array}{ll}
     0  & \mbox{ if $J \neq \tilde{I}$ } \\
 (\det\, \Omega) \, e_{1} \wedge ...\wedge e_{n} & \mbox{ if $J = \tilde{I}$ }
       \end{array}
\right . $

where $S=(s_{1},...,s_{m})$, $T=(t_{1},...,t_{m})$,
and the last but one  equality holds because $ \Omega_{S,I}= 0 $ for
$ S \in {\cal E}$ and $I \in {\cal I}$.

{\bf b)}
Let    $I, J \in {\cal E}$.
We remark that, since neither $I$ nor $J$  contain $1$, $I$ and $J$ have some
 index in common.    Let  $I=(i_{1},...,i_{m})$ and
 $J=(j_{1},...,j_{m})$  with $l =i_{r}=j_{s}$.
We have:
\begin{eqnarray*}
& &
(\det \Omega) (A^{-1}C)_{IJ} e_{1} \wedge ... \wedge e_{n} \stackrel{a)}{=}
 (D^{t}C)_{IJ} e_{1} \wedge ... \wedge e_{n}=
 (\sum_{K  \in { \cal I} }  \Omega _{\tilde{K},I}
\Omega _{ K,J})e_{1} \wedge ... \wedge e_{n} =  \\
& = &\overline{v}_{i_{1}} \wedge ...\wedge \overline{v}_{i_{m}} \wedge (
v_{j_{1}} \wedge... \wedge v_{j_{m}}- \overline{v}_{j_{1}} \wedge... \wedge
\overline{v}_{j_{m}})
 =  \overline{v}_{i_{1}} \wedge ...\wedge \overline{v}_{i_{m}} \wedge
v_{j_{1}} \wedge... \wedge v_{j_{m}}  =  \\
& = & \overline{v}_{i_{1}} \wedge ...\wedge \overline{v}_{i_{m}} \wedge
v_{j_{1}} \wedge...\wedge v_{j_{s-1}} \wedge  \Omega_{1,l} e_{1}
\wedge v_{j_{s+1}} \wedge ... \wedge v_{j_{m}}  = \\
& = &  \overline{v}_{i_{1}} \wedge ...\wedge \overline{v}_{i_{m}} \wedge
\overline{v}_{j_{1}} \wedge...\wedge \overline{v}_{j_{s-1}} \wedge
\Omega_{1,l}e_{1}
\wedge\overline{v}_{j_{s+1}} \wedge ... \wedge \overline{v}_{j_{m}} =  \\
& = &  \left\{ \begin{array}{ll}
0     & if \, I\, and\, J\, have\, more\, than \,one\,index \, in\, common \\
\epsilon \, \frac{\Omega_{1,l}}{\Omega_{1,1}} \, \det \, \Omega \,
 e_{1} \wedge ... \wedge e_{n}                  &
if\, I \,and\, J \,have\, only\, l\,
in\, common
\end{array}          \right .
\end{eqnarray*}
where $\epsilon $ is the sign of the permutation taking $(i_{1},...,i_{m},
j_{1},...,j_{s-1},1,j_{s+1},...,j_{m})$ in $(1,...,n)$.
\hfill $ \Box $

{\bf Lemma 2.3.} {\it
Let $m \in {\bf N} $ and $n=2m$.
Consider  the set  of positive scalar products on $R^{n}$ up to conformal
equivalence.
The map, defined on this set, associating to the class
of a positive definite scalar
product its operator  $\ast $ on $\wedge^{m} ({\bf R}^{n})^{ \vee }$,
is injective.}

{\sc Proof.}
 We observe that,
 given a scalar product on  ${\bf R}^{n}$, two elements of ${\bf R}^{n}$, $v$ and
 $w$, are perpendicular iff there exists a  $m$-subspace of ${\bf R}^{n}$
perpendicular to $w$ and containing $v$; thus $v$ and
 $w$ are perpendicular iff $\exists \; \alpha \in \wedge^{m-1} {\bf R}^{n}$,
$\alpha $ simple (i.e. of the kind: $v_{1} \wedge ...\wedge v_{m-1}$)
such that $w \wedge \ast (\alpha \wedge v)=0 $  and $ \alpha \wedge v \neq 0$.
Thus $\ast$ determines the conformal structure.
 \hfill $\Box$

\subsection*{2.b. Proof of Theorem A}

{\bf Remark 2.4.} {\it
The set    $\{({\bf R}^{n}/ {\bf Z}^{n},g) \; |  \; g \;
  flat \;  metric \; on \; {\bf R}^{n}/{\bf Z}^{n}\}/ conf. \, equivalence\;$
 is in bijection with the set  $P_{n}$
of symmetric positive definite matrices
$n \times n$ with determinant $1$,  thus with $ {\cal T}_{n}$.
         The set    ${\cal F}_{n}= \{({\bf R}^{n}/ {\bf Z}^{n},g) \; |  \; g \;
  flat \;  metric \; on \; {\bf R}^{n}/{\bf Z}^{n}\}/ \sim \; =
P_{n}/ SL(n, {\bf Z})$ (where $A \in SL(n, {\bf Z})$ acts on $P_{n}$  by
$P \mapsto A^{t} P A $) and we endow it with the quotient topology induced by
the set above.}

\vspace{0.1cm}

{\bf Proposition 2.5.} {\it
Let  $n := 2m= 2(2k+1)$
and $N:= \frac{1}{2} {\footnotesize \left( \begin{array}{c}
                                                 n  \\
                                                 m
\end{array} \right)}$.

Let $ L \in GL(n, {\bf R})$;
let $\wedge ^{2k+1} (L^{-1})^{t}  =$
{\footnotesize $ \left( \begin{array}{cc}
                                                  A( L) & C( L) \\
                                                  B( L) & D( L)
\end{array} \right)$}
in the  sym. lex. order;
set
{\footnotesize $ \left( \begin{array}{c}
                                               X( L)  \\
                                               Y( L)
\end{array} \right) $} $=$
{\footnotesize  $ \left( \begin{array}{cc}
                                              A( L) &  -B( L) \\
                                              B( L) &   A ( L)
\end{array} \right)^{-1} $}
{\footnotesize $ \left( \begin{array}{c}
                                               C( L) \\
                                               D( L)
\end{array} \right); $}
define $Z( L)=X( L)+iY( L) \in M(N \times N, {\bf C} )$.

  Let   $det \, L > 0$.
Call $\{ dx_{1},..., dx_{n}\}$
 the standard basis of $({\bf R}^{n})^{\vee}$.
Choose the sym. lex. ordered basis
$\{ dx_{I} = dx_{i_{1}} \wedge ... \wedge dx_{i_{2k+1}}
\}_{I \in {\cal I} \cup {\cal E}}$
 as symplectic basis of
$H^{m}({\bf R}^{n}/ {\bf Z}^{n},{\bf Z})$ for
$ - \int \cdot \wedge \cdot$.
We have that $Z(L) =  \hat{T} ({\bf R}^{n} /  {\bf Z}^{n}, L^{t}L)$
(the orientation of $ {\bf R}^{n} / {\bf Z}^{n} $
is given by the standard one of ${\bf R}^{n}$).}

{\sc Proof.}
Let $b_{i}$ be  the columns of  $L^{-1}$, they are an orthonormal basis
of $ {\bf R}^{n}$ for the metric $ L^{t} L$.

