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La Selle-sur-le-Bied est une commune française située dans le Gâtinais, dans le département du Loiret en région Centre-Val de Loire.

Le , la commune absorbe celle de Saint-Loup-de-Gonois en prenant le statut administratif de commune nouvelle.

Géographie

Localisation et communes limitrophes

La commune de La Selle-sur-le-Bied se trouve dans le quadrant nord-est du département du Loiret, dans la région agricole du Gâtinais pauvre. À vol d'oiseau, elle se situe à   d'Orléans, préfecture du département, à  de Montargis, sous-préfecture, et à  de Courtenay, ancien chef-lieu du canton dont dépendait la commune avant . La commune fait partie du bassin de vie de Ferrières-en-Gâtinais dont elle est éloignée de .

Les communes les plus proches sont : Louzouer (), Courtemaux (), Mérinville (), Thorailles (), Chantecoq (), Griselles (), La Selle-en-Hermoy (), Pers-en-Gâtinais () et La Chapelle-Saint-Sépulcre ().

Géologie et relief

Géologie
La commune se situe dans le sud du Bassin parisien, le plus grand des trois bassins sédimentaires français. Cette vaste dépression, occupée dans le passé par des mers peu profondes et des lacs, a été comblée, au fur et à mesure que son socle s'affaissait, par des sables et des argiles, issus de l'érosion des reliefs alentours, ainsi que des calcaires d'origine biologique, formant ainsi une succession de couches géologiques.

Les couches affleurantes sur le territoire communal sont constituées de formations superficielles du Quaternaire et de roches sédimentaires datant du Cénozoïque, l'ère géologique la plus récente sur l'échelle des temps géologiques, débutant il y a  d'années, et du Mésozoïque, anciennement appelé Ère secondaire, qui s'étend de . La formation la plus ancienne est de la craie blanche à silex remontant à la période Crétacé. La formation la plus récente est des alluvions récentes des lits mineurs remontant à l'époque Holocène de la période Quaternaire. Le descriptif de ces couches est détaillé dans  les feuilles « n°329 - Château-Landon » et « n°330 - Chéroy » de la carte géologique au  du département du Loiret et leurs notices associées.

Relief
La superficie cadastrale de la commune publiée par l'Insee, qui sert de références dans toutes les statistiques, est de . La superficie géographique, issue de la BD Topo, composante du Référentiel à grande échelle produit par l'IGN, est quant à elle de . L'altitude du territoire varie entre  et .

Hydrographie
La Cléry, affluent du Loing, traverse le territoire de la commune d'est en ouest.

Toponymie
Le nom de la localité est attesté sous la forme Cella en ll56.

Celle :  « cellule, ermitage, dépendance d'un monastère », probablement « petit monastère ».

Bied est une variante du mot bief, canal apportant l'eau à un moulin, également cours d'eau (aujourd'hui la Cléry).

Histoire

Entre le  et le , plus de  espagnols fuyant l'effondrement de la république espagnole devant les troupes de Franco, arrivent dans le Loiret. Devant l'insuffisance des structures d'accueil d'Orléans,  d'accueil ruraux sont ouverts, dont un à La Selle-sur-le-Bied. Les réfugiés, essentiellement des femmes et des enfants (les hommes sont désarmés et retenus dans le Sud de la France), sont soumis à une quarantaine stricte, vaccinés, le courrier est limité, et le ravitaillement, s'il est peu varié et cuisiné à la française, est cependant assuré. Une partie des réfugiés rentrent en Espagne, incités par le gouvernement français qui facilite les conditions du retour, ceux préférant rester sont regroupés au camp de la verrerie des Aydes, à Fleury-les-Aubrais.

Le , la commune fusionne avec Saint-Loup-de-Gonois pour former la commune nouvelle éponyme de La Selle-sur-le-Bied dont la création est actée par un arrêté préfectoral du .

Urbanisme

Typologie
La Selle-sur-le-Bied est une commune rurale, car elle fait partie des communes peu ou très peu denses, au sens de la grille communale de densité de l'Insee.
La commune est en outre hors attraction des villes.

Occupation des sols
L'occupation des sols de la commune, telle qu'elle ressort de la base de données européenne d'occupation biophysique des sols Corine Land Cover (CLC), est marquée par l'importance des territoires agricoles (84,9 % en 2018), une proportion sensiblement équivalente à celle de 1990 (86,4 %). La répartition détaillée en 2018 est la suivante :
terres arables (77,3 %), forêts (11,2 %), zones agricoles hétérogènes (7,6 %), zones urbanisées (4 %).

L'évolution de l'occupation des sols de la commune et de ses infrastructures peut être observée sur les différentes représentations cartographiques du territoire : la carte de Cassini (), la carte d'état-major (1820-1866) et les cartes ou photos aériennes de l'IGN pour la période actuelle (1950 à aujourd'hui).

Planification
La Selle-sur-le-Bied fait partie de la communauté de communes du Betz et de la Cléry créée le  et entrée en vigueur le . Parmi les diverses compétences attribuées à cette intercommunalité figure l'élaboration et le suivi du Plan local d'urbanisme intercommunal (PLUi) portant sur l'ensemble du territoire de la communauté de communes. Le document est approuvé par le conseil communautaire le . Pour répondre à l'obligation de mettre en conformité le PLUi avec la loi du  dite « Grenelle II » et la loi pour l'accès au logement et un urbanisme rénové du , dite loi « Alur » mais aussi avec le schéma de cohérence territoriale du Montargois en Gâtinais et pour tenir compte de l'évolution du périmètre (Intégration de la commune de Saint-Loup-d'Ordon au ), l'assemblée prescrit la révision du PLUi par délibération du .

Logement

Voies de communication et transports

Infrastructures routières
La commune historique est traversée par trois routes départementales : la RD 32, la RD 36 et la RD 146. La RD 32, qui relie Nargis à Courtenay, supporte en 2014 à l'ouest du bourg un trafic de  et à l'est de . La RD 36, qui relie Château-Renard à Bazoches-sur-le-Betz et la RD 146, qui relie la commune à Chevry-sous-le-Bignon, sont des routes à faible trafic.

Transports en commun
Aucune ligne régulière du réseau Ulys, le réseau interurbain de transport par autocar du Conseil départemental du Loiret, ne dessert la commune. Par contre la commune bénéficie du service Ulys Proximité qui emmène les usagers de la commune vers un arrêt desservi par la ligne régulière la plus proche. À compter du , la compétence des services de transports routiers interurbains, réguliers et à la demande est transférée des départements aux régions, et donc localement du département du Loiret à la région Centre-Val de Loire, consécutivement à la loi NOTRe du .

Risques majeurs
La commune de la Selle-sur-le-Bied est vulnérable à différents aléas naturels : climatiques (hiver exceptionnel ou canicule), mouvements de terrains ou sismique (sismicité très faible). Elle est également exposée à un risque technologique : le risque de transport de matières dangereuses.
Entre 1989 et 2019, vingtarrêtés ministériels ayant porté reconnaissance de catastrophe naturelle ont été pris pour le territoire de la commune  : vingt pour des inondations et coulées de boues et un pour des mouvements de terrains.

Risques naturels

Le territoire de la commune peut être concerné par un risque d'effondrement de cavités souterraines non connues. Une cartographie départementale de l'inventaire des cavités souterraines et des désordres de surface a été réalisée. Il a été recensé sur la commune plusieurs effondrements de cavités.

Par ailleurs le sol du territoire communal peut faire l'objet de mouvements de terrain liés à la sècheresse. Le phénomène de retrait-gonflement des argiles est la conséquence d'un changement d'humidité des sols argileux. Les argiles sont capables de fixer l'eau disponible mais aussi de la perdre en se rétractant en cas de sècheresse. Ce phénomène peut provoquer des dégâts très importants sur les constructions (fissures, déformations des ouvertures) pouvant rendre inhabitables certains locaux. Celui-ci a particulièrement affecté le Loiret après la canicule de l'été 2003. Une grande partie du territoire de la commune est exposée à un aléa  face à ce risque, selon l'échelle définie par le Bureau de recherches géologiques et minières (BRGM).

Depuis le , la France dispose d'un nouveau zonage sismique divisant le territoire national en cinq zones de sismicité croissante . La commune, à l'instar de l'ensemble du département,  est concernée par un risque très faible.

Risques technologiques

La commune est exposée au risque de transport de matières dangereuses, en raison du passage sur son territoire d'une canalisation de gaz et d'un itinéraire routier structurant supportant un fort trafic (l'autoroute A19).

Politique et administration

Découpage territorial

Bloc communal : Commune et intercommunalités

La paroisse et bourg de La Selle-sur-le-Bied acquiert le statut de municipalité avec le  décret du  de l'Assemblée Nationale puis celui de « commune », au sens de l'administration territoriale actuelle, par le décret de la Convention nationale du 10 brumaire an II (). Il faut toutefois attendre la loi du  sur l'organisation municipale pour qu'un régime juridique uniforme soit défini pour toutes les communes de France, point de départ de l'affirmation progressive des communes face au pouvoir central.

Aucun événement de restructuration majeure du territoire, de type suppression, cession ou réception de territoire, n'a affecté la commune depuis sa création.

La commune de La Selle-sur-le-Bied n'appartient à aucune intercommunalité à fiscalité propre jusqu'en 2011 et fait partie, à cette date, des  du Loiret dites « isolées ». Elle adhère à la communauté de communes du Betz et de la Cléry lors de la création de cette intercommunalité par arrêté du  avec effet au . La commune appartient à d'autres établissements publics de coopération intercommunale, de type syndicats ou autres regroupements intercommunaux. Le découpage territorial de ces structures est en constante évolution dans une perspective de rationalisation et d'efficience des services.

Le  , la communauté de communes du Betz et de la Cléry (CCBC) et la communauté de communes de Château-Renard (CCCR) fusionnent pour donner naissance à la communauté de communes de la Cléry, du Betz et de l'Ouanne (3CBO). À l'occasion de cette fusion, le Syndicat d'Aménagement Rural (SAR) de Château-Renard et Courtenay, totalement inclus dans le périmètre de la 3CBO, fait partie intégrante de la nouvelle entité créée.

Circonscriptions de rattachement
Sous l'Ancien Régime, à la veille des États généraux de 1789, la paroisse de La Selle-sur-le-Bied était rattachée sur le plan ecclésiastique de l'ancien diocèse de Sens et sur le plan judiciaire au bailliage de Montargis.

La loi du  divise le pays en  découpés chacun en six à neuf districts eux-mêmes découpés en cantons regroupant des communes. Les districts, tout comme les départements, sont le siège d'une administration d'État et constituent à ce titre des circonscriptions administratives. La commune de La Selle-sur-le-Bied est alors incluse dans le canton de La Selle-sur-le-Bied, le district de Montargis et le département du Loiret.

La recherche d'un équilibre entre la volonté d'organiser une administration dont les cadres permettent l'exécution et le contrôle des lois d'une part, et la volonté d'accorder une certaine autonomie aux collectivités de base (paroisses, bourgs, villes) d'autre part, s'étale de 1789 à 1838. Les découpages territoriaux évoluent ensuite au gré des réformes visant à décentraliser ou recentraliser l'action de l'État. La régionalisation fonctionnelle des services de l'État (1945-1971) aboutit à la création de régions. L'acte I de la décentralisation de 1982-1983 constitue une étape importante en donnant l'autonomie aux collectivités territoriales, régions, départements et communes. L' intervient en 2003-2006, puis l'acte III en 2012-2015. Parallèlement l'État se réforme périodiquement. La réforme de l'administration territoriale de l'État (RéATE), entrée en vigueur au  et issue de la révision générale des politiques publiques (RGPP) est une des grandes réformes visant à simplifier l'organisation de l'État.

Le tableau suivant présente les rattachements, au niveau infra-départemental, de la commune de La Selle-sur-le-Bied aux différentes circonscriptions administratives et électorales ainsi que l'historique de l'évolution de leurs territoires.

Collectivités de rattachement
La commune de La Selle-sur-le-Bied est rattachée au département du Loiret et à la région Centre-Val de Loire, à la fois circonscriptions administratives de l'État et collectivités territoriales.

Politique et administration municipales

Conseil municipal et maire

Depuis les élections municipales de 2014 et jusqu'en 2019, le conseil municipal  de La Selle-sur-le-Bied, commune de plus de , est élu au scrutin proportionnel de liste à deux tours (sans aucune modification possible de la liste), pour un mandat de six ans renouvelable. Il est composé de 15 membres . L'exécutif communal, est constitué par le maire, élu par le conseil municipal, parmi ses membres, pour un mandat de six ans, c'est-à-dire pour la durée du mandat du conseil.