The matrix representing the ordered basis
$\{dx_{I}\}_{I \in {\cal I} \cup {\cal E}}$  in function of
the sym. lex. ordered basis of
$H^{m}({\bf R}^{n}/ {\bf Z}^{n},{\bf R})$,
$\{b_{I}^{\vee}\}_{I \in {\cal I} \cup {\cal E}}$
 is $\wedge ^{2k+1} (L^{-1})^{t}=$
{\footnotesize $ \left( \begin{array}{cc}
                                                  A & C \\
                                                  B & D
\end{array} \right)$}.
Then the matrix expressing
$\{dx_{I}, \ast dx_{I} \}_{I \in {\cal I}}$
   in function of
the sym. lex. ordered basis
$\{b_{I}^{\vee}\}_{I \in {\cal I} \cup {\cal E}}$
is
{\footnotesize $ \left( \begin{array}{cc}
                                              A  &  -B \\
                                              B  &   A
\end{array} \right)$}.
Thus the matrix expressing $\{dx\}_{I \in  {\cal E}}$
  in function of  $\{dx_{I}, \ast dx_{I} \}_{I \in {\cal I}}$
is:
{\footnotesize $ \left( \begin{array}{cc}
                                              A  &  -B \\
                                              B  &   A
\end{array} \right)^{-1} $}
{\footnotesize $ \left( \begin{array}{c}
                                               C \\
                                               D
\end{array} \right). $}
   \hfill $\Box$

\vspace{0.1cm}

{\sc Proof of Theorem A.} In proving ii), iii) and v) we use Prop. 2.5
 and its notation.

{\bf i)}
The map $\hat{T}$ is injective  by  Lemma 2.3 and because
the harmonic forms on tori are the translations-invariant forms.

{\bf ii) }
Observe that, if $L \in {\cal T}_{n}$, then
 $ \wedge ^{2k+1} (L^{-1})^{t} =
  {\footnotesize \left( \begin{array}{cc}
                            A(L) &  C(L) \\
                               0 &   (A(L)^{t})^{-1}
\end{array} \right)}$
by Lemmas 2.1 and 2.2.

Then
 $X(L)= A(L)^{-1} C(L)$ and
 Lemma 2.2 implies the  claim for $X$ .

Besides  $Y(L)=  A^{-1}(L) (A^{-1}(L))^{t}$,
where  $A(L)_{I,J}= \det ((L^{-1})^{t})_{I,J} $;
 then we have
$Y(L)=  A^{-1}(L) (A^{-1}(L))^{t} =  A(L^{-1}) (A(L^{-1}))^{t}$
where $A(L^{-1})_{I,J}= \det (L^{t})_{I,J} $.
Taking $ E = A(L^{-1}) $ and $ T = L^{t} $ we conclude.

{\bf iii)} Let $G$ be the modular group.
If $F_{\sigma} := \{ x \in {\cal H}_{N} \mid \sigma (x) =x \} $
 for   $\sigma \in G^{\ast } := G - \{ Id \}$,
we have to prove that ${\cal T}_{n}- Z^{-1}(\cup_{\sigma \in G^{\ast }}
 F_{\sigma})$ is a open dense subset of ${\cal T}_{n}$.
The openness follows from the fact  that
$G$  acts properly and discontinously on  ${\cal H}$
(see \cite{L-B} p. 218).
To prove the density, it is sufficient (by Baire's theorem)
to  prove that $ \forall \sigma  \in G^{\ast}$,
the set ${\cal T}_{n} - Z^{-1}( F_{\sigma})$
is a open dense subset of ${\cal  T}_{n}$; since
$Z^{-1}( F_{\sigma})$ is defined by
polynomial equations,  we have only to prove that
${\cal T}_{n} - Z^{-1}( F_{\sigma}) \neq \emptyset$ i.e.
that $Z({\cal T}_{n}) \subset F_{\sigma}$ implies $\sigma = Identity$.

Let $\sigma \in G $ be the  map $Z \mapsto (MZ+N)(PZ+Q)^{-1}$.
Obviously $Z \in F_{\sigma}$  iff
 $ZPZ+ZQ=MZ+N$.

Let $L \in {\cal T}_{n}$; for every $c \in {\bf R}^{+}$, let
$\tilde{L}(c)$ be the matrix obtained from $L$
by  multiplying $L_{1,1}$
by $c^{-4k-1}$ and the other entries of $L$ by $c$;
 we have  $Z(\tilde{L}(c))=c^{4k+2}Z(L)$; since
  $Z(\tilde{L}(c) ) \in F_{\sigma}$ $\forall c \in {\bf R}^{+}$,
 we   have
$c^{8k+4} Z(L)PZ(L)+c^{4k+2}(Z(L)Q-MZ(L))-N=0$
 $\forall c \in {\bf R}^{+} $, $ \forall L \in {\cal T}_{n}$;
 thus:

1) $N=0$  \hspace{1cm}
2) $Z(L)PZ(L)=0$  $\forall \;  L \in {\cal T}_{n}$ \hspace{1cm}
3) $Z(L)Q-MZ(L)=0$ $\forall \; L \in {\cal T}_{n}$

Hence:  $Q=(M^{t})^{-1}$ by 1) and  $\sigma \in G $;
 $P=0$, $M=Q$ by 2) and 3), taking $L=I$. Thus $M$ is
orthogonal.
From 3), we have: $Y(L)M=MY(L)$ and  this
implies $M$ diagonal (take  $L $ in such way that
$Y(L)$ diagonal with the elements
 of the diagonal different one from another).
Being $M$  orthogonal and diagonal,
  $M$ must be  diagonal  with only $\pm 1$ on the diagonal;
still from $Y(L)=MY(L)M^{-1}$, taking $L$ in such way that
$Y(L)$ not diagonal, we obtain $M=I$.

{\bf iv)} It follows right away from i) and iii), since
  $G$ acts properly and discontinously on  ${\cal H}$.

{\bf v) } Let us consider two diagonal matrices
$\in {\cal T}_{n}$, one inverse of the other:
$F$ and $F^{-1}$.

We have: $ C(F)   =  B(F) = C(F^{-1})   =  B(F^{-1}) =0  $  and
$ A(F)   =  A(F^{-1})^{-1} $,  $D(F)=  D(F^{-1})^{-1}$.
Thus $Y(F^{-1})=Y(F)^{-1}$ and $X(F^{-1})=X(F)=0$.
Thus $Z(F)=-Z(F^{-1})^{-1}$; so the $Z$'s differ by a modular map.
Thus Lazzeri's Jacobians of $ ({\bf R}^{n} /  {\bf Z}^{n},  F^{2})$ and
of $ ({\bf R}^{n} / {\bf Z}^{n} , (F^{-1})^{2})$ are isomorphic p.p.a.v.'s.
But  we can choose  $F$ in such way  that
$\not \exists  A \in SL(n, {\bf Z}) $ s.t. $ A^{t} F^{2} A = (F^{-1})^{2} $.
 \hfill $ \Box $

\section*{3. The case of K\"{a}hler manifolds}

   We recall some facts on the  operators $ \ast $, $ C $ and $L$
(see for instance  \cite{Ch}, \cite{Wei}, \cite{G-H}, \cite{Wel}, \cite{Gre}).

\vspace{0.1cm}

Let  $M$ be a complex manifold of complex dimension
$m$; Weil's operator $C$ is defined on
the forms of bedegree $(a,b)$ by $C \eta = i^{a-b} \eta $ and then  is
extended by linearity to arbitrary forms;  observe that $C$
 takes real forms to  real forms.