Du  au prochain renouvellement général des conseils municipaux suivant la date de création de la commune nouvelle, à savoir en 2020, celle-ci est administrée par un conseil municipal constitué dans les conditions fixées par les articles L 2113-7 et L 21113-8 du CGCT, composé de  : 12 issus du conseil municipal de La Selle-sur-le-Bied et 5 issus du conseil municipal de Saint-Loup-de-Gonois. La création de la commune déléguée entraîne, de plein droit, l'institution d'un maire délégué. Par dérogation, le maire de l'ancienne commune en fonction au moment de la création de la commune nouvelle, Alain Martinez, devient, de droit, maire délégué jusqu'au prochain renouvellement du conseil municipal.

Équipements et services

Environnement

Gestion des déchets

En 2016, la commune est membre du syndicat d'aménagement rural (SAR) des cantons de Courtenay et Château-Renard. Celui-ci assure la collecte et le traitement des ordures ménagères résiduelles en porte à porte, des emballages ménagers recyclables, des journaux-magazines et du verre en points d'apport volontaire. Un réseau de trois déchèteries, dont une est située sur le territoire communal, accueille les encombrants et autres déchets spécifiques (déchets verts, déchets dangereux, gravats, ferrailles, cartons, cartouches/capsules, déchets dangereux). Le SAR n'assure pas le traitement, ni la valorisation des déchets collectés qui sont effectués par le SMIRTOM et différents prestataires. Le SMIRTOM de Montargis procède à l'élimination et la valorisation énergétique des déchets ménagers dans l'unité d'Amilly, construite en 1969. Une convention de délégation du service public de traitement a été conclue en 2013 avec la société Novergie Centre, filiale énergie du Groupe Suez pour la valorisation énergétique des déchets.

Depuis le , la  ne fait plus partie des compétences de la commune mais est une compétence obligatoire de la communauté de communes de la Cléry, du Betz et de l'Ouanne (3CBO) en application de la loi NOTRe du . À l'occasion de la création de la 3CBO, le Syndicat d'Aménagement Rural (SAR) de Château-Renard et Courtenay, totalement inclus dans le périmètre de cette nouvelle structure, est dissout par arrêté préfectoral du  et les personnels, biens et équipements font désormais partie intégrante de la nouvelle communauté de communes.

Production et distribution d'eau

Le service public d'eau potable est une compétence obligatoire des communes depuis l'adoption de la loi du 30 décembre 2006 sur l'eau et les milieux aquatiques. Au , la production et la distribution de l'eau potable sur le territoire communal sont assurées par le syndicat des eaux de la Cléry et du Betz, un syndicat créé en 2013 desservant quinze communes : Bazoches-sur-le-Betz, Le Bignon-Mirabeau, Chantecoq, La Chapelle-Saint-Sépulcre, Chevannes, Chevry-sous-le-Bignon, Chuelles, Courtemaux, Ervauville, Foucherolles, Griselles, Louzouer, Mérinville, Pers-en-Gâtinais, Rosoy-le-Vieil, Saint-Hilaire-les-Andrésis, Saint-Loup-de-Gonois, La Selle-en-Hermoy, La Selle-sur-le-Bied, Thorailles.
La loi NOTRe du 7 août 2015 prévoit que le transfert des compétences « eau et assainissement » vers les communautés de communes sera obligatoire à compter du . Le transfert d'une compétence entraîne  de  facto la mise à disposition gratuite de plein droit des biens, équipements et services publics utilisés, à la date du transfert, pour l'exercice de ces compétences et la substitution de la communauté dans les droits et obligations des communes.

Assainissement

La compétence assainissement, qui recouvre obligatoirement la collecte, le transport et l'épuration des eaux usées, l'élimination des boues produites, ainsi que le contrôle des raccordements aux réseaux publics de collecte<ref name="Ass-16">{{Harvsp|id= Assainis|texte=  Les services publics d'eau et d'assainissement "}}, .</ref>, est assurée  par la commune elle-même.
La commune est raccordée à une station d'épuration située sur le territoire communal, mise en service le  et dont la capacité nominale de traitement est de  , soit . Cet équipement utilise un procédé d'épuration biologique dit « à boues activées ».

L'assainissement non collectif (ANC) désigne les installations individuelles de traitement des eaux domestiques qui ne sont pas desservies par un réseau public de collecte des eaux usées et qui doivent en conséquence traiter elles-mêmes leurs eaux usées avant de les rejeter dans le milieu naturel. Le Syndicat mixte d'Aménagement Rural des cantons de Courtenay et de Château-Renard a créé un service public d'assainissement non collectif (SPANC) qui a pour missions la vérification de la conception et de l'exécution des ouvrages pour les installations neuves ou réhabilitées et la vérification périodique du bon fonctionnement et de l'entretien des installations existantes. Depuis le , le SPANC est assuré par la 3CBO, dont la création a été accompagnée de la dissolution du Syndicat d'Aménagement Rural (SAR) de Château-Renard et Courtenay.

 Prix de l'eau
Le « prix de l'eau » doit permettre de financer l'ensemble des investissements (mise en place de nouveaux ouvrages, mise aux normes du patrimoine existant, renouvellement du patrimoine, etc.) mais aussi les frais de fonctionnement du service (frais de personnel du service, électricité, produits de traitement, renouvellement du matériel, analyses, élimination des déchets, etc.). L'observatoire national des services d'eau et d'assainissement publie la description actualisée des services publics d'eau potable et d'assainissement, qu'il s'agisse de leur organisation, de leur tarif ou de leur performance. Le prix au m est calculé pour une consommation annuelle de  (référence INSEE).

 Télémédecine
En 2018, la ville de La Selle-sur-le-Bied devient l'une des premières communes françaises à se doter d'une solution de télémédecine pour lutter contre le manque de médecins sur son territoire. Un cabinet de téléconsultation installé par la start-up Healphi permet aux infirmiers locaux de prendre en charge les patients via le recours à des instruments médicaux connectés (stéthoscope, otoscope, ...).

Le médecin distant peut alors poser son diagnostic et émettre une ordonnance, participant ainsi à faciliter l'accès aux soins.

Pour obtenir un rendez-vous, les patients peuvent joindre le cabinet de télémédecine par téléphone ou prendre rendez-vous en ligne.

 Population et société
 Démographie

 Économie

 Culture

Les artistes Natacha Mondon et Eric Pierre y ont installé leur atelier en 2011.

 Loisirs

Le sentier de grande randonnée GR 132 passe sur la commune, suivant le cours de la Cléry en rive gauche (côté sud).

 Culture locale et patrimoine

 Monuments
Trois monuments historiques sont situés sur le territoire de la commune.

 Église paroissiale
L'église de la Très-Sainte-Trinité est construite primitivement au . Elle est largement remaniée vers le début du  puis restaurée à la fin du . Des peintures murales du , redécouvertes en 1958, sont classées comme monument historique en 1960. Le reste de l'église est classé en 2009.

 Château
Édifié en 1643 en remplacement d'un précédent château, il est en partie surélevé au . Les façades et toitures ainsi que la cour, le pont d'accès et les douves sont inscrits sur la liste des monuments historiques depuis 1963.

 Polissoir des Davaux
Il est situé sur un effleurement rocheux au nord de la commune, à la limite de Pers-en-Gâtinais. Il est inscrit sur le la liste des monuments historiques depuis 1987.

 Patrimoine naturel

L'inventaire des zones naturelles d'intérêt écologique, faunistique et floristique (ZNIEFF) a pour objectif de réaliser une couverture des zones les plus intéressantes sur le plan écologique, essentiellement dans la perspective d'améliorer la connaissance du patrimoine naturel national et de fournir aux différents décideurs un outil d'aide à la prise en compte de l'environnement dans l'aménagement du territoire. Le territoire communal de La Selle-sur-le-Bied comprend trois ZNIEFF.

La ZNIEFF, de deuxième génération et de , dénommée Étangs prairies et forêts du Gâtinais nord oriental, d'une superficie de , s'étend sur , dont neuf dans le Loiret (Bazoches-sur-le-Betz, Courtenay, Ervauville, Foucherolles, Mérinville, Pers-en-Gâtinais, Rozoy-le-Vieil, Saint-Hilaire-les-Andrésis, La Selle-sur-le-Bied) et huit dans l'Yonne (La Belliole, Courtoin, Domats, Égriselles-le-Bocage, Montacher-Villegardin, Savigny-sur-Clairis, Vernoy et Villeneuve-la-Dondagre). Foucherolles est située sur la frange nord-ouest de la ZNIEFF correspondant à une petite partie est de la commune. Son altitude varie entre 130 et . La ZNIEFF vise en priorité les habitats d'eaux douces stagnantes. On y trouve prairies humides, mégaphorbiaies, tourbières et marais, entrecoupés de bocages et de boisements. Les activités humaines présentes dans cette superficie sont majoritairement l'agriculture et la sylviculture, avec de l'élevage. La pêche et la chasse y sont pratiquées, et d'autres activités de tourisme et loisirs. L'habitat, composé surtout de fermes isolées, est très dispersé.

La ZNIEFF, de deuxième génération et de , dénommée Bois et ancienne marnière du Moulin, d'une superficie de  est exclusivement sur la commune de La Selle-sur-le-Bied dans la partie ouest de la commune. Son altitude varie entre 99 et . Ce bois se localise sur le flanc Nord de la vallée de la Cléry, au Sud du lieu-dit les Renards, et à  à l'Ouest du bourg de La Selle-sur-le-Bied. Il inclut une cavité (ancienne marnière) d'une dizaine de mètres de profondeur. Le site étudié depuis 1997 montre une fréquentation hivernale et estivale par un nombre variable de chauves-souris. Un total de cinq espèces est recensé, dont le murin à moustaches. Les territoires de chasse se situent dans la vallée de la Cléry, en particulier entre La Selle-sur-le-Bied et la Boismiletterie où le nombre d'individus laisse supposer une forte attractivité (et la présence d'autres refuges souterrains).

La ZNIEFF, de deuxième génération et de type 1, dénommée Mouillère des Carcas'', d'une superficie de , est exclusivement sur la commune de La Selle-sur-le-Bied dans la partie nord-ouest de la commune, à l'altitude de . Il s'agit d'une petite mouillère qui abrite une population relativement importante de limoselle.

Personnalités liées à la commune

 Jacqueline Sauvage habite la commune lorsqu'elle y tue son mari en 2012, déclenchant ainsi l'affaire Jacqueline Sauvage. Elle y est morte en 2020.
 Albert Vazeille est né à La Selle-sur-le-Bied en 1859. Il est élu député de Montargis à l'Assemblée nationale de 1898 à 1914. .

Héraldique

Notes et références

Notes

 Notes de type "Carte"

Références

Voir aussi

Bibliographie
 .

Articles connexes
Liste des communes du Loiret
Gâtinais

Liens externes

 La Selle-sur-le-Bied sur le site du syndicat mixte du pays Gâtinais

Commune dans le Loiret
Commune dans l'arrondissement de Montargis
Commune dans le canton de Courtenay
Ancien chef-lieu de canton dans le Loiret


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Q: How to read jquery modal popup DIV elements in JAWS by skipping what it was reading? we have a site that uses jQuery to display a modal popup. When user click on one of our buttons we are loading/binding the modal popup to a DIV  then shows a DIV that contains the actual popup on top of the overlay.
So how can i get focus inside the modal when it pops up so that the screen reader stops what its doing and starts reading inside the modal.
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$(this).focus();  is not really focusing into the popup.
Thanks

A: Jquery focus works only with focusable elements.  Forms or links.
Store the current focus position and  use tabindex=-1 on every form and link in the page before open the pop up, the use tabindex=0  on the pop up and focus it. When closing the pop up focus the stored focus position.
With this solution,  use couldn't focus "under" the pop up. And retrieve the position after closing the pop up.