If $M$ is  compact
and of  K\"{a}hlerian type,  then, $ \forall q \in {\bf N}$ odd,
the operator $C$ defines a complex structure on $H^{q}(M,{\bf C})$ (and
thus on $H^{q}(M,{\bf R})$) depending only on the complex structure of $M$;
the $i$-eigenspace of $C$ is
$J_{q} (M) = \oplus_{a+b=q, a-b \equiv 1 \; (4)} H^{a,b}(M,{\bf C})$.

\vspace{0.1cm}

Now suppose  that $(M,g)$
 is a hermitian manifold and let
$\Omega $ be the $(1,1)$-real form associated to $g$; let
 $L$ be the operator on the set of forms defined by
 $L \eta = \Omega  \wedge \eta $ and $ \Lambda $
the adjoint operator of $L$.
A form $ \eta $ such that $ \Lambda  \eta =0 $ is said primitive.
Every form $\omega $ of degree $q$ can be written uniquely in the form
$\omega = \sum_{r \geq (q-m)^{+}} L^{r} \omega_{r}$, where $\omega_{r}$ is a
primitive form of degree $q-2r$.
If  $g$ is of K\"{a}hler
then $L$ defines an operator on $  H^{q}(M, {\bf C})$;
 a class $[\omega] \in H^{q}(M,{\bf C})$
is said primitive if $[\Lambda \omega]= 0$ and a class $[\omega ] \in
H^{q}(M,{\bf C})$ can be written uniquely in the form
$[\omega] = \sum_{r \geq (q-m)^{+}} L^{r} [\omega_{r}]$, where $[\omega_{r}]$
 is a  primitive class of degree $q-2r$.

\vspace{0.1cm}

{\bf Warning 3.1.} Here the operator $\ast
$, defined as usual  on  real forms, is extended to complex forms by
${\bf C}$-linearity as in \cite{Ch} and \cite{Wei}.

\vspace{0.2cm}

{\bf Lemma 3.2.} {\it
  Let $(M, g)$ be a hermitian manifold of complex
 dimension $m$.
Let $ \eta $ be  a $m$-form on $M$. If we write
$ \eta = \sum_{r \geq
0} L^{r} \eta_{r} $ with $\eta_{r}$ primitive $(m-2r)$-form,
 we have that:
\[ \ast \eta = \sum_{r \geq 0} (-1)^{\frac{m^{2}+m}{2}+r} L^{r} C \eta_{r},
\;\;\;\;\;\;\;\;\;\;\;
  C \eta = \sum_{r \geq 0} L^{r} C \eta_{r}. \]
Then we observe that we can decompose the space of the m-forms in two parts:

$\{ \eta = \sum_{r \geq 0,\, r \, even}
L^{r} \eta_{r} $ with $\eta_{r}$ primitve $(m-2r)$-form$  \}$ and

$\{ \eta = \sum_{r \geq 0, \,r \,odd}
L^{r} \eta_{r} $ with $\eta_{r}$ primitve $(m-2r)$-form$ \}$;

on the first part $ \ast =  (-1)^{\frac{m^{2}+m}{2}} C$,
on the second one  $ \ast =  (-1)^{\frac{m^{2}+m}{2}+1} C$. }

{\sc Proof.}
In \cite{Ch} p. 26 or \cite{Wel} the following formula is proved:
{\it if $\omega $ is a primitive $p$-form on $M$  and
 $r \leq m-p$ ($m= \dim _{\bf C} M$), then}
\(\;\;
 \ast L^{r} \omega = (-1)^{\frac{p(p+1)}{2}} \frac{r!}{(m-p-r)!} L^{m-p-r}
 C \omega . \)

Applying it, with $p=m-2r$, to each term $L^{r} \eta_{r} $
of the sum $ \eta = \sum_{r \geq 0} L^{r} \eta_{r} $,  we obtain:
\[ \ast \eta = \ast \sum_{r \geq 0} L^{r} \eta_{r} = \sum_{r \geq
0} \ast L^{r} \eta_{r} =  \sum_{r \geq 0} (-1)^{\frac{m^{2}+m}{2}+r} L^{r}
 C \eta_{r}. \]

Since $C L=L C$, we have:

\vspace{0.1cm}

\hspace{5cm}
$ C \eta = \sum_{r \geq 0} C L^{r}  \eta_{r}
=\sum_{r \geq 0} L^{r} C \eta_{r}.$
 \hfill $\Box$

\vspace{0.2cm}

{\bf Corollary 3.3.} {\it
Let $(M, g)$ be a compact  K\"{a}hler  manifold
of complex dimension $m$. Consider the operators
$ \ast $ and $C$ on $H^{m}(M,{\bf C})$.
We have that $\ast =C$ on $K$ and $ \ast =-C$ on $K'$
(see Not. 1.2).}

\vspace{0.2cm}

Let $(M,g)$ be a compact K\"{a}hler  manifold of complex dimension $m$.
We recall the definitions of Weil's and Griffiths' Jacobians
(see for instance \cite{Ch}, \cite{Wei}, \cite{G-H}, \cite{Gre}).

$ \bullet $  Suppose that the 1-1 real
form $\Omega $ associated to the metric is
rational; let $p \in {\bf N}$ with $p \leq m-1$;

 $p^{th}$ {\it Weil's Jacobian}   is an abelian variety so defined:

the torus $H^{2p+1}(M,{\bf R})/(H^{2p+1}(M,{\bf Z})/torsion)$ with
the complex structure given by $C$  and the polarization whose real part is
${\cal R}(\alpha, \beta)   =\int \alpha \wedge \ast \beta$.

$ \bullet $ Let $p \in {\bf N} $, $p \leq m-1$;
$p^{th}$ {\it Griffiths' Jacobian} is the following complex torus:

the torus $H^{2p+1}(M,{\bf C})/ (F^{p+1}(M) +H^{2p+1}(M,{\bf Z}))$
$=\overline{F^{n-p+1}(M)}/ \pi (H^{2p+1}(M,{\bf Z})) $
 with the complex structure given by $i$.

\vspace{0.15cm}

{\sc Proof of Theorem B.}
 We recall that $H^{m}(M,{\bf C})= K \oplus K'$  and that,
by Corollary 3.3, $ \ast= - C$ on $K$ and  $\ast =  C$ on $K'$.
To prove the Theorem  we have only to observe that
 $H^{m}(M,{\bf R})=(H^{m}(M,{\bf R}) \cap K) \oplus
(H^{m}(M,{\bf R}) \cap K')$
and $J_{m}=(J_{m} \cap K) \oplus  (J_{m} \cap K')$. \hfill $\Box$

\vspace{0.15cm}

Thus $k^{th}$-Griffiths', $k^{th}$-Weil's,
 Lazzeri's Jacobians of a K\"{a}her manifold $(M,g)$
of dimension $m= 2k+1$
 are the same real tori with a different complex structure and the
``change'' of the complex structure depends on the complex structure of $M$
for Griffiths' J.- Weil's J. and on the class of the $(1,1)$-form associated
to $g$ for Lazzeri's J.- Weil's J..