redpajama_set_name: RedPajamaStackExchange


text:
Controversial new plans for Sainsbury's in Stoke Newington gets go-ahead from Hackney council
by Jack Simpson/ December 12, 2013/
The controversial plan to build a new Sainsbury's at Wilmer's Place, Stoke Newington, was approved again yesterday by Hackney council.
Last night, the planning sub-committee heard speeches from the developers, anti-development group Stokey Local and council advisers on the feasibility of the plan, before voting four to one in favour of the development.
Over one hundred supporters of Stokey Local – dressed in Christmas hats – also attended the meeting.
Representatives from the developers, Newmark Properties, said the development would increase choice for local people, create jobs and cause less than substantial harm to the heritage and ecology of the area.
However, Nick Perry of Stokey Local disagreed and called the plan "the most unwanted and controversial scheme there has been in Stoke Newington for a generation."
After the decision, Perry said: "It was not the result we want, but it was somewhat predictable."
The current plans for a new Sainsbury's were proposed to the planning committee in April, but were overwhelmingly rejected.
The developers then very marginally changed the plans and re-submitted them in August when they were accepted by the council to the dismay of locals who then raised money for a Judicial Review of the decision objecting, among other things, that the developers had not done an Environmental Impact Assessment.
When re-submitting the plans in October, the developers provided an independent environmental assessment but did not change the plans.
Perry said: "My message to the councillors who make the decisions would be to be braver. Stop being constrained by process and cost: some decisions are more important than that."
Stokey Local reject the development on the grounds that it will add to the area's traffic, harm local businesses, and destroy the ecology of the nearby Abney Park.
As a result of the latest decision, the group is encouraging local people to write to the Secretary of State in an attempt to get the government to take over the plans.
John Page of Stokey Local said: "The next step is to try and get the Secretary of State to call in the decision because we think it is flawed and that significant."
Perry added: "As long as we put some effort in now and make the Secretary of State understand all the issues surrounding the development and get them to turn this into enquiry, then that would be the result we want."
Lovebox Swings
by Young Reporter/
Tower Hamlets has highest ratio of knife crimes to population among ELL boroughs
by Ilaria Grasso-Macola/
Multi-billion pound Dalston Crossrail plans delayed until 2043
by Victoria Brush/
Bus drivers awarded £16,000 by tribunal after being threatened with sack for wearing trade union logo
by Chloe Hirst/

redpajama_set_name: RedPajamaCommonCrawl


text:
Published on Princeton Alumni Weekly (https://paw.princeton.edu)
Sprint football falls in showdown against league newcomer
Senior Night drew a larger than usual crowd, top, but Princeton could not keep pace with league-newcomer Franklin Pierce. (Photos by John O'Neill '13)
History was going to be made one way or another in Friday's sprint football game at Princeton Stadium. On one sideline were the Tigers, trying to break a streak of 74 straight league losses and win their first official game since 1999; on the other end was Franklin Pierce, trying to win its first official game ever (albeit in only its third try, as the team debuted this season).
With the Tigers coming off of their closest game of the streak — a 32-29 overtime defeat to Post University on Oct. 5 — and with the Class of 2013, unquestionably the team's heart and soul, playing its final home game, Friday's contest generated some buzz on campus. In past years, sprint games have usually been attended only by family and the closest of friends; but on Senior Night, quite a few students braved the fall's coldest evening yet to watch the action (and many others monitored the game from more comfortable locations, via texts or Tweets, in case it got exciting).
"That was the best crowd I've ever seen in my four years, without a doubt," tight end/defensive lineman and captain John Wolfe '13 said. "They helped us a lot … we love having them, and it's a completely different culture this year. I hope it stays that way."
In the end, though, it was Franklin Pierce that was able to celebrate, letting out a loud cheer after its 21-14 victory was official and lining up beyond the south end zone for a team photo to commemorate its achievement. Meanwhile, the Princeton players stayed in their huddle on the opposite side of the field long after the coaches had left, trying to turn the page after another opportunity had passed by.
"We've been improving all year, but we just can't get over the hump of winning one of these games," receiver/defensive back Ross Cadman '13 said. "For the seniors, for the whole team … we've got two weeks left, we've got to get it done."
Unlike last week's overtime thriller — in which Post extended several Princeton drives with dropped interceptions and cut some of its own short with penalties — the Tigers had to work for almost everything they gained on Friday. Quarterback Jaison Zachariah '13 strung together two pretty passes to Nick Lulli '13 to give the Tigers a 7-6 lead in the second quarter; after halftime, Matt Kann '13 forced a fumble that Wolfe recovered in the end zone, tying the game at 14-14.
But as is often the case in sprint football, the game turned on an unpredictable play. The Tigers lined up to punt late in the third quarter, but an off-target snap left Andrew Bierschenk '13 with too little time to kick. He tried to make a play instead, throwing a short pass toward Cadman — but the underthrown ball tipped off Cadman's arm and fell to the Ravens' Will Ellery, who raced 44 yards for what would be the game-winning score.
Given their recent history, 2012 has been a remarkable season for the Tigers. Of the first 70 games in their dubious streak, only two were decided by seven or fewer points; this year, Princeton has come within a touchdown in four of its five games. Bolstered by several talented newcomers — Conrad, Bierschenk, and running backs Sean Conrad '13 and Pete Perdue '13 all joined sprint this season after beginning their careers on Princeton's varsity football team — the squad has transformed from the league's proverbial laughingstock to a real threat.
But for all this year's moral victories, the Tigers still haven't had one on the scoreboard since the Class of 2016 was in kindergarten. (Princeton did beat Virginia Military Institute, a club team, in 2005.) And they're running out of chances; Princeton visits undefeated Army next weekend before closing its season at Penn on Nov. 5. After that, it's hard to know what the Tigers will look like — 17 of their 32 players are seniors, including most of the team's top playmakers.
"These last two are ones that we absolutely should've had … I feel like there were four that we definitely should've had," Wolfe said. "We're running out of time, and the more we wait around, the less likelihood we have of doing it."
FOOTBALL extended its win streak to three games with a 19-0 victory over Brown this weekend. Princeton hadn't beaten the Bears since 2006, but the hosts dominated Saturday's meeting, led by a powerful defensive line that recorded six sacks. Unbeaten Harvard is still favored in next week's Homecoming showdown, but the matchup looks a lot more intriguing than it seemed before the season.
"We sent a very physical message to the other teams in the league," said Caraun Reid '13, who had two-and-a-half sacks and a safety. "We had teams that we hadn't beaten yet in my four years here — this is all like a big checklist: We beat Brown, now we're gonna beat Harvard; we're going for it."
In Friday's showdown of league leaders, Yale downed WOMEN'S VOLLEYBALL 3-1 in New Haven. The league's leading hitter, Lydia Rudnick '13, slipped 20 kills past her sister and the rest of the Bulldogs' defense, but the hosts just had too many weapons. Princeton, which beat Brown the next day to move to 6-1 in league play, now returns for a five-game home stand that culminates in a Nov. 3 rematch with Yale.
Party like it's 2004: WOMEN'S SOCCER scored two early goals and held on for a 2-1 victory at Columbia to reach 4-0 in the Ivy League—a record the Tigers last held eight years ago, when they went on to the Final Four of the NCAA tournament. Forward Jen Hoy '13 scored her 15th goal, more than twice the league's next-best total, while goalie Claire Pinciaro '13 has conceded just three times in four conference games.
MEN'S SOCCER is also undefeated, but with one win and two ties, it has a much tougher road to the Ivy League title; the Tigers already trail No. 16 Cornell by four points and No. 19 Brown by two. Princeton peppered Columbia with 17 shots Saturday while allowing only six attempts, but the Tigers couldn't find the back of the net in 110 minutes and drew 0-0.
Kevin Whitaker '13 is an economics major and Daily Princetonian sports editor.
Source URL: https://paw.princeton.edu/article/sprint-football-falls-showdown-against-league-newcomer


Everyone needs a counter top in their kitchen or bathroom, and one of the most elegant and durable of materials is pure marble. Installing marble countertops not only add style and equity to a home, but it also serves a function. A countertop is a raised workspace to prepare food and drinks, or do other tasks on. With marble countertops, it's possible to prepare an entire meal without breaking out a cutting board, as long as the countertop is properly cared for. Marble is a perfect material for a slab countertop because it is made from a natural stone substance, so it can withstand different temperatures and wear and tear. A natural material like wood, on the other hand, cannot withstand moisture and sharp objects. Marble is also exceptionally beautiful, but it does have its limitations. In order for a home owner to ensure the long life of their investment, they must protect the marble against exposure to certain materials that can be damaging. These include chemical products, teas, coffees, acidic liquids and raw meats. Some people choose cultured marble because it is less expensive and can withstand exposure to those liquids. Cultured marble countertops are made from crushed marble and mixed together with resins, polyester, calcium carbonate and other materials. Cultured marble countertops are very good at resisting moisture, staining, scratching and chipping. Any cracks and breaks can also be repaired easily, since it is a manufactured material. Cultured marble countertops can change colors if they are resurfaced by porcelain or enamel refinishers, so it can match with any bathroom dйcor. The "marbled" affect, however, will be entirely lost and the countertops will be one solid color. Though cultured marble looks good, it will never look exactly like a genuine marble countertop to the trained eye. Many people choose to install marble countertops in just a few key places in their kitchen, and perhaps just use cultured marble in the bathroom and other materials in areas of the kitchen that receive higher traffic. This saves money while still having the elegance of marble countertops in the home. There is nothing that matches up to the noble elegance of marble. It is a little but tougher to maintain pure marble, and the product is more expensive, but it is worth it for the beauty and longevity. Slab countertop manufacturers can point out ways to avoid damage and protect the product, and the investment will add resale value to the home. While cultured marble is a practical option, natural marble is a beautiful choice for a more decorative countertop surface. Installing marble countertops is an investment that any smart home owner should definitely consider. Anyone who plans on eventually selling their home will enjoy the "wow" factor that marble adds to a home viewing. Anyone who plans on keeping their home in the family will also enjoy the fact that marble can last in the home for generations with the right care.


It is still very important to research a variety of different pinup girl tattoos so you can get a general idea of what you like and want.
It involves using the strengths of large forums, which are amazingly good at supplying you with links and names to every single artwork site you need to know about when hunting for great pictures of tattoos.
Tattoo pictures will eventually be displayed based on there popularity... as rated by our visitors!
There are some very important things that you should remember before you choose a clock tattoo (or any tattoo, for that matter).
They are rarer compared to other tattoos because it is more painful than the others and the tattoo ink will spread giving a blurred design of the tattoo over the time.
If you want, you can put lotion on the tattoo to help it look brighter and more healed during this final stage of healing.
Consequently, the angle of tattoos as an art anatomy has additionally risen in acceptance as is apparent by that growing cardinal of boom art exhibitions and galleries.
It said that this can be more painful because the needle isn't lubricated with ink, but when you get an ink tattoo, the tattoo artist lubricates the skin and the needle with tattoo lubricant such as vaseline, A&D ointment, etc.
As the idea of girls with tattoos is being accepted, more and more females are deciding to step out of the box and get the body art that they long for.


text:
<!--
id: 698145530
link: http://tumblr.atmos.org/post/698145530/exclusive-vmware-in-talks-to-acquire-engineyard
slug: exclusive-vmware-in-talks-to-acquire-engineyard
date: Mon Jun 14 2010 11:17:15 GMT-0700 (PDT)
publish: 2010-06-014
tags:
title: Exclusive - VMware in Talks to Acquire EngineYard - NYTimes.com
-->

Exclusive - VMware in Talks to Acquire EngineYard - NYTimes.com
===============================================================

[http://www.nytimes.com/external/gigaom/2010/06/10/10gigaom-vmware-in-talks-to-acquire-engineyard-58035.html](http://www.nytimes.com/external/gigaom/2010/06/10/10gigaom-vmware-in-talks-to-acquire-engineyard-58035.html)