\vspace{0.2cm}

{\scriptsize
\begin{tabular}{|c||c|c|}
\hline
              &  $H^{m}(M,{\bf R})/(H^{m}(M,{\bf Z})/torsion)$ &
 $J_{m}(M)/ \pi(H^{m}(M,{\bf Z}))$ \\ \hline \hline
$k^{th}$ G.J. &  $C|_{\{\eta + \overline{\eta }\mid\;
\eta \in J_{m}(M) \cap \overline{F^{k+1}(M)}\}}
\oplus -C|_{\{\eta + \overline{\eta } \mid \;
\eta \in J_{m}(M) \cap F^{k+1}(M)\}}$ &
$i|_{J_{m}(M) \cap   \overline{F^{k+1}(M)}}
 \oplus -i|_{J_{m}(M)\cap   F^{k+1}(M)}$
\\ \hline
$k^{th}$ W. J. & $C$ &  $i$
\\ \hline
L. J.   & $C|_{H^{m}(M,{\bf R}) \cap K} \oplus
-C|_{H^{m}(M,{\bf R}) \cap K'}$ &  $i|_{J_{m}(M) \cap K}
\oplus -i|_{J_{m}(M) \cap K'}$
\\ \hline
\end{tabular}
}

\vspace{0.2cm}

\vspace{0.2cm}

{\bf Definition 3.4.} {\it Let $(M, g)$ be a compact K\"{a}hler
 manfold of complex dimension $ m =2k+1$. Set
$J'_{m}(M) := (K(M,g) \cap J_{m}(M)) \oplus \overline{(K'(M,g) \cap
J_{m}(M))}$}.

{\bf Corollary 3.5.} {\it Let $(M, g)$ be a compact K\"{a}hler
 manfold of complex dimension $ m =2k+1$; we have
  $T(M,g) = J'_{m}(M)/ \pi_{J'_{m}(M), \overline{J'_{m}(M)} }
 (H^{m}(M,{\bf Z}))$ with the complex structure given by
$i$ and the imaginary part of the polarization
$(\alpha, \beta) = - \int_{M} (\alpha + \overline{\alpha}) \wedge
(\beta + \overline{\beta })$.}

{\bf Remark 3.6.}
Let  $(M,g)$ be  a compact K\"{a}hler manifold of complex
dimension $m=2k+1$; seeing $T(M,g)$ as $J'_{m}(M)/ \pi(H^{m}(M,{\bf Z}))$,
 we can define a Abel map for  $k$-cycles:

let $Z_{0}$ be a $k$-cycle in $M$ and set
{\em $B(Z_{0}) := \{ Z$   $k$-cycle  homologous to  $Z_{0}\}$};
let  $\{\psi_{1},....\psi_{l}\}$ be a basis of
$J_{m}'(M)$:
Abel's map $\mu : B(Z_{0}) \longrightarrow T(M) $ is so defined:
if $Z \in B(Z_{0})$ and $C$ is a $(2k+1)$-chain with $\partial  C =Z-Z_{0}$,
\[\mu(Z):=(\int_{C}\psi_{1},....,\int_{C}\psi_{l}).\]

One can  easily see that the definition is good (in an analogous way as in
\cite{Li} p.131) and
by the same calculation   in \cite{Gri} p.826, one can  see that if
 $\{Z_{\lambda}\}_{\lambda \in B}$ is a family of effective
$k$-subvarieties  with $B$ not singular and $Z_{0}=Z_{\lambda_{0}}$,
the map $\mu(\lambda)=\mu(Z_{\lambda})$ is not  holomorphic in general:
let $C_{\lambda}$ be  a chain such that $\partial  C_{\lambda}
=Z_{\lambda}-Z_{0}$; we have that $\int_{C_{\lambda}}\psi_{i}$
is $0$ unless $\psi_{i} $ is of kind $(k+1,k)$ or $(k,k+1)$ and that
$\int_{C_{\lambda}}\psi_{i}$ is holomorphic in the first case
and antiholomorphic in the second case.

\vspace{0.2cm}

{\bf Notation 3.7.}
{\it
Let $M$ be a complex compact manifold
 of  dimension $m$; consider a smooth deformation  of
the complex structure $ {\cal M } \rightarrow  \Delta $,
  ($\Delta $ polycylinder $\ni 0$);
we  call $ M_{t}$ the fibre over $t$ and let  $ \phi $
be a $C^{\infty}$ trivialization: ${\cal M} \rightarrow M \times \Delta$
(possibly restricting  $\Delta $);
$\phi$ induces diffeomorphism $ \phi_{t}:$ $M_{t}\longrightarrow M$.
     Let $ \rho: T_{0}(\Delta ) \longrightarrow H^{1}( \Theta)$ be the
 Kodaira-Spencer map, where $T_{0}(\Delta )$ is the holomorphic tangent
 space to $\Delta $ in $0$ and $\Theta= \theta (T^{10}(M))$.
Suppose that $M$ is of  K\"{a}hlerian type
 (and then also $M_{t}$  by  Theorem 15 in \cite{K-S}).}

We recall the definition of  Griffiths' and Weil's period maps.

  $\bullet $   The Griffiths period map
${\cal G}_{q}:  \Delta \longrightarrow G_{{\bf C}}(f_{q}^{q},H^{q}(M,{\bf C})) \times
... \times G_{{\bf C}}(f^{q-\nu }_{q} ,H^{q}(M,{\bf C})) $
(possibly restricting $\Delta $) is the map:
 \[t \mapsto ({\cal G}^{q}_{q}(t),...,{\cal G}^{q-\nu}_{q}(t)), \]
          where
${\cal G}^{p}_{q}(t)=  (\phi_{t}^{-1})^{\ast} F^{p}_{q}(M_{t})$,
$f_{q}^{p}= \dim F_{q}^{p}(M)$ and $\nu=[\frac{q-1}{2}]$.

 $\bullet $ If $q$ is an odd positive integer, the Weil period map
${\cal W}_{q}:  \Delta \longrightarrow G_{{\bf C}}(b_{q}/2,H^{q}(M,{\bf C})) $
(possibly restricting $\Delta $) is the map
\[ t \, \mapsto \,  {\cal W}_{q}(t)= (\phi_{t}^{-1})^{\ast} J_{q}(M_{t}).
 \]
Let $\phi (t) $ be a smoothly varying harmonic $(q-r,r)$ form on $M_{t}$.
Call $\phi = \phi (0)$. We have  (see \cite{Gri} p. 812 or \cite{Gre}
p. 33):

\vspace{0.1cm}

$  \frac{\partial \phi(t) }{\partial t}|_{t=0}
=  \rho ( \frac{\partial  }{\partial t}) \cdot \phi$,
which  is  of type  $(q-r-1,r+1)$,   \hspace{2.5cm} (1)

\vspace{0.1cm}

$  \frac{\partial \phi(t) }{\partial \overline{t}}|_{t=0}
= \overline{ \rho ( \frac{\partial  }{\partial t})} \cdot \phi$,
which is  of type $(q-r+1,r-1)$,   \hspace{2.5cm} (2)

\vspace{0.1cm}

where $\cdot$ is the contraction.