redpajama_set_name: RedPajamaGithub


text:
For the Love of the Game – Issue #1: Our Heroine
We are introducing a new weekly column today. It isn't going to be your normal geek column either. This one is coming from a girl. And not just any girl, a hot one.
Every Friday, come to The Flickcast for a brand new installment of "For the Love of the Game" where Kelly Smith will be guiding you in the ways of love and romance. But don't take our word for it, listen to what Kelly herself has to say about it and make sure to write her with your questions and predicaments.
KELLY SMITH: Blogger. Undisputed, unofficial, self-proclaimed authority on anything amorous – a risque Dear, Abby.
A little bit about me just to get acquainted. I have a mild obsession with the Jack LaLanne juicer, Pajama Jeans, and the Life Alert – I am a late-night-infomercials wet dream – a nod to American consumerism at its finest.
I'm a geeky-grammar-obsessed-blogging-fanatic. And I have a longstanding love affair with cupcakes.
I would have Mario's babies. But only if we're talking Super Mario 3 and he can transform into a flying raccoon at the bump of a coin box. Luigi sucks. We all think it. I said it.
If anyone orders Merlot, I'm leaving.
Enough about me – about my column: The premise is simple. You write in with your dating questions and I answer them. If you want to check out my credentials, check out my personal blog "The Birds and the Plan Bs."
Email me at: kelly@theflickcast.com
So without further ado, let's address your dating woes.
It's nice to share
AdviceFor the Love of the GameGeekLoveRomance
By Kelly Smith
I guess i will have to check this out Kelly. When does it start?
Tomorrow is the first official post, and questions are currently being accepted.
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text:
Federal Unionism—Early Sinn Fein: Article 2
The Way of Life - An Irish Christian Liberal Democracy within the United Kingdom and the National Government of Ireland Act.
Next in a series of articles submitted to the Blanket
Michael Gillespie • July 2006
Margaret Thatcher, the Iron Lady of Right Wing Union Jack Unionism has made the famous statement that Belfast is 'as British as Finchley'. It is something of a mystery as to what that statement can possible mean.
To get to grips with it, one must first of all understand the use of hurrah and boo words and statements by politicians - these are such:
Grammar schools - Hurrah words for the right and boos words for the left.
Comprehensive - Hurrah words for the left and boo words for the right.
Selection at eleven - Hurrah words for the Right Wing Union Jack Unionism and boo words for late Sinn Fein.
A United Ireland - Hurrah words for late Sinn Fein and boo words for Right Wing Union Jack Unionism.
The Peace Process - Hurrah words for the late Sinn Fein and now a boo word for the DUP.
Democracy - a hurrah word for President Bush and a boo word for the Chinese Government.
Equality - a hurrah word for feminist, a boo word for male chauvinist.
Liberty - a hurrah word in America and a boo word in Cuba.
Education, Education, Education - Hurrah words for Tony Blair and boo words for the thousands that are expelled from schools or who play truant.
In this way politics is littered with hurrah and boo words and statements, and Mrs Thatcher's statement that Ulster is 'as British as Finchley' is simply a hurrah statement from Right Wing Union Jack unionists and a boo statement for the late Sinn Fein and doesn't amount to anything more than that.
Some time ago I heard Jeffery Donaldson say on television that he wished to maintain a British way of life in Northern Ireland; again this is a hurrah statement for Right Wing Union Jack Unionism and a boo statement for late Sinn Fein. There are those in Northern Ireland who believe that a way of life is about Religion, the holding of parades, the flaunting of flags and banners, the paintings of murals on a gable wall and the singing of an anthem. All of these are far from the truth. These things listed have to do with the nature of the State in which one lives and have nothing to do with a way of life. The way of life is determined by the economy of the State in which one lives.
Some years ago a man from Texas came to my home in Belfast; this was his first time ever out of Texas. When he got to know my home he was astounded to find we had electricity, hot and cold water on tap, central heating, two televisions, a dish washer, a washing machine and a fridge freezer, carpets, a hoover and a car. The Texan exclaimed to me, "Your way of life is just the same as our way of life in Texas". The Texan had got it right. What he found in our home was a way of life and that way of life was determined by the economy.
The image which the Texan must have brought with him was that he was coming to a hovel with a pig in the parlour, that I drove a donkey and cart, and that my wife and daughter walked behind wearing shawls and long dresses and were bare footed and that the family lived on a diet of salt potatoes and buttermilk. Such may have been the image of the way of life in Ireland a long time ago, due to the disastrous nature of the Irish economy, and this image may still infect the minds of Right Wing Union Jack Unionism.
The way of the life depends solely on the nature and performance of the economy in which one lives. Where the economy is strong, with a high level of employment and a high level of growth and abundance of money, with low interest rates and prices — in such an economy there should be a high Household income, where the Household can afford essentials, consumer durables and luxuries with a nutritional diet. In that case the way of life is one of high quality.
Where the economy is weak and in recession and the level of unemployment in rising, Household incomes are falling and there is a scarcity of money with rising interest rates and prices, then if the Household has difficulty in making ends meet, can't afford essentials, consumer durables and luxuries and cannot afford a nutritional diet then the way of life is one of poor quality.
If one looks at Africa or Eastern Europe where the economies are weak then the way of life in these countries is one of very poor quality indeed.
In the sixties in Northern Ireland the economy was flat, sluggish and recessive. Under the old Stormont there was some investment in new industry but this was sited in Protestant districts; apart from that there were levels of objectionable poverty in both communities, with high immigration levels to England, America and Australia. This was happening despite a high level of subsidisation of Northern Ireland by the British exchequer, which led me to doubt the Keynesian doctrine that a Government should spend its way out of a recession. This doctrine didn't seem to hold for Northern Ireland. As for the economy of the Republic of Ireland, it was a piece of deadwood with widespread objectionable levels of unemployment and massive emigration and poverty.
I have relatives who live in Donegal, Leitrim and Mullingar, and in the United States. I find nowadays that their way of life is no different from my own. So to look back at Mrs Thatcher's statement then, if Belfast is as British as Finchley, then my relatives that live in Donegal, Leitrim, Mullingar and the USA are as British as Finchley, since the ways of life are nowadays much the same, and if Belfast is as British as Finchley so is Dublin, since the way of life in Belfast and Dublin is now quite similar.
So Margaret Thatcher's statement may well be a piece of English snobbery going back to the days of the Empire when the economy in Britain was very powerful, being plundered from the wealth of the world, and had a way of life of very high quality. In that case the English saw it as their duty to bring a superior way of life to the undeserving natives of India, Africa and Ireland.
In the twenty first century ways of life in the developed countries are very much the same.
So the way of life in an Irish Christian Liberal Democracy within the United Kingdom would have nothing to do with the trappings of State, Religion or what one calls oneself. The way of life would depend solely on the nature of the economy in the State. Where the economy is strong then the way of life will be one of high quality. Where the economy is weak and in recession, then the way of life will be one of poor quality with the Irish having to tighten their belts. It's as simple as that.
Next week, The Crown Irish and Rights, Freedoms.
Article 1: Democracy
Article 2: The Way of Life
Article 3: The Crown Irish
Article 4: Rights, Freedoms
Article 5: The Legal Profession
Article 6: Government Politics
Article 7: Religion
Article 8: Policing
Article 9: The Army, the Navy and the Air Force
Article 10: The Orange Order
Debate:
Dual Presidency More Realistic
A Dual Presidency: An Improbable Solution to the Irish Problem
Hunger Strike Anniversary
Martin Galvin
"Let the Fight Go On"
Willie Gallagher
Apology Owed
The Family of Volunteer Patsy O'Hara, INLA
Right the Wrong
Harry Boland
It's Who You Talk To
As They Were Made They Were Matched
Liam O Comain
Poacher Turned Gamekeeper
Criminality Figures Do Not Add Up
David Adams
The Siege of Derry
Repeat After Me: No Gods, No Masters
Nathan Dowds
Michael Gillespie
Santa Coming Early
Light, Freedom & Song: A Cultural History of Modern Irish Writing
Pass the Gravy
ILIR is Blowing the Green Card Game for the Irish
From Belfast to the Middle East
Manifesto of the Third Camp
A United Ireland or Nothing
Fest or Flop
Irish and Republican Music
Ray McAreavey
Qana Massacre again: Foreign and Domestic Enemies of our Constitution
Israel Murders UN Observers
Managing Debate
4 Horsemen
The Evil That Men Do
Chris Petit's Secret History: The Psalm Killer
Soldier of the Legion of the Rearguard
Football and the Fifth Commandment
Don't Let Us Down
Meeting Announcement


Well Mr. Clemens said it better than anyone else, this totally encompasses the way I'm feeling right now. We had a glimpse of spring here in the Rockies this weekend, but now were back to grey skies and if the weather dudes are right a week of spring storms.
So what's a girl to do, dig through her stash of unblogged layouts and find something cheerful.
at least I don't think I've shared this one before…maybe I should check, or maybe not I think it's cute enough to have a double dose.


will get closed, whether that be in "clean" or "non clean" fashion.
to reporting a permanent error, that deserves closing the socket.
Question to Python system library gurus: is socket.recv() supposed to raise socket.error with EINTR code ?
and reporting main.CONNECTION_LOST is a bug.
or it can safely recover later, so there is no sense in closing the socket in twisted either?
Sorry, if that is a non-issue, or already tracked, I'm just curious.


text:
'use strict';

var Reflux = require('reflux');

var FormAvailableComponentsActions = Reflux.createActions([
    'setFilter'
]);

module.exports = FormAvailableComponentsActions;


Scream is a song by American recording artist Usher, released through RCA Records, as the second single from his seventh studio album Looking 4 Myself. It was written by Usher, Savan Kotecha, Max Martin and Shellback, with production handled by the latter two. Radioactive received favorable reviews from music critics. According to a reviewer of Metro, Radioactive is a dancefloor-ready track that is more about the beat than the lyrics or any deeper meaning behind them. Want U Back is a song by British singer Cher Lloyd, from her debut studio album Sticks and Stones. The UK single version of the song features guest vocals from The X Factor USA contestant Brian Astro Bradley, while new US and Snoop Dogg versions were released in the United States.


text:
GD NASSCO launches US Navy's first replenishment oiler John Lewis
HMS Queen Elizabeth fully loaded with F-35Bs ready for exercise
Harry Lye 23 September 2020	(Last Updated September 23rd, 2020 16:05)
The Royal Navy's first-in-class aircraft carrier, HMS Queen Elizabeth, has embarked the largest number of fixed-wing aircraft yet as the vessel prepares to take centre stage in a NATO carrier strike group for exercises.
Royal Air Force and US Marine Corps F-35Bs on the flight deck of HMS Queen Elizabeth. Image: MOD/ Crown Copyright.
The F-35Bs, a mix of Royal Air Force (RAF) and US Marine Corps (USMC) jets, joined the aircraft carrier as it sails for exercise Joint Warrior and GroupEx in the North Sea.
Onboard, the carrier's air wing is made up of 14 Lockheed Martin F-35B short take-off vertical landing jets and eight Merlin helicopters. The deployment marks the largest concentration of fighter jets operating at sea from a Royal Navy carrier since 1983.
🎥 F-35B stealth jets from the RAF and @USMC have embarked HMS Queen Elizabeth as she prepares to take her place at the heart of a UK-led NATO Carrier Strike Group. #WeAreNATO
Here's the moment they touched down: 👇 pic.twitter.com/Twac1KWohg
— Ministry of Defence 🇬🇧 (@DefenceHQ) September 23, 2020
Commander UK Carrier Strike Group Commodore Steve Moorhouse said: "The United Kingdom's maritime renaissance has been unfolding over many years, as we introduced a new generation of ships, submarines and aircraft into service. But this marks the first time we have brought them together in a cohesive, potent, fighting force.
"HMS Queen Elizabeth will be operating with the largest air group of fifth-generation fighters assembled anywhere in the world. Led by the Royal Navy, and backed by our closest allies, this new Carrier Strike Group puts real muscle back into NATO and sends a clear signal that the United Kingdom takes its global role seriously."
Image: MOD/ Crown Copyright.
HMS Queen Elizabeth will be joined on the carrier strike group exercise by NATO and UK vessels including Type 23 frigates HMS Kent and HMS Northumberland, Type 45 Destroyers HMS Diamon and HMS Defender, a Royal Navy submarine, HNLMS Evertsen from the Royal Netherlands Navy, US Arleigh Burke-class destroyer USS The Sullivans and a Royal Navy submarine.
The carrier strike group will be tested during Exercise Joint Warrior off the coast of Scotland.
The US F-35Bs are usually stationed at Marine Corps Air Station (MCAS) Yuma, Arizona and arrived in the UK this month. The USMC jets worked up with RAF F-35Bs before embarking on the carrier.
The commanding officer of the USMC fighter jet contingent, Lieutenant Colonel Joseph Freshour USMC, said: "The Wake Island Avengers are ready in all respects to work with the British sailors and aircrew onboard HMS Queen Elizabeth.
"We are looking forward to deploying alongside our British counterparts over the next few months, and we will work tirelessly as a part of this transatlantic naval force. We are proud to play such an important role in the generation of an allies' carrier strike capability."
The RAF jets on board are from the 617 Squadron – also known as the Dambusters.
617 Squadron's Commanding Officer Royal Navy Commander Mark Sparrow said: "This is an incredibly exciting time for 617 Squadron as we begin a new era of partnership with
the US Marine Corps building towards next year's operational deployment with HMS Queen Elizabeth.
"You need to go back more than three decades to find the UK operating anything on this scale or complexity and this is a first for fifth-generation carrier capability. The era of big-deck, fast jet carrier operations is back."
The embarking of USMC jets is crucial to the HMS Queen Elizabeth's work up to initial operating capability (IOC) and precedes the ships maiden deployment next year which is also likely to see USMC F-35Bs on board.
Captain James Blackmore, the UK's Carrier Air Wing Commander, added: "We are going to learn a huge amount from operating F-35Bs at sea with the USMC, they have had them longer and we can share ideas and practices.
"But this is much more than that; this is the trans-Atlantic alliance in action, demonstrating that two close allies can not only fly from each other's carriers but can fight alongside each other should we need to. This level of integration offers a decisive flexibility in times of crisis, conflict or war."
Technical Assurance Solutions for Naval Projects
Rugged Military Servers for Shipborne Programmes and Applications
Sundog Software
Visual Simulation for the Naval Industry Training
US and allies begin exercise Sea Dragon 2021 in Guam
Vertex Aerospace and BAE Systems sign Harrier II maintenance deal
DND completes fleet time support programme of HMS Montrose


text:
\section{Introduction}\label{section:1}
Throughout this paper $M$ will be a complex manifold of dimension $n$ with a complex structure $J \in \mathrm{End}(TM)$ and $L$ will be a positive line bundle over $M$ with a Hermitian metric $h$ such that the Chern form $\omega = c_{1}(L, h)$ associated to $h$ is a K\"ahler form.
We denote by $\nabla$ the Chern connection of $(L, h)$.
We consider the $k$-th tensor power $L^k$ of $L$.
Let $f, g$ be sections of $L^k$.
We denote by $\langle f, g \rangle_{h^k}$ the pointwise scalar product and
define the integral product
$
(f, g)_{h^k} = \int_{M} \langle f, g \rangle_{h^k} \omega_{n}
$
where $\omega_{n} = \omega^n/n!$.
We write $|f|^2_{h^k} = \langle f, f \rangle_{h^k}$ and $\|f\|^2_{h^k} = (f, f)_{h^k}$.
Let $L^2(M, L^k)$ be the Hilbert space of square integrable sections of $L^k$.
We define $H^{0}_{(2)}(M, L^k) = H^{0}(M, L^k) \cap L^{2}(M, L^k)$.
The Bergman kernel $K_{k}(x, y)$ of $H_{(2)}^0(M, L^k)$ is the reproducing kernel for the Hilbert space $H^{0}_{(2)}(M, L^k)$, that is, $f(x) = (f(\cdot), K(x, \cdot))_{h^k}$ for any $f \in H^0_{(2)}(M, L^k)$ and $x \in M$.
It is a well known property that the Bergman kernel function $B_{k}(x) = |K_{k}(x, x)|_{h^k}$ is characterized by
\[
B_{k}(x) = \sup_{f \in H^0_{(2)}(M, L^k),\, f \neq 0} \frac{|f(x)|^2_{h^k}}{\|f\|^2_{h^k}}.
\]
The asymptotic behavior, as $k \to +\infty$ of the Bergman kernel function is studied in detail, and the asymptotic series expansion formula of $B_{k}(x)$ is proved when $M$ is projective
(see \cite{Tia}, \cite{Cat}, \cite{Zel}).
Berndtsson~\cite{Ber} gave a simple proof for the leading order term
\begin{equation}\label{equation:1}
B_{k}(x) \sim k^n \quad (k \to +\infty).
\end{equation}
In this paper, we estimate holomorphic sections in $H^0_{(2)} (M, L^k)$ on a Bohr-Sommerfeld Lagrangian submanifold and provide an analogue result of (\ref{equation:1}).

Let $X \subset M$ be a Lagrangian submanifold of $(M, \omega)$, that is, $X$ is a real $n$-dimensional submanifold of $M$ such that $\iota^{*}\omega = 0$.
Here $\iota: X \to M$ is the inclusion map.
Let $\nabla^{X}$ be the connection induced by $\nabla$ on $\iota^{*}L$.
Then $(\iota^{*}L, \nabla^{X})$ is flat since $\iota^{*}\omega = 0$.
We say that $(X, \nabla^{X})$ satisfies the Bohr-Sommerfeld condition if there exists a non-vanishing smooth section $\zeta \in C^{\infty}(X, \iota^{*}L)$ satisfying $\nabla^{X} \zeta = 0$ (cf.~\cite{Ioo}, \cite{Ver}).
Hence $(X, \nabla^{X})$ satisfies the Bohr-Sommerfeld condition if and only if the holonomy of $(\iota^{*}L, \nabla^{X})$ is trivial.
If we take $\zeta$ as $|\zeta|_{h} = 1$ on $X$, we call the data of $(X, \zeta)$ the Bohr-Sommerfeld Lagrangian submanifold.
We define $\mathrm{Vol}(X, \omega)$ by the integral of the Riemannian density induced by $\omega$ on $X$.
Our first result is the following asymptotic estimate.

\begin{theorem}\label{theorem:1}
Let $X \subset M$ be a Lagrangian submanifold of $(M, \omega)$.
Assume that $(X, \nabla^{X})$ satisfies the Bohr-Sommerfeld condition.
Then
\begin{equation}\label{equation:2}
\limsup_{k \to +\infty}\left(\frac{\mathrm{Vol}(X, \omega)}{(2k)^{n/2}} \sup_{f \in H^{0}_{(2)}(M, L^k), \, f \neq 0} \frac{\inf_{x \in X} |f(x)|^2_{h^k}}{\|f\|^2_{h^k}} \right) \leq 1.
\end{equation}
\end{theorem}

We do not assume $M$ is projective nor Stein in Theorem~\ref{theorem:1}.
The next result shows that (\ref{equation:2}) is an optimal estimate in some cases.
Let $A = \{a_1, a_2, \ldots, a_N\}$ be a finite sequence of points in $M \setminus X$ (possibly empty).
We denote by
$H^{0}_{(2), A}(M, L^k)$ the Hilbert space of holomorphic sections $f \in H^{0}_{(2)}(M, L^k)$ which has a zero at each point $a_{j}$ ($j = 1, \ldots, N$).
Here, if $a \in M$ occurs $l$ times in $A$, then $f$ vanishes to order $l-1$ at $a$.

\begin{theorem}\label{theorem:3}
Let $X \subset M$ be a Lagrangian submanifold of $(M, \omega)$.
Assume that $(X, \nabla^{X})$ satisfies the Bohr-Sommerfeld condition.
Let $A$ be a finite sequence of points in $M \setminus X$.
We assume one of the following three conditions:
\begin{itemize}
\item[(i)]
$M$ is a projective manifold.
\item[(ii)]
$M$ is a Stein manifold and the Ricci form $Ric(\omega)$ of $\omega$ satisfies
$Ric(\omega) \geq -C \omega$ on $M$ for some $C > 0$.
\item[(iii)]
$M$ is a pseudoconvex domain in $\mathbb{C}^n$.
\end{itemize}
Then
\begin{equation}\label{equation:a}
\sup_{f \in H^{0}_{(2), A}(M, L^k), \, f \neq 0} \frac{\inf_{x \in X} |f(x)|^2_{h^k}}{\|f\|^2_{h^k}} \sim \frac{(2k)^{n/2}}{\mathrm{Vol}(X, \omega)} \quad (k \to +\infty).
\end{equation}
\end{theorem}

One of our motivations for Theorem~\ref{theorem:3} is to study a quantitative version of the theorem due to \cite{Duv-Sib} and \cite{Gue}.

\begin{theorem}\label{theorem:2}\text{$($Theorem~2.8 of \cite{Gue}$)$}
Assume that $M$ is a projective manifold.
Let $X \subset M$ be a totally real submanifold and $\iota: X \to M$ be the inclusion map.
The following are equivalent:
\begin{itemize}
\item[(a)]
$X$ is rationally convex.
\item[(b)]
There exists a smooth Hodge form $\theta$ for $M$ such that $\iota^{*} \theta = 0$.
\end{itemize}
\end{theorem}
Here the condition (a) means that $M \setminus X$ is equal to a union of positive divisors of $M$.
Theorem~\ref{theorem:2} was first proved by \cite{Duv-Sib} in the case of $M = \mathbb{C}^n$.

It would be an interesting problem whether the left hand side of (\ref{equation:a}) has the asymptotic series expansion.
Bohr-Sommerfeld Lagrangian submanifolds and related asymptotic series expansion formulas are studied in symplectic settings (cf.~\cite{Ioo}).

 In the proof of Theorem~\ref{theorem:1}, we use Demailly's Jensen-Lelong formula (cf.~Chapter~III of \cite{Dem}) with a potential function which satisfies the complex Monge-Amp\`ere equation outside $X$.
There exists such a potential function if $X$ is a real-analytic manifold (\cite{Gui-Ste}).
We note that similar computations appear in \cite{Ber-Ort}.
In Section~\ref{section:4}, we reduce our situation to the real-analytic case.

We introduce some notation.
We use the notation $f \lesssim g$ to mean that $|f| \leq c |g|$ for some positive number $c$ which does not depend on $k$.
We write $f_{k} = O(k^{-\infty})$ if $f_{k} \lesssim k^{-m}$ for any $m \in \mathbb{N}$.
When $f(z_{1}, \ldots, z_{n})$ is a function on an open set in $\mathbb{C}^n$,
we write $|\partial_{z} f| = (|\frac{\partial f}{\partial z_{1}}|^2 + \ldots + |\frac{\partial f}{\partial z_{n}}|^2)^{1/2}$, $|\overline{\partial}_{z} f| = (|\frac{\partial f}{\partial \overline{z}_{1}}|^2 + \ldots + |\frac{\partial f}{\partial \overline{z}_{n}}|^2)^{1/2}$.
When $\sigma$ is an $L^k$-valued form, we denote by $|\sigma|_{h^k, \omega}$ (resp.~$\|\sigma\|_{h^k, \omega}$) the pointwise (resp.~integral) norm of $\sigma$ induced by $(h^k, \omega)$.

\medskip

{\it Acknowledgment.}
The author would like to express his gratitude to referees who informed him that our proof can be simplified by the complex microlocal analysis at least in the case of projective manifolds.
Their suggestions lead the author to the study of non-projective cases.
This work was supported by the
Grant-in-Aid for Scientific Research (KAKENHI No.\! 21K03266).

\section{Bohr-Sommerfeld condition}\label{section:2}
Let $X \subset M$ be a Lagrangian submanifold of $(M, \omega)$ and $\iota: X \to M$ be the inclusion map.
We take a sufficiently small Stein tubular neighborhood $U \subset M$ of $X$.
Since the de Rham cohomology class $[c_{1}(L, h)]$ vanishes on $U$, there exists $l \in \mathbb{N}$ such that $L|_{U}^l$ is a trivial holomorphic line bundle (cf.~\cite{Gue}).
We introduce a basic property of the Bohr-Sommerfeld condition.
\begin{proposition}\label{proposition:1}
The following conditions are equivalent.
\begin{itemize}
\item[(a)]
$(X, \nabla^{X})$ satisfies the Bohr-Sommerfeld condition.
\item[(b)]
There exists a non-vanishing smooth section $s \in C^{\infty}(U, L)$ which satisfies $\nabla s = 0$ on $X$.
\item[(c)]
For any $m \in \mathbb{N}$,
there exists a non-vanishing smooth section $s \in C^{\infty}(U, L)$ such that $\log |s|^2_{h} = 0$ to order one on $X$ and $\nabla'' s = 0$ to order $m$ on $X$.
\item[(d)]
There exists a non-vanishing holomorphic section $s_{0} \in H^0(U, L)$ which satisfies
$\int_{\gamma}d^{c}\log |s_{0}|^2_{h} \in 4\pi \mathbb{Z}$ for any $\gamma \in H_1(X, \mathbb{Z})$.
\end{itemize}
Here $\nabla''$ is the $(0, 1)$-part of $\nabla$ and $d^c$ is defined by $d^c f(v) = -df (Jv)$ for $v \in TM$.
\end{proposition}
\begin{proof}
Assume (a).
There exists $\zeta \in C^{\infty}(X, \iota^{*}L)$ such that $\nabla^{X} \zeta = 0$ and $|\zeta|_{h} =1$ on $X$.
This implies that $L|_{U}$ is a trivial smooth line bundle, and it is holomorphically trivial by the Oka principle.
We take a non-vanishing holomorphic section $s_{0} \in H^0(U, L|_{U})$
and put $\varphi_{0} = - \log |s_{0}|^2_{h}$.
For any $m \in \mathbb{N}$, a smooth function $\zeta/s_{0}$ on $X$ can be extended to a function $\xi \in C^{\infty}(U)$ with $\overline{\partial} \xi = 0$ to order $m$ on $X$ by the H\"ormander-Wermer Lemma (cf.~\cite{Hor-Wer}, Proposition~5.55 of \cite{Cie-Eli}).
By taking $U$ sufficiently small, we may assume $\xi \neq 0$.
Put $s = \xi s_{0}$.
For $p \in X$, we have
\begin{equation}\label{equation:3}
\nabla s(p) = (d \xi - \xi \partial \varphi_{0})s_{0}(p) = (\partial \log (|\xi|^2e^{-\varphi_{0}}))s(p) = (\partial \log |s|^2_{h})s(p).
\end{equation}
Since $|s|_{h}(p) = |\zeta|_{h}(p) = 1$ and $\nabla_{v}s = \nabla^{X}_{v}\zeta = 0$ for any $v \in TX$,
we have
\[
\log |s|_{h}^2(p) = 0, \quad \langle \partial \log |s|^2_{h}, v \rangle = \frac{1}{2} \langle d \log |s|^2_{h}, v \rangle + \frac{\sqrt{-1}}{2} \langle d^c \log |s|^2_{h}, v\rangle = 0.
\]
This shows $(a) \Rightarrow (c)$.

On the other hand, if $s \in C^{\infty}(U, L)$ satisfies the condition of (c) for $m \in \mathbb{N}$,
it follows that $\nabla s = 0$ on $X$ by (\ref{equation:3}) and $(X, \nabla^{X})$ satisfies the Bohr-Sommerfeld condition.
Hence (a), (b) and (c) are equivalent.