Thus we have that, while ${\cal G}_{q}$ is holomorphic and
$Im \, ( d {\cal G}_{q}) (0)$ is in  $\oplus_{r=0...\nu}
Hom(H^{q-r,r},H^{q-r-1,r+1})$,
the  map ${\cal W}_{q}$ is not holomorphic; precisely, if $ \phi \in
F^{q-r}_{q}(M)$

\vspace{0.1cm}

$ (\frac{\partial}{\partial t}{\cal G}_{q}^{q-r})(0)(\phi)=
 \pi_{{\footnotesize \overline{F_{q}^{r+1}(M)} ,F_{q}^{q-r}(M)}}
(\rho ( \frac{\partial  }{\partial t}) \cdot \phi)
    = \rho ( \frac{\partial  }{\partial t}) \cdot \phi, $

\vspace{0.1cm}

$ (\frac{\partial  }{\partial \overline{t}}{\cal G}_{q}^{q-r})(0)(\phi)=
\pi_{\overline{F_{q}^{r+1}(M)} ,F_{q}^{q-r}(M)}
(\overline{ \rho ( \frac{\partial  }{\partial t})} \cdot \phi)
 = 0 $;

\vspace{0.1cm}

while, if q is odd and  $ \phi \in J_{q}(M) $

\vspace{0.1cm}

$ (\frac{\partial  }{\partial t}{\cal W}_{q})(0)(\phi)=
 \pi_{\overline{J_{q}(M)} ,J_{q}(M)}
(\rho ( \frac{\partial  }{\partial t}) \cdot \phi)
= \rho ( \frac{\partial  }{\partial t})\cdot \phi,$

\vspace{0.1cm}

$ (\frac{\partial}{\partial \overline{t}} {\cal W}_{q})(0) (\phi)=
 \pi_{\overline{J_{q}(M)} ,J_{q}(M)}
  (\overline{ \rho ( \frac{\partial  }{\partial t})} \cdot \phi)
  = \overline{ \rho ( \frac{\partial  }{\partial t})} \cdot \phi $.

\vspace{0.2cm}

{\bf Corollary C.}
{\it  Let  $(M,g)$ be  a compact K\"{a}hler  manifold   of odd complex
dimension $m$; let $\Omega \in H^{2}(M,{\bf R})$ be the class of the $(1,1)$-form
 associated to $g$;
 consider a smooth deformation  of the complex structure
$ {\cal M } \rightarrow  \Delta $
for which we use Notation 3.7.
Let $\Delta' = \{ t \in \Delta \mid
\phi_{t}^{\ast}(\Omega) \in H^{2} (M_{t}, {\bf R})$ {\it can be represented
by  a K\"{a}hler form for $M_{t} \}$}; suppose $\Delta'$ be a
neighbourhood of $0$.

{\bf a)} For $t \in \Delta' $ let
$g_{t}$ be  a K\"{a}hler metric on $M_{t}$ whose (1,1)-form is of class
$\phi_{t}^{\ast}(\Omega)$;
  if we fix a symplectic basis of $H^{m}(M,{\bf Z})$,
we have that   the map
$  t  \mapsto  Z(T(M_{t}, g_{t})) $
defined from  a neighbourhood of $0$ in
$ \Delta'$ to ${\cal H}$, associating to $t$
 the matrix in ${\cal H}$ representing
 $T(M_{t},g_{t})$ with that choice of the symplectic basis,
 is
holomorphic in $0$ iff $ \rho (T_{0}(\Delta)) \cdot
(
K(M,g) \cap J_{m}(M)
\oplus
\overline{ (      K'(M,g) \cap J_{m}(M)    )    } )=0$
(where $ \cdot $ is the contraction),
 (observe that $  t  \mapsto  Z(T(M_{t}, g_{t})) $
is the composition of the map $t \mapsto (\phi_{t}^{-1})^{\ast}
g_{t} $ with $\hat{T}$).

{\bf b)} Suppose now that the canonical bundle of $ M$  is trivial
and $ \rho$ is injective.
For $t \in \Delta' $ let
$g_{t}$ be  a K\"{a}hler metric on $M_{t}$ whose (1,1)-form is of class
$\phi_{t}^{\ast}(\Omega)$, for instance
the  K\"{a}hler-Einstein metric whose (1,1)-form is of class
$\phi_{t}^{\ast}(\Omega)$
($\exists !$  by  Calabi-Yau's theorem
 (see \cite{S.P.}); fix a symplectic basis of $H^{m}(M,{\bf Z})$;
 then the differential in $0$ of    the map
$  t  \mapsto  Z(T(M_{t}, g_{t})) $ (defined above) is injective.}

{\sc Proof.} We set
${\cal W}'_{m}(t)= (\phi_{t}^{-1})^{\ast} J'_{m}(M_{t}).$
 By (1) and (2), we have that, if $ \phi \in J'_{m}(M)$, then

$ (\frac{\partial  }{\partial t}{\cal W}'_{m})(0)(\phi)=
 \pi_{\overline{J'_{m}(M)} ,J'_{m}(M)}
(\rho ( \frac{\partial  }{\partial t}) \cdot \phi)
= \rho ( \frac{\partial  }{\partial t})\cdot \phi,$

\vspace{0.05cm}

$ (\frac{\partial}{\partial \overline{t}} {\cal W}'_{m})(0) (\phi)=
 \pi_{\overline{J'_{m}(M)} ,J'_{m}(M)}
  (\overline{ \rho ( \frac{\partial  }{\partial t})} \cdot \phi)
  = \overline{ \rho ( \frac{\partial  }{\partial t})} \cdot \phi $.

See  $T (M_{t},g_{t})=  {\cal W}_{m}'(t)/
\pi_{  {\cal W}_{m}'(t), \overline{   {\cal W}_{m}'(t)  } }
(H^{m}(M,{\bf Z})) $  with  the polarization
whose imaginary part is $(\alpha, \beta) =
- \int_{M} (\alpha + \overline{\alpha}) \wedge (\beta + \overline{\beta })$.
         Choosen a symplectic (for $-\int \cdot \wedge \cdot$)
basis $\{\gamma_{i}\}$ of
$H^{m}(M,{\bf Z})$ and  a  basis $\{\omega_{j}(t)\}$
of ${\cal W}_{m}'(t)$, let
{\footnotesize $ \left( \begin{array}{c}
                                               E(t) \\
                                               F(t)
\end{array} \right) $} be the matrix expressing $ \omega_{j} (t) $
in function of $ \gamma_{i}$.
Then  $Z(T (M_{t},g_{t}))= - \overline{E(t)F(t)^{-1}}$.

 Thus $Z(T (M_{t},g_{t}))$ is  holomorphic in $t$
iff $\overline{{\cal W}_{m}'(t)} \in G_{{\bf C}}(b_{m}/2,H^{m}$ $(M, {\bf C}))$
 is holomorphic in $t$.

If  the canonical bundle of $M$ is trivial,  the maps
$ H^{1}(\Theta) \rightarrow Hom ( H^{n,0}, H^{n-1,1})$ and
$ \overline{H^{1}(\Theta)} \rightarrow Hom ( H^{0,n}, H^{1,n-1})$ given
 by the contraction are injective (see \cite{Gri} p. 844).
Then, as in \cite{Gri},
if also   $\rho$  is injective,   the maps   $( d{\cal W}_{m})(0)$ and
$( d{\cal W}_{m}')(0)$ are injective   and we obtain b).
\hfill $ \Box$

\section*{4. Lazzeri's Jacobian of a bundle}

{\bf Definition 4.1.} {\it
  An element $\lambda$ of a lattice $\Lambda$ is
said primitive if  $ \not \exists \lambda' \in \Lambda$ such that
$\lambda \in {\bf Z}\lambda'$.}

\vspace{0.1cm}

{\bf Proposition D.} {\it
Let $(M,g_{M})$ and $(N,g_{N})$ be riemannian compact oriented manifolds
of  dimension $2(2k+1)$, resp. $2(2s)$.
Let $p:F \rightarrow M$ be a bundle with fibre $N$
and   structure  group  $G \subset$Diff$(N)$ and suppose
$g_{N} $ to be $G$-invariant.
We consider on $F$ the metric induced by $g_{M}$ and  $g_{N}$.
Suppose  $\exists \;\; \lambda \in H^{2s}(N,{\bf Z})/torsion$ such that
$\lambda \neq 0$, $\ast_{N} \lambda = \lambda$
and $\lambda$ $G$-invariant.