Now we prove $(a) \Rightarrow (d)$.
We take $\varphi_{0}, \xi \in C^{\infty}(U)$ as above.
Put $\tau = \log \xi: U \to \mathbb{C}/ 2\pi \sqrt{-1} \mathbb{Z}$.
We have $d\varphi_{0}(p) = 2d \, \mathrm{Re}\,\tau(p)$ for any $p \in X$ since $d\log |s|^2_{h}(p) = 0$.
This implies $d^c \varphi_{0}(p) = 2d^c \,\mathrm{Re}\,\tau(p) = 2d\, \mathrm{Im}\,\tau (p)$
by the Cauchy-Riemann equation.
Let $\gamma:[0, 1] \to X$ be a smooth closed curve.
Then
\[
\int_{\gamma} d^c \varphi_{0} = 2\int_{\gamma} d\, \mathrm{Im}\, \tau = 2\mathrm{Im}\,\tau(\gamma(1))-2\mathrm{Im}\, \tau(\gamma(0)) \in 4\pi \mathbb{Z}
\]
and (d) holds.

Conversely, we assume (d).
Put $\varphi_{0} = -\log |s_{0}|^2_{h}$ and
define smooth function $g$ on X by $g = \exp(\frac{1}{2}\varphi_{0}+\frac{\sqrt{-1}}{2}\int d^{c}\varphi_{0})$.
Using H\"ormander-Wermer Lemma, we extend $g$ to a function $\tilde{g} \in C^{\infty}(U)$ with $\overline{\partial}\tilde{g} =0$ to order $m \in \mathbb{N}$ on $X$.
Put $\tilde{g} = e^{G}$ locally.
On $TX$, we have
\[
d (2\mathrm{Re}\,G - \varphi_{0}) = d(\varphi_{0} - \varphi_{0}) = 0,
\]
\[
d^c (2\mathrm{Re}\, G - \varphi_{0}) = 2 d \,\mathrm{Im}\, G - d^c \varphi_{0} = d \int d^c \varphi_{0} -d^c \varphi_{0} = 0.
\]
This shows that $s = \tilde{g}s_{0} \in C^{\infty}(U, L)$ satisfies the condition of $(c)$.
\end{proof}

Assume there exists a non-vanishing holomorphic section $s_{l, 0} \in H^{0}(U, L^l)$.
Put $\varphi_{0} = -\frac{1}{l}\log |s_{l, 0}|^2_{h}$.
For any $\varepsilon > 0$ and any $q \in \mathbb{N}$, there exists $\varphi_{0}' \in C^{\infty}(U)$ and $l' \in \mathbb{N}$
such that $|\varphi_{0}-\varphi_{0}'|_{C^{q}} < \varepsilon$ on $U$, $\varphi_{0} = \varphi_{0}'$ on $X$, $\iota^{*}dd^{c}\varphi'_{0} = 0$ and that $l'\int_{\gamma} d^c \varphi_{0}' \in 4\pi \mathbb{Z}$ for any $\gamma \in H_1(X, \mathbb{Z})$ (see the proof of Lemma~3.2 of \cite{Duv-Sib}).
Hence we can approximate $h$ by a Hermitian metric $h'$ such that the Chern connection associated to $(ll'L, ll'h')$ induces the connection which satisfies the Bohr-Sommerfeld condition on $X$.

\section{Reduction to the real-analytic case}\label{section:4}
Let $M$ be a complex manifold of dimension $n$.
Let $X \subset M$ be a Lagrangian submanifold of $(M, \omega)$ such that $(X, \nabla^{X})$ satisfies the Bohr-Sommerfeld condition.
Let $U \subset M$ be a sufficiently small Stein tubular neighborhood of $X$.
We take a non-vanishing section $s_{0} \in H^{0}(U, L)$ and put $\varphi_{0} = -\log |s_{0}|^2_{h}$.
Take $m \in \mathbb{N}$ to be sufficiently large
and take $s \in C^{\infty}(U, L)$ which satisfies the condition of (c) in Proposition~\ref{proposition:1}, that is, one-jet of $\log |s|_{h}^2$ vanishes on $X$ and $\nabla'' s = 0$ to order $m$ on X.
We write $s = \xi s_{0}$ for $\xi \in C^{\infty}(U)$.
Put $\varphi = -\log |s|_{h}^2$.
Let $f_{k} \in H^{0}_{(2)}(M, L^k)$ which satisfies $\|f_{k}\|_{h^k} = 1$ and let $u_{k} \in C^{\infty}(U)$ such that $f_{k} = u_{k}s^k$ on $U$.
By Whitney's theorem (\cite{Whi2}), there exists a real-analytic manifold $Y$ which is diffeomorphic to  $X$.
Let $N$ be a Stein complexification of $Y$ (see \cite{Gra}).
By shrinking $N$ and $U$ if necessary,
there exists a diffeomorphism $\Phi: N \to U$ such that $\Phi(Y) = X$ and that the $m$-jet of $\overline{\partial}\Phi$ vanishes on $Y$ (cf.~Proposition~5.55 of \cite{Cie-Eli}).
Put $\frac{1}{4\pi}dd^c (\varphi \circ \Phi) = \omega'$ and $\omega'_{n} = (\omega')^n/n!$.
We have $\omega' = \frac{1}{4\pi}\Phi^{*} dd^c \varphi = \Phi^{*}\omega$ on $Y$
and $Y$ is a Lagrangian submanifolds of $(N, \omega')$.
We note that $\varphi \circ \Phi$ is a strictly plurisubharmonic function near $Y$.
Let $d_{Y}(z)$ be the distance from $z \in N$ to $Y$ with respect to the Riemannian metric induced by $\omega'$.
Put $W_{k} = \{z \in N\, |\, d_{Y}(z) < \frac{2\log k}{\sqrt{k}}\}$.
\begin{lemma}\label{lemma:4}
\[
\int_{W_{k}} |\overline{\partial}(u_{k}\circ\Phi)|^2_{\omega'} e^{-k\varphi \circ \Phi}\omega'_{n} = O(k^{-m+4}).
\]
\end{lemma}
\begin{proof}
Assume that $k \in \mathbb{N}$ is sufficiently large number.
For simplicity we abuse notation and write $|\cdot|$ for the norm of a form.
Because $\overline{\partial}(u_{k}\xi^k) = 0$, we have $\overline{\partial}u_{k} = -ku_{k}\xi^{-1}\overline{\partial} \xi$.
Then
\[
|\overline{\partial} (u_{k} \circ \Phi)|^2
\lesssim | \partial u_{k}|^2 |\overline{\partial}\Phi|^2 + |\overline{\partial} u_{k}|^2 |\overline{\partial} \overline{\Phi}|^2
\lesssim  | \partial u_{k}|^2 |\overline{\partial}\Phi|^2 + k^2 |u_{k}\overline{\partial}\xi|^2 |\partial \Phi|^2.
\]
It follows that
\[
k^2 \int_{W_{k}} |u_{k} \overline{\partial}\xi|^2 |\partial \Phi|^2 e^{-k\varphi \circ \Phi} \omega'_{n}
\lesssim
k^2 \int_{\Phi(W_{k})} |u_{k}|^2 |\overline{\partial}\xi|^2 e^{-k\varphi} \omega_{n}
=
O(\frac{ (\log k)^{2m}}{k^{m-2}}).
\]
Let $\chi \in C^{\infty}(\mathbb{R})$ be a function such that $0 \leq \chi \leq 1$, $\chi = 1$ on $(-\infty, 1/2]$ and that $\chi = 0$ on $[1, +\infty)$.
Put $\tilde{\chi}_{k} = \chi (\frac{\sqrt{k}d_{Y}}{4\log k})$.
Then
\[
\int_{W_{k}} |\partial u_{k}|^2 |\overline{\partial} \Phi|^2 e^{-k\varphi \circ \Phi} \omega'_{n} \leq \int \tilde{\chi}^2_{k} |\partial u_{k}|^2 |\overline{\partial} \Phi|^2 e^{-k\varphi \circ \Phi} \omega'_{n}.
\]
Denote by $I_{k}$ the right hand side of the above inequality.
We have
\begin{align*}
I_{k} & \lesssim \int \tilde{\chi}^2_{k} |\overline{\partial} \Phi|^2 e^{-k \varphi} \partial u_{k} \wedge \overline{\partial u_{k}} \wedge \omega^{n-1} \\
& \lesssim \int \tilde{\chi}^2_{k} |\overline{\partial} \Phi|^2 e^{-k \varphi} |u_{k}| |\partial \overline{\partial} u_{k}| \omega_{n}
+ k  \int  \tilde{\chi}^2_{k} |\overline{\partial} \Phi|^2 e^{-k \varphi} |u_{k}| |\partial u_{k}| \omega_{n}   \\
& \qquad
+ \int |\partial \left((\tilde{\chi}^2_{k} |\overline{\partial} \Phi|^2) \circ \Phi^{-1} \right)| e^{-k \varphi}|u_{k}| |\partial u_{k}| \omega_{n} \\
& \lesssim
k \left( \int \tilde{\chi}^2_{k} |\overline{\partial} \Phi|^2 e^{-k \varphi} |{u}_{k}|^2 \omega_{n}
+  \int \tilde{\chi}^2_{k} |\overline{\partial} \Phi|^2 e^{-k \varphi} |u_{k}| |\partial u_{k}| \omega_{n} \right.\\
& \qquad
+\left. \int \tilde{\chi}_{k} |\overline{\partial} \Phi|^2 e^{-k \varphi} |u_{k}| |\partial u_{k}| \omega_{n}\right)
+ \int \tilde{\chi}^2_{k} |\overline{\partial} \Phi| |D \overline{\partial} \Phi|e^{-k \varphi} |u_{k}| |\partial u_{k}| \omega_{n}
\end{align*}
where $D$ is a differential operator of order one.
We have
\[
k  \int \tilde{\chi}^2_{k} |\overline{\partial} \Phi|^2 e^{-k \varphi} |{u}_{k}|^2 \omega_{n}
= O(\frac{(\log k)^{2m}}{k^{m-1}}).
\]
Put $W'_{k} = \{w \in N\, |\, d_{Y}(w) < \frac{4\log k}{\sqrt{k}}\}$.
The Cauchy-Schwarz inequality implies
\begin{align*}
& k\int \tilde{\chi}^2_{k} |\overline{\partial} \Phi|^2 e^{-k \varphi} |u_{k}| |\partial u_{k}| \omega_{n}
\leq k \int \tilde{\chi}_{k} |\overline{\partial} \Phi|^2 e^{-k \varphi} |u_{k}| |\partial u_{k}| \omega_{n}  \\
\leq &
k \left( \int_{\Phi(W'_{k})}|\overline{\partial} \Phi|^2 |u_{k}|^2 e^{-k \varphi} \omega_{n} \right)^{1/2} \left(\int \tilde{\chi}^2_{k} |\partial u_{k}|^2 |\overline{\partial} \Phi|^2 e^{-k \varphi} \omega_{n} \right)^{1/2}
\lesssim
\frac{(\log k)^{m}}{k^{m/2-1}} I_{k}^{1/2}
\end{align*}
and
\begin{align*}
\int \tilde{\chi}^2_{k} |\overline{\partial} \Phi| |D \overline{\partial} \Phi| |u_{k}| |\partial u_{k}|e^{-k \varphi}\omega_{n}
& \leq
\left(\int \tilde{\chi}^2_{k} |D\overline{\partial} \Phi|^2 |u_{k}|^2 e^{-k\varphi}\omega_{n} \right)^{1/2}
\left(\int \tilde{\chi}^2_{k} |\overline{\partial} \Phi|^2 |\partial u_{k}|^2 e^{-k\varphi}\omega_{n} \right)^{1/2} \\
& \lesssim
\frac{(\log k)^{m-1}}{k^{m/2-1/2}} I_{k}^{1/2}.
\end{align*}
Hence there exists $c > 0$ which does not depend on $k$ and
\[
I_{k} \leq c \left( \frac{1}{k^{m-2}} + \frac{1}{k^{m/2-2}} I_{k}^{1/2} \right)
\leq
c \left(\frac{1}{k^{m-2}} + \frac{2c}{k^{m-4}} + \frac{1}{2c}I_{k} \right).
\]
We have $I_{k} = O(k^{-m+4})$ and we complete the proof.
\end{proof}

For sufficiently large $k \in \mathbb{N}$, $W_{k}$ is a Stein manifold.
By regarding $u_{k} \circ \Phi$ as a $\bigwedge^n T^{(1, 0)}N|_{W_{k}}$-valued $(n, 0)$-form,
there exists $v_{k} \in C^{\infty}(W_{k})$ such that $\overline{\partial}v_{k} = \overline{\partial} (u_{k}\circ \Phi)$ and that $\int_{W_{k}} |v_{k}|^2 e^{-k \varphi \circ \Phi}\omega'_{n} = O(k^{-m+4})$ (cf.~\cite{Dem}).
Let $\beta_{k} = u_{k}\circ \Phi -v_{k}$.
Then $\beta_{k}$ is a holomorphic function on $W_{k}$.
If $m > 4$, we have
\begin{equation}\label{equation:5}
\limsup_{k \to +\infty}\int_{W_{k}}|\beta_{k}|^2 e^{-k\varphi \circ \Phi} \omega'_{n} \leq 1.
\end{equation}

Our next task is to estimate $|v_{k}|$ on $Y$.
We first estimate $u_{k}$ near $X$.
\begin{lemma}\label{lemma:5}
Let $r > 0$ be a sufficiently small number.
Let $c > 0$ be a positive number which is larger than the $C^1$-norm of $\varphi_{0}$ on a neighborhood of $X$.
Let $D$ be a differential operator of order one with bounded coefficients.
Then there exists $\delta > 0$ which does not depend on $r$ and the following estimates hold:
\begin{itemize}
\item[(i)]
$\sup_{\{z \in M, d_{X}(z) < \delta r\}} |u_{k}|^2 e^{-k\varphi} \lesssim e^{ckr}/r^{2n}, $

\item[(ii)]
$\sup_{\{z \in M, d_{X}(z) < \delta r\}} |Du_{k}|^2 e^{-k\varphi} \lesssim e^{4ckr}/r^{2+2n} +
k^2e^{ckr}/r^{2n}.  $

\end{itemize}
\end{lemma}
\begin{proof}
We take open sets $\{V_{i}\}_{i=1}^l$, $\{V'_{i}\}_{i=1}^l$ ($V_{i} \subset  \subset V'_{i}$) in $M$ such that $X \subset \bigcup_{i=1}^l V_{i}$ and that there exists a holomorphic coordinate on $V'_{i}$.
Let $B_{i}(p, r) \subset V'_{i}$ be the Euclidean ball of center $p \in V'_{i}$ and radius $r$ in $V'_{i}$.
We may assume that $B_{i}(p, 2r) \subset V'_{i}$ for any $p \in X \cap V_{i}$ by taking $r$ sufficiently small.
Let $p \in X \cap V_{i}$.
Since $u_{k}\xi^k$ is a holomorphic function, the mean value inequality implies
\[
\sup_{B_{i}(p, r)} |u_{k}\xi^k|^2e^{-k\varphi_{0}} \lesssim \frac{e^{ckr}}{r^{2n}} \int_{B_{i}(p, 2r)} |u_{k}\xi^k|^2 e^{-k\varphi_{0}}d\mu_{i, \mathrm{Leb}} \lesssim \frac{e^{ckr}}{r^{2n}}
\]
where $d\mu_{i, \mathrm{Leb}}$ is the Lebesgue measure on $V'_{i}$.
Since
$
\sup_{B_{i}(p, r)}|D(u_{k}\xi^k)| \lesssim \frac{1}{r} \sup_{B_{i}(p, 2r)} |u_{k}\xi^k|,
$
we have
\[
\sup_{B_{i}(p, r)}|D(u_{k} \xi^k)|^2 e^{-k\varphi_{0}} \lesssim  \frac{e^{2ckr}}{r^2} \sup_{B_{i}(p, 2r)}|u_{k}\xi^k|^2 e^{-k\varphi_{0}}.
\]
Hence
\[
\sup_{B_{i}(p, r)} |D u_{k}|^2 e^{-k\varphi} \lesssim \frac{e^{2ckr}}{r^{2}} \sup_{B_{i}(p, 2r)}|u_{k}|^2e^{-k\varphi}  + k^2 \sup_{B_{i}(p, r)}  |u_{k}|^2e^{-k\varphi}
\lesssim \frac{e^{4ckr}}{r^{2+2n}} + \frac{k^2e^{ckr}}{r^{2n}}.
\]
If we take $\delta$ sufficiently small, $\bigcup_{i=1}^l \bigcup_{p \in X \cap V_{i}} B_{i}(p, r)$ contains $\{z \in M\,|\, d_{X}(z) < \delta r \}$ for any sufficiently small $r$.
This completes the proof of (i), (ii).
\end{proof}

\begin{lemma}\label{lemma:6}
We have
\[
|v_{k}|^2 = O(k^{2n-2m}) + O(k^{2n-m+4})
\]
uniformly on $Y$.
\end{lemma}
\begin{proof}
Let $q \in Y$ and $W_q \subset N$ be a sufficiently small neighborhood of $q$ and $(w_{1}, \ldots, w_{n})$ be a holomorphic coordinate on $W_q$.
Let $\delta > 0$ be as in Lemma~\ref{lemma:5}.
Then there exists $\delta' > 0$ which does not depend on $q \in Y$ such that $\Phi (B(q, \delta' r)) \subset \{z \in M\, |\, d_{X}(z) < \delta r\}$ for any $0 < r << 1$.
Here $B(q, \delta'r)$ is the Euclidean ball of center $q$ and radius $\delta' r$ in $W_q$.
We take $r$ to be a positive number which depends only on $k$ such that $B(q, \delta'r) \subset W_{k} = \{z \in N \,|\, d_{Y}(z) < \frac{2\log k}{\sqrt{k}}\}$.
Let $c' > 0$ be a positive number which is larger than the $C^1$-norm of $\varphi \circ \Phi$ on $W_q$.
By Lemma~15.1.8 of \cite{Hor}, we have
\begin{align*}
|v_{k}(q)|^2 &\lesssim (\delta' r)^2 \sup_{B(q, \delta'r)} |\overline{\partial}_{w} v_{k}|^2 + \frac{1}{(\delta'r)^{2n}} \int_{B(q, \delta'r)} |v_{k}|^2 d\mu_{\mathrm{Leb}} \\
& \lesssim r^2 e^{c'\delta'kr} \sup_{B(q, \delta' r)} |\overline{\partial}_{w}v_{k}|^2e^{-k\varphi \circ \Phi} + r^{-2n}e^{c'\delta'kr} k^{-m+4}.
\end{align*}
Let $(z_{1}, \ldots, z_{n})$ be a holomorphic coordinate on a neighborhood of $\Phi (q)$.
Then
\begin{align*}
|\overline{\partial}_{w}v_{k}| = |\overline{\partial}_{w}(u_{k} \circ \Phi)| \leq |\partial_{z} u_{k}| |\overline{\partial}_{w} \Phi| + |\overline{\partial}_{z} u_{k}| |\partial_{w} \Phi|
\lesssim
|\partial_{z}u_{k}||\overline{\partial}_{w} \Phi| + k\big|u_{k}\overline{\partial}_{z}\xi\big|.
\end{align*}
By Lemma~\ref{lemma:5}, we have
\begin{align*}
& |v_{k}(q)|^2 \\
 \lesssim & r^{-2n} e^{c'\delta' kr} ((e^{4ckr} + r^2k^2 e^{ckr} )\sup_{B(q, \delta' r)}|\overline{\partial}_{w} \Phi|^2 + r^2k^2 e^{ckr}\sup_{B(q, \delta' r)}|\overline{\partial}_{z}\xi|^2 ) + r^{-2n}e^{c'\delta'kr}k^{-m+4} \\
\lesssim &
r^{-2n + 2m}e^{c'\delta' kr}  \left(e^{4ckr} + r^2k^2e^{ckr} \right) + r^{-2n} e^{c' \delta' kr} k^{-m+4}.
\end{align*}
If we take $r = k^{-1}$, we have $B(q, \delta'r) \subset W_{k}$ for sufficiently large $k$ and
$|v_{k}(q)|^2 = O(k^{2n-2m}) + O(k^{2n-m+4})$.
Since $Y$ is compact, the above estimates hold uniformly on $Y$.
\end{proof}

\section{Monge-Amp\`ere equation and Demailly's Jensen-Lelong formula}\label{section:5}
We use the same notation as in Section~\ref{section:4}.
Since $N$ is Stein, we may assume that $N$ is a real-analytic submanifold in a higher dimensional Euclidean space.
Then one can approximate $\varphi \circ \Phi$ by analytic functions in the $C^l$-norm for any $l \in \mathbb{N}$ on a neighborhood of $Y$ (\cite{Whi}).
Let $\varepsilon > 0$ be a sufficiently small number.
Let $\varphi_{\varepsilon}$ be a real analytic function on a neighborhood of $Y$ such that
\begin{equation}\label{equation:7}
|\varphi_{\varepsilon}-\varphi\circ\Phi|_{C^2} < \varepsilon.
\end{equation}
Let $ds^2$ be the real-analytic Riemannian metric on $Y$ induced by $\frac{1}{4\pi}dd^c \varphi_{\varepsilon}$.
By the fundamental result of Guillemin and Stenzel~\cite{Gui-Ste},
there exists the real-analytic strictly plurisubharmonic function $\rho$ on a neighborhood of $Y$ such that $0 \leq \rho \leq 1$, $\rho^{-1}(0) = Y$, the K\"ahler form $\frac{1}{4\pi}dd^c \rho$ induces the Riemannian metric $ds^2$ on $Y$ and that
$
(dd^c \sqrt{\rho})^n = 0
$
outside $Y$.
We note that $\rho$ depends on $\varepsilon > 0$.
By Shrinking $N$ if necessary, we may assume that $\rho$ is defined on $N$.
Since $\rho$ is strictly plurisubharmonic and $\sqrt{\rho}$ is a solution of the Monge-Amp\`ere equation, $\partial \overline{\partial}\sqrt{\rho}$ has $(n-1)$ positive eigenvalues and does not have negative eigenvalues on $N \setminus Y$.
Because $\sqrt{\rho}$ attains its minimum at $Y$,
we have that $\sqrt{\rho}$ is plurisubharmonic on $N$ (cf.~Theorem~5.6 of \cite{Lem-Szo}).