We  define a map
$e_{\lambda}:T(M) \rightarrow T(F)$
(we omit the metrics).
If $ F = M \times N$ we can define $e_{\lambda} $ simply as
the  map  induced by the map $H^{2k+1}(M,{\bf R}) \rightarrow
H^{2k+1+2s}(M \times N,{\bf R})$ defined by
$ \eta \mapsto \eta \wedge \lambda  $.
More generally
define $\Lambda \in H^{2s}(F,{\bf R})$ in the following way:
if $U$ a trivializing open subset of $M$,
$\Lambda|_{p^{-1}(U)}:= \pi^\ast \lambda$, where
$\pi $ is the composition  of a $C^{\infty}$ trivialization
$ p^{-1}(U) \rightarrow U \times N$  with the projection
$U \times N \rightarrow N$;
 suppose $\Lambda $ is an integral form;
define $E_{\lambda}: H^{2k+1}(M,{\bf R})
\rightarrow H^{2k+1+2s}(F,{\bf R})$
by $E_{\lambda}(\eta)= p^{\ast} \eta \wedge \Lambda $;
  the map $E_{\lambda}$ defines a map  $e_{\lambda}:T(M) \rightarrow T(F)$.

The  map $e_{\lambda}$ is holomorphic and, if $ \theta_{F}$ and $ \theta_{M}$
are the polarizations of $T(F)$ and of $T(M)$, then $ e_{\lambda}^{\ast}
\theta_{F}= (\int_{N} \lambda \wedge \ast \lambda) \theta_{M}$.
Besides $e_{\lambda}$ is  injective if one of the following conditions holds:
a) $\int_{N} \lambda \wedge \ast \lambda= 1$
b) $ F =M \times N$  and $\lambda$ is primitive
c) $ H^{\ast}(N , {\bf Z}) $ is free and $G$-invariant.}

\vspace{0.1cm}

\vspace{0.1cm}

{\sc Proof.}
$\bullet $ The fact that $\ast_{N} \lambda = \lambda$ implies  at once that
    $e_{\lambda}$ is holomorphic and
 that  $e_{\lambda}^{\ast } \theta_{F} =
(\int_{ N } \lambda  \wedge \ast \lambda) \theta_{M}$:

let $\{U_{\alpha} \}_{\alpha}$  be a trivializing covering of $M$ and let
$\psi_{\alpha}$ be a partition of the unity for this covering; let
$\phi_{\alpha}$ be the partition of the unity for the covering of $F$ $\{p^{-1}
(U_{\alpha})\}$  defined by $\phi_{\alpha}(y)=\psi_{\alpha}(p(y))$;
let $\omega_{1}, \omega_{2}  \in H^{2k+1}(M,{\bf R})$:
\hspace{-1cm}\begin{eqnarray*}
 & & \int_{F} E_{\lambda}(\omega_{1}) \wedge  E_{\lambda}(\omega_{2}) =
 \sum_{\alpha} \int_{p^{-1}(U_{\alpha})} \phi_{\alpha} E_{\lambda}
 (\omega_{1})
  \wedge  E_{\lambda}(\omega_{2}) = \\
& & \sum_{\alpha} \int_{U_{\alpha} \times N} \phi_{\alpha} \omega_{1} \wedge
  \lambda  \wedge  \omega_{2} \wedge \lambda =
 \sum_{\alpha} \int_{U_{\alpha} } \psi_{\alpha} \omega_{1}
 \wedge  \omega_{2} \int_{ N } \lambda  \wedge \ast \lambda =
(\int_{ N } \lambda  \wedge \ast \lambda)
\int_{M } \omega_{1} \wedge  \omega_{2}
\end{eqnarray*}

$\bullet$ If a) holds then the map $e_{\lambda}$ is injective
since it is  a homomorphism  of p.p.a.v.'s and preserves the polarization.
It is easy to verify that  b) implies
 $e_{\lambda}$  injective. Finally, if c) holds, then
 $e_{\lambda}$ is  injective by the theorem of Leray-Hirsch
(see \cite{Sp} p.258). \hfill $ \Box$

\vspace{0.15cm}

{\bf Remark 4.2.} Suppose  $F = M \times N$.
Observe that $T(M \times N)$ is the product of the
abelian subvarieties $A_{t}$ corresponding to
$(H^{t}(M,{\bf R}) \otimes  H^{2k+1+2s -t}(N,{\bf R})) \oplus
(H^{2(2k+1)-t}(M,{\bf R})$  $ \otimes  H^{2s-2k-1 +t}(N,{\bf R}))$
 for  $t \neq 2k+1$,
 and  the abelian subvariety $A_{2k+1}$  corresponding to
$H^{2k+1}(M,{\bf R})$ $  \otimes  H^{2s}(N,{\bf R})$.
           Obviuosly  $ Im \; e_{\lambda} \subset A_{2k+1}$.

The abelian subvarieties $A_{t}$
for $t \neq 2k+1$ are represented in the Siegel upper half
space by a matrix $Z$ with real part equal to $0$; in fact:

 take as a complex basis of
$(H^{t}(M,{\bf R}) \otimes  H^{2k+1+2s -t}(N,{\bf R})) \oplus
(H^{2(2k+1)-t}(M,{\bf R}) \otimes  H^{2s-2k-1 +t}(N,{\bf R}))$
a basis $\{r_{i}\} $ of  $H^{t}(M,{\bf Z}) \otimes  H^{2k+ 1 + 2s -t}(N,{\bf Z} )$;
we can complete $\{r_{i}\} $ to a  symplectic
basis $\{r_{i},s_{i}\}$ of the lattice taking  as $\{s_{i}\}$  a suitable
 basis of $H^{2(2k+1)-t}(M,{\bf Z}) \otimes  H^{2s-2k-1 +t}(N,{\bf Z})$;
since each $s_{i}$ is a real linear combination
of the $\ast r_{i}'s$, $Z$ is imaginary.

\vspace{0.15cm}

{\bf Remark 4.3.} If $N$ is K\"{a}hler and the 1-1 form $\Omega$ associated to
the metric is integral, i.e. $N$ is projective,
the hypothesis ``$ \lambda \in H^{2s}(N,{\bf Z})/torsion$,
$\ast_{N} \lambda = \lambda$'' is satisfied if $\lambda = [\Omega^{s}]$.