Put $\omega_{\rho} = \frac{1}{4\pi}dd^c \rho$.
Let $q \in Y$ and $W_{q} \subset N$ be a small open neighborhood of $q$.
Let $x = (x_{1}, \ldots, x_{n})$ be a smooth coordinate on $W_{q} \cap Y$ and $\{e_{1}, \ldots, e_{n}\}$ be an orthonormal basis of $TY$ on $W_{q} \cap Y$ with respect to the Riemannian metric induced by $\omega_{\rho}$.
Let $J_{N} \in \mathrm{End} (TN)$ be the complex structure of $N$.
Then $(J_{N}e_{1}, \ldots, J_{N}e_{n})$ induces a frame of the normal bundle $TN/TY$ on $W_{q} \cap Y$.
We have $\langle e_{1}, \ldots, e_{n}\rangle^{\bot} = \langle J_{N}e_{1}, \ldots, J_{N}e_{n}\rangle$.
Let $(x, y) = (x_{1}, \ldots, x_{n}, y_{1}, \ldots, y_{n})$ be the coordinate on $W_q$ associated to $(J_{N}e_{1}, \ldots, J_{N}e_{n})$,
that is, $(x_{1}, \ldots, x_{n}, 0, \ldots, 0)$ corresponds to a point in $W_{q} \cap Y$ and
$(x_{1}, \ldots, x_{n}, y_{1}, \ldots, y_{n})$ corresponds to $\exp_{(x, 0)} (y_{1}J_{N}e_{1} + \cdots + y_{n}J_{N}e_{n}) \in N$.
Then
$
\rho (x, y) = 2\pi \sum_{i=1}^n y_{i}^2 + O(|y|^3)
$
on $W_q$.
On the other hand, we have
\[
\varphi \circ \Phi (x, y) = \sum_{i=1}^{n} a_{ij}(x) y_{i}y_{j} + O(|y|^3)
\]
where $a_{ij}(x) = 2\pi \omega' (\partial/\partial_{y_{i}}, J_N \partial/\partial_{y_{j}})(x, 0)$.
Here $\omega' = \frac{1}{4\pi}dd^c \varphi \circ \Phi$.
By (\ref{equation:7}), there exist $c_{1}, c_{2}, c_{3} > 0$ which do not depend on $k$ or $\varepsilon$ such that
\begin{equation}\label{equation:8}
(1-c_{1}\varepsilon) \varphi \circ \Phi \leq \rho \leq (1+c_{1}\varepsilon) \varphi \circ \Phi,
\end{equation}
\begin{equation}\label{equation:9}
(1-c_{2}\varepsilon) \omega'_{n} \leq \omega_{\rho, n} \leq (1+c_{2}\varepsilon) \omega'_{n},
\end{equation}
\begin{equation}\label{equation:10}
(1-c_{3}\varepsilon) \mathrm{Vol}(X, \omega) \leq \mathrm{Vol}(Y, \omega_{\rho}) \leq (1+c_{3}\varepsilon) \mathrm{Vol}(X, \omega)
\end{equation}
on a neighborhood of $Y$.
Here $\omega_{\rho, n} = \omega_{\rho}^n/n!$ and $\mathrm{Vol}(Y, \omega_{\rho})$ is defined by the integral of the Riemannian density induced by $\omega_{\rho}$ on $Y$.
We define $B_{\sqrt{\rho}}(r) = \{z \in N\, |\, \sqrt{\rho}(z) \leq r \}$, $S_{\sqrt{\rho}}(r) = \{z \in N\, |\, \sqrt{\rho}(z) = r \}$.
For sufficiently large $k \in \mathbb{N}$, we have $B_{\sqrt{\rho}}(\frac{\sqrt{2\pi}\log k}{\sqrt{k}}) \subset W_{k} = \{z \in N\,|\, d_{Y}(z) < \frac{2\log k}{\sqrt{k}}\}$.
By Demailly's Jensen-Lelong formula (see \cite{Dem}),
we have
\begin{align*}
\inf_{y \in Y} |\beta_{k}(y)|^2 \int_{S_{\sqrt{\rho}}(r)} d^c \sqrt{\rho} \wedge (dd^c \sqrt{\rho})^{n-1}
& = \inf_{y \in Y} |\beta_{k}(y)|^2 \int_{Y} (dd^c \sqrt{\rho})^n \leq \int_{Y} |\beta_{k}|^2 (dd^c \sqrt{\rho})^n \\
& \leq \int_{S_{\sqrt{\rho}}(r)} |\beta_{k}|^2 d^c \sqrt{\rho} \wedge (dd^c \sqrt{\rho})^{n-1}
\end{align*}
for $0 < r < \frac{\log k}{\sqrt{k}}$.
Here $(dd^c \sqrt{\rho})^n$ is the Monge-Amp\`ere measure whose support is contained in $Y$ (see \cite{Bed-Tay}, \cite{Dem}).
Since $d^c \sqrt{\rho} = \rho^{-1/2}\frac{d^c \rho}{2}$ and
$dd^c \sqrt{\rho} = \rho^{-1/2}\frac{dd^c \rho}{2}-\rho^{-3/2}\frac{d\rho \wedge d^c\rho}{4}$,
we obtain
\[
\inf_{y \in Y} |\beta_{k}(y)|^2 \int_{S_{\sqrt{\rho}}(r)} e^{-kb\rho} \rho^{-1/2}d^{c} \rho \wedge (dd^c \rho)^{n-1} \leq \int_{S_{\sqrt{\rho}}(r)} |\beta_{k}|^2 e^{-k b\rho} \rho^{-1/2}d^c \rho \wedge (dd^c \rho)^{n-1}
\]
for any $b > 0$.
Thus,
\begin{align*}
& \inf_{y \in Y} |\beta_{k}(y)|^2 \int_{B_{\sqrt{\rho}}(\frac{\sqrt{2\pi}\log k}{\sqrt{k}})} e^{-kb\rho} \rho^{-1}d\rho \wedge d^{c} \rho \wedge (dd^c \rho)^{n-1} \\
= & 2\inf_{y \in Y} |\beta_{k}(y)|^2 \int_{0}^{\frac{\sqrt{2\pi}\log k}{\sqrt{k}}} dr \int_{S_{\sqrt{\rho}}(r)} e^{-kb\rho} \rho^{-1/2} d^c \rho \wedge (dd^c \rho)^{n-1} \\
\leq & 2\int_{0}^{\frac{\sqrt{2\pi}\log k}{\sqrt{k}}} dr \int_{S_{\sqrt{\rho}}(r)} |\beta_{k}|^2 e^{-kb\rho}\rho^{-1/2} d^c \rho \wedge (dd^c \rho)^{n-1} \\
= & \int_{B_{\sqrt{\rho}}(\frac{\sqrt{2\pi}\log k}{\sqrt{k}})} |\beta_{k}|^2 e^{-kb\rho} \rho^{-1} d\rho \wedge d^c \rho \wedge (dd^{c}\rho)^{n-1}.
\end{align*}
Since $(dd^c \sqrt{\rho})^n = 0$,
we have
\[
n \rho^{-1}d\rho \wedge d^c \rho \wedge (dd^c \rho)^{n-1} =  2 (dd^c \rho)^n.
\]
Hence we have
\begin{equation}\label{equation:11}
\inf_{y \in Y} |\beta_{k}(y)|^2 \int_{B_{\sqrt{\rho}}(\frac{\sqrt{2\pi}\log k}{\sqrt{k}})} e^{-kb\rho} \omega_{\rho, n}
\leq \int_{B_{\sqrt{\rho}}(\frac{\sqrt{2\pi}\log k}{\sqrt{k}})} |\beta_{k}|^2 e^{-kb\rho} \omega_{\rho, n}.
\end{equation}

\begin{lemma}\label{lemma:7}
\[
\int_{B_{\sqrt{\rho}}(\frac{\sqrt{2\pi}\log k}{\sqrt{k}})} e^{-kb\rho} \omega_{\rho, n} \sim \frac{\mathrm{Vol}(Y, \omega_{\rho})}{(2kb)^{n/2}} \qquad (k \to +\infty).
\]
\end{lemma}
\begin{proof}
Let $\{q_{i}\}_{i = 1}^{l} \subset Y$ and $W_{i} \subset N$ be a small neighborhood of $q_{i}$ such that $Y \subset \bigcup_{i=1}^l W_{i}$ and that there exist
non-negative smooth functions $\lambda_{i} \in C^{\infty}_{0}(W_{i})$ which satisfy $\sum_{i=1}^{l}\lambda_{i} = 1$ on a neighborhood of $Y$.
We take a smooth coordinate $(x, y)$ on $W_i$ as in the first part of this section.
Since $\rho(x, y) = 2\pi \sum_{i=1}^{n} y_{i}^2 + O(|y|^3)$,
there exists $c_{4} > 1$ which does not depend on $k$ and satisfies
\[
\{z \in W_q \, |\, |y(z)| < c_{4}^{-1}\frac{\log k}{\sqrt{k}} \} \subset   B_{\sqrt{\rho}}(\frac{\sqrt{2\pi}\log k}{\sqrt{k}}) \subset \{z \in W_q \, |\, |y(z)| < c_{4}\frac{\log k}{\sqrt{k}}\}
\]
for sufficiently large $k$.
We have
\begin{align*}
& \int_{|y| < c_{4}^{-1} \frac{\log k}{\sqrt{k}}} \lambda_{i}e^{-kb\rho}\omega_{\rho, n}  = \int_{|y| < c_{4}^{-1} \frac{\log k}{\sqrt{k}}} \lambda_{i}(x, y) e^{-2\pi b k|y|^2 + kO(|y|^3)} (1+ O(|y|)) d\mu_{Y}dy \\
= & \frac{1}{k^{n/2}}\int_{|y| < c_{4}^{-1}\log k} \lambda_{i}(x, \frac{y}{\sqrt{k}}) e^{-2\pi b |y|^2} \left(1 + O\left(\frac{(\log k)^3}{\sqrt{k}}\right) \right) d\mu_{Y}dy
\end{align*}
where $d\mu_{Y}$ is the Riemannian density on $Y$ induced by $\omega_{\rho}$.
Since
\[
\int_{c_{4}^{-1} \log k \leq |y| < c_4 \log k}\lambda_{i}(x, 0) e^{-2\pi b |y|^2} d\mu_{Y}dy = O(k^{-\infty}),
\]
it follows that
\begin{align*}
\int_{B_{\sqrt{\rho}}(\frac{\sqrt{2\pi} \log k}{\sqrt{k}})} \lambda_{i} e^{-kb\rho}\omega_{\rho, n}
= &\frac{1}{k^{n/2}} \left(1 + O\left(\frac{(\log k)^3}{\sqrt{k}}\right) \right) \int \lambda_{i}(x, 0) e^{-2\pi b |y|^2} d\mu_{Y} dy + O(k^{-\infty}) \\
= & \frac{1}{(2kb)^{n/2}} \left(1 + O\left(\frac{(\log k)^3}{\sqrt{k}}\right) \right)  \int \lambda_{i}(x, 0) d\mu_{Y} + O(k^{-\infty}).
\end{align*}
Hence we obtain $\int_{B_{\sqrt{\rho}}(\frac{\sqrt{2\pi} \log k}{\sqrt{k}})} e^{-kb\rho}\omega_{\rho, n} = \frac{1}{(2kb)^{n/2}} \left(1 + O\left(\frac{(\log k)^3}{\sqrt{k}}\right) \right) \mathrm{Vol}(Y, \omega_{\rho})$.
\end{proof}
Now put $b = \frac{1}{1-c_{1}\varepsilon}$.
By (\ref{equation:5}), (\ref{equation:8}) and (\ref{equation:9}), we have
\[
\limsup_{k \to +\infty}\int_{B_{\sqrt{\rho}}(\frac{\sqrt{2\pi} \log k}{\sqrt{k}})} |\beta_{k}|^2 e^{-kb \rho} \omega_{\rho, n} \leq 1 + c_{2}\varepsilon.
\]
Then (\ref{equation:10}), (\ref{equation:11}) and Lemma~\ref{lemma:7} imply
\[
\limsup_{k \to \infty} \inf_{y \in Y} |\beta_{k}(y)|^2 \frac{\mathrm{Vol}(X, \omega)}{(2k)^{n/2}} \leq \frac{1 + c_2 \varepsilon}{(1-c_1 \varepsilon)^{n/2}(1-c_3 \varepsilon)}.
\]
By Lemma~\ref{lemma:6}, we have
\begin{align*}
\limsup_{k \to +\infty}\frac{\inf_{x \in X}|f_k(x)|^2_{h^k}\mathrm{Vol}(X, \omega)}{(2k)^{n/2}} \leq &\limsup_{k \to +\infty}\inf_{y \in Y} \left((1+ \varepsilon) |\beta_k(y)|^2 + \left(1 + \frac{1}{\varepsilon} \right)|v_{k}(y)|^2\right) \frac{\mathrm{Vol}(X, \omega)}{(2k)^{n/2}} \\
\leq & \frac{(1 + \varepsilon)(1+c_2 \varepsilon)}{(1-c_1 \varepsilon)^{n/2}(1-c_3\varepsilon)}.
\end{align*}
for sufficiently large $m$.
This completes the proof of Theorem~\ref{theorem:1} since $\varepsilon$ is any positive number.

\section{Estimate from below}\label{section:3}
Let $M$ be a complex manifold of dimension $n$.
Let $X \subset M$ be a Lagrangian submanifold of $(M, \omega)$ such that $(X, \nabla^{X})$ satisfies the Bohr-Sommerfeld condition.
We take $U \subset M$, $m \in \mathbb{N}$, $s_0 \in H^{0}(U, L)$, $s \in C^{\infty}(U, L)$ and $\xi, \varphi_{0}, \varphi \in C^{\infty}(U)$ as in Section~\ref{section:4}.
Let $p \in X$ and $V \subset M$ be a small neighborhood of $p$.
We take a smooth coordinate $x = (x_{1}, \ldots, x_{n})$ on $V \cap X$ and take a local frame $(e_{1}, \ldots, e_{n})$ of $TX$ on $V \cap X$.
We take a local coordinate system $(x, y) = (x_{1}, \ldots, x_{n}, y_{1}, \ldots, y_{n})$ as in the first part of Section~\ref{section:5}.
Since $\omega = -\frac{\sqrt{-1}}{2\pi}\partial \overline{\partial} \log |s_{0}|^2_{h} = \frac{1}{4\pi}dd^c \varphi + O(|y|)$,
we have
$\varphi(x, y) = 2\pi \sum_{i=1}^n y_i^2 + O(|y|^3)$.
Let $d_{X}(z)$ be the distance from $z \in M$ to $X$.
Let $\chi \in C^{\infty}(\mathbb{R})$ be a function such that $0 \leq \chi \leq 1$, $\chi = 1$ on $(-\infty, 1/2]$ and that $\chi = 0$ on $[1, +\infty)$.
Define $\chi_{k}(z) = \chi(\frac{\sqrt{k}d_{X}(z)}{\log k})$ for $z \in M$.
Put $s_{k} = \chi_{k} s^k \in C^{\infty}(U, L^k)$ for $k \in \mathbb{N}$.
\begin{lemma}\label{lemma:1}
\[
\|\nabla'' s_{k}\|^2_{h^k, \omega} = O(k^{2-m}).
\]
\end{lemma}
\begin{proof}
Assume that $k \in \mathbb{N}$ is sufficiently large.
We have
\begin{align*}
|\nabla'' s_{k}|^2_{h^k, \omega} & = |\xi^k \overline{\partial}\chi_{k} +k\chi_{k}\xi^{k-1}\overline{\partial}\xi|^2_{\omega} e