 \section*{{\bf\large Appendix}}

Let  $M= {\bf R}^{2(2k+1)}/ \Lambda $ be a torus and $g_{0}$ a singular flat
 metric.
We prove now that the definition of $T(M,g_{0})$ we gave at the end of
Introduction does not depend on the metric $g$ extending $g_{0}$:

let $\{e_{1},..., e_{n}\} $ be a basis  of ${\bf R}^{n}$ with $n =2(2k+1)$,
orthonormal for $g$ with $ < e_{1},..., e_{r} >  = (\ker g_{0})^{\perp_{g}} $
and $ < e_{r+1},..., e_{n} > = \ker g_{0}$;

let $g'$ be another metric extending $ g_{0}$ in the  above sense;
let $\{v_{1},..., v_{n}\} $ be a basis  of ${\bf R}^{n}$
orthonormal for $g'$ with $ < v_{1},..., v_{r} >  =
 (\ker g_{0})^{\perp_{g'}} $   and $ < v_{r+1},..., v_{n} > = \ker g_{0}$;

let
$ \left( \begin{array}{cc}
                                                  A & 0 \\
                                                  C & E
\end{array} \right) $ be
the matrix  expressing the basis $ \{v_{i}\}$ in function of
the basis $\{ e_{i} \}$;

observe that the matrix $A$ is orthogonal, in fact $ g_{0}=
   \left( \begin{array}{cc}
                                                  I & 0 \\
                                                  0 & 0
\end{array} \right) $ both in the basis $\{e_{ 1},...,e_{r} ,e_{r+1},
...,  e_{n}\} $
 and in the basis $\{v_{1},...,v_{r} , v_{r+1},...,  v_{n}\} $;

thus $ \left( \begin{array}{cc}
                                                  A & 0 \\
                                                  C & E
\end{array} \right)^{t}   \left( \begin{array}{cc}
                                                  I & 0 \\
                                                  0 & 0
\end{array} \right)  \left( \begin{array}{cc}
                                                  A & 0 \\
                                                  C & E
\end{array} \right) = \left( \begin{array}{cc}
                                                  I & 0 \\
                                                  0 & 0
\end{array} \right) $; then $A$ is orthogonal;

consider the basis    $\{v'_{1},..., v'_{n}\} $ of ${\bf R}^{n}$ that  is
expressed by the matrix
$ \left( \begin{array}{cc}
                                                  A^{-1} & 0 \\
                                                  0 & I
\end{array} \right) $ in function of the basis
$\{v_{1},..., v_{n}\} $;

the matrix expressing $\{v'_{1},..., v'_{n}\} $ in function of
$\{e_{1},..., e_{n}\} $ is
$ \left( \begin{array}{cc}
                                                  I  & 0 \\
                                                  C A^{-1}  & E
\end{array} \right) $;

thus   $\{v'_{1},..., v'_{n}\} $ is a basis orthonormal for $g'$ such that
$<v'_{1},..., v'_{r}> = \ker g_{0}^{\perp_{g'}} $ and
$<v'_{r+1},..., v'_{n}> = \ker g_{0} $;

the map  $ e_{I}^{\vee} \mapsto (v'_{I})^{\vee}$ is an isomorphism
between $T(M, g_{0})$ built by $g$ and $T(M, g_{0})$ built by $g'$.

\vspace{1cm}

{\small {\bf Acknowledgements.}
I   thank prof. F. Catanese for suggesting me to study
Lazzeri's Jacobian and for  many
helpful   discussions and suggestions on it.
I thank also M. Manetti for some corrections to  the first part of this work.}

{\footnotesize

redpajama_set_name: RedPajamaArXiv


perfect for any friend or family member.
adding an extra dimension to the attractiveness of it.
The oil burner's neat tidy shape, the wider base makes it very stable.
It also has small holes around the top to let out the excess heat.
inside the tray so it is an ideal gift for someone who has never used one.


Today I've got some japanese Rock . I love Japan and there are some really great bands over there.
Only after that, I discovered Versailles because I listend to this song.
I fell in love with Camijo-kun's voice immediately.
74 mySQL Anfragen in 0,559 Sekunden.


Aikido Seishin Kai was formed in 1996 and has clubs in Manchester and Stoke-on-Trent. The Manchester club is based in Dukinfield.
We practice the traditional style of Aikido, as taught by Morihei Ueshiba or O Sensei.
Andrew Hodgin 5th Dan and Gavin Taylor 4th Dan are instructors at the club.
We are an inclusive, family-friendly club, made up of male and female members from different ages and backgrounds.
Training at the HITT Station every Monday and Thursday between 20:30-21:30.
Would you like more information on hours, classes and programs available at Aikido Seishin Kai? Please contact the school with your questions.
Update the details, Write a Review or Report Error of Aikido Seishin Kai.
This Chart shows the Martial Arts Styles offered most in Manchester, UK area. View a list of Dojo in Manchester, or enter your address below to find schools closest to you.


Learn the most effective way to value those relationships that are valuable. Focus energy consistently to develop and maintain productive, reciprocal relationships. Whether you are growing a network to expand membership or solidifying the relationships among your talented team members, it is all about currency. Fun and interactive activities that leave the audience feeling great about the good they do.


This week my son and I tackled the front view boy. I figured I'd show you the step by step process to show you how easy it is.
First, start with a circle. Cut it into four equal parts and then cut the lower half into four equally spaced sections. I go freehand. Dude likes to use a ruler.
Next, add a frame for the chin. Manga likes a pointy chin. I'll probably start putting my own spin on this soon.
Next, add the eyes in the third lower section. Eyebrows on the line above, each slightly towards the outer circle. The nose line just to the left of the center line, right on the outer circle mark, and do your best on the ears. Dude and I agree ears are EVIL.
Next, the dreaded Manga Hair. I drew this directly from the diagram. I don't get the flow of the hair yet. Dude does better than me on this aspect.
Next, darken and add a little clothing if desired. I decided to round the chin at this point as well. Also add a shade under the chin. Once your lines are nice and dark, erase the light lines of the "cheat" circle originally drawn.
That's it. Break it down a step at a time, and it's really not that hard.
This entry was posted in Artwork and tagged Animation, Anime, art lessons, art with kids, Arts, Comics, Drawing, Fandom, learning to draw, Manga, pencil drawing, Shōnen manga, United States. Bookmark the permalink.
Thanks. You should… and don't wait as long as I did. Take some time while the kiddies are napping.
Ha! That's very funny. I actually like the look of some of it. The more adult-contemporarty characters look fun, but more complicated. I'm taking it one step at a time, but would like to get there.
Wow good, or Wow bad?
Colour … is good too!
Fun! Thanks for the step by step.
Wonderful, isn't it, when your child mirrors your genes? I think it's amazing being part of this miracle. Nice to hear you little guy is still into the drawing and is being challenged. Heartwarming to read about.


text:
Business & Orgs
Slice of Rye
After Rye
Along for the Rye'd
Dressing on the Side
Right in Our Backyard
Beyond Rye
Day Trippin'
Dining & Wining
Billing Inquires
Our newsletter will keep you up-to-date on the latest RYE happenings.
Home Home & Garden	The March of a Millennial
The March of a Millennial
by admin June 3, 2017 0 comment
By Robin Jovanovich
Katherine Doherty grew up on the other side of Westchester, in the river town of Hastings-on-Hudson, where her family has lived for generations. Kat, as she is known, always wanted to be an architect, but her mother always thought she would make a great teacher. So that's the career path she took.
After earning a B.S. in Elementary Education, with a concentration in Psychology, and an M.S. in Special Education, Kat taught first and second grade and Special Ed in Valhalla. While she loved teaching young children, in short order she had two of her own.
When she moved to Rye five years ago ("I like to think of myself as bi-coastal because my roots are still in Hastings"), Kat followed the arc of many her age. "I gave up a salaried career and decided to wing it!"
A self-described "house nerd", Doherty became a real estate associate at Houlihan Lawrence. "I love looking at layouts, seeing how spaces are designed and decorated, and then reimagining them. I think being passionate is a quality that draws people to you." She added, "It's not just social media that brings you buyers and sellers."
While many realtors are quick to tell you that buyers in our neck of the woods only want "new and done", Doherty is seeing signs of a shift among young families. "Yes, everyone is looking for walkability, but more and more want charm and character."
The challenge in the current market, she noted, is that a lot of buyers are slow to pull the trigger. "Before the first open house, potential buyers have already done their research — and walked or driven by the house multiple times. Some of them are Facebook 'stalkers' who've researched the sellers!"
She derives the most satisfaction from working with buyers and helping them find a home where they'll be happy and want to stay.
And she said she's already learned from the best. "Barbie Haynes is among the best. She's that authentic, relatable agent, in addition to being a wonderful woman." She added, "I've already had that teachable moment: You can still be an honest, ethical person in this very competitive business."
Doherty's the first to admit that being a realtor and the mother of young children (and class mom at both Rye Presbyterian Nursery School and Milton School) requires a lot of juggling. "It takes a village. Luckily, Rye is an amazing village."
On a recent Friday on which she had an appointment with a buyer, she got a flat tire. "Joe from Rye Brook Service came to get my car. My friend sent her husband to get me. I borrowed their car and made it to the appointment with five minutes to spare," she recalled with a smile.
"Right now I'm trying to get my friends from high school to move here. That's one of my dreams."
Downsize, Upsize, or Move to a Country Cottage
Giving Clutter the Heave-Ho
Earning That First Green Badge of Honor
Sam Bridge, a Business With Deep Roots
The Lazy Gardener's Guide to Success
Oh, Give Me a Home & Garden
REALTOR"S ROW
The Jay Estate, the Jewel in Our Backyard
A Garden Hat-Trick
Pathways to Happiness
When a Realtor Goes to Sell Her Own...
Conduct Unbecoming of a School District December 23, 2020 Howard Husock
A Whole-Hearted Thank You to Mayor Cohn December 17, 2020 admin
Why I was removed from the Race, Inclusivity & Community Task Force December 17, 2020 admin
This Is Our Opportunity November 16, 2020 admin
Disappointed in Paper's One-Sided Reporting on School Task Force November 9, 2020 admin
The Rising COVID Count in Westchester
VINTAGE RYE: Squash Racquets, Anyone?
Tom Hanks Returns
Making Every Moment Count in the New Year
7 Sky Meadow Farm, Purchase: The Weekend's Featured Listing
Subscribe to our Newsletter for new posts, news & updates. Let's stay connected!
Your opinion matters. Let us know what you think about your community.
For 22 years, The Rye Record has reported on the community — its people, organizations, businesses, schools, athletic feats.
14 Elm Place, Suite 206
njovanovich@ryerecord.com
Online ad sizes, rates & specifications
@2019 - The Rye Record. All Right Reserved. | Term & Conditions | Privacy Policy


text:
Fan-made app playing around with Swift, Cocoa, and SwiftDate.

![](screenshot.png)

Game of Thrones will be simulcast around the world on 24 April, at 9:00 pm Eastern Standard time.  If you are in the UK, then it will be on April 25th, at 2:00 am.

**Usage:**

An archived app is in the app folder.  Otherwise clone or download the repo, then install the SwiftDate CocoaPod.

**License**
Code is MIT.  See LICENSE for details.

The Game of Thrones true type font has it's own separate license as indicated on the [fonts for free](http://www.fonts4free.net/game-of-thrones-font.html) website.

Audio file from [Wikipedia](https://en.wikipedia.org/wiki/File:Game_of_Thrones_Main_Title_sample.ogg).

Game of Thrones, all images and other related files © HBO, and are *not* MIT licensed.

redpajama_set_name: RedPajamaGithub


Happy Friday! What better way to celebrate the end of the work-week than with felted heart pins?!
I'm dashing out to the post office to send these little friends off to their new homes.
Megan, I love you. You are so amazing. I want to make every single thing that you post. I also want to live next door to you and chat over a steaming cup of postum every morning.
Awww HPerk! That makes me so happy and sad at the same time! I'm sitting here drinking Inka right this moment (it's my Postum stand-in) and wiiiiiiiiishing we were neighbors. Are you coming down here anytime soon? I'll shower you in handmade love. And keychains.
Ok, so how do you make these lil cuties?? Please tell.


This high-potency, hypoallergenic blend of 5 Lactobacillus species and 5Bifidobacterium species is designed to safely meet the metabolic and intestinal support needs of infants up to the age of 2 years. All Lactobacillus species in Nutrivene Baby Biotic produce predominantly L(+)-lactic acid that can be safely metabolized by infants. An infant's gastrointestinal tract is not fully mature at birth and cannot metabolize the D(-) isomer of lactic acid produced by many lactobacilli. Each species in Baby Biotic has been selected based on the scientific literature documenting its safety and crucial importance to the normal development of an infant's gastrointestinal and immune systems. This product is designed to support a normal infant microflora that can be easily disrupted by caesarean section delivery, formula feedings, antibiotics, toxin exposures, and a maternal diet lacking in fresh fruits and vegetables.
Nutrivene Baby Biotic is available in a high potency powder providing 5+ billion CFUs per 1 scoop. Baby Biotic powder can be mixed with expressed breast milk, formula, or solid food at room temperature as appropriate.
• L. rhamnosus increases colonic B. breve populations and promotes healthy bifidobacteria diversity in infants. L. rhamnosus is important for the normal maturation of an infant's immune system.
• L. casei is critical for the proper differentiation of immune dendritic cells as the infant's immune system learns to distinguish between harmless antigens and toxins and pathogens. It increases the number of intestinal cells secreting IgA. L. casei contains enzymes that facilitate the break down of milk proteins and polypeptides as well as gluten.
• L. paracasei produces high amounts of L(+)-lactic acid and antagonizes pathogens such as rotavirus, Staphylococcus aureus, and Clostridium difficile.
• L. gasseri is native to the human gut and normally present in human breast milk. It produces hydrogen peroxide and bacteriocins inhibitory to Clostridium, Listeria, andEnterococcus.
• L. salivarius normally colonizes the intestinal tract, enhances calcium absorption, and promotes normal intestinal barrier function.
Specific Bifidobacterium species are normally predominant in an infant's gut microflora and constitute 95% of the anaerobic bacteria cultured from the stool of healthy, breast-fed infants. Nutrivene Baby Biotic contains species that are appropriate to an infant's gastrointestinal tract. Bifidobacteria ferment an array of oligosaccharides including those found in breast milk generating the vital short-chain fatty acids such as butyrate, which are essential to healthy colon maturation and function.
• B. breve promotes growth of B cells in Peyer's patches in the ileum. It antagonizes rotavirus and reduces intestinal populations of anaerobic Bacteroides and Clostridium.
• B. infantis is normally found in infants and its presence reduces the risk of necrotizing enterocolitis in premature or low birth weight babies.
• B. lactis enhances normal immune function by increasing numbers of total, helper, and activated T cells. It modifies gut microflora to support against allergic inflammation and increases IgA response to toxins.
• B. bifidum suppresses total and allergen-specific IgE secretion and its populations are often low in infants with allergies.
• B. longum regulates the immune response to respiratory antigens and inhibits neutrophil elastase, which is important to normal innate immune functions.
Suggested Use: For children over 12 months: 1 scoop, once or twice daily. For infants 6 to 12 months: 1 scoop daily. Mix product in infant formula, milk, juice, or solid food at room temperature as appropriate.
Nutrivene Baby Biotic contains 66 grams per container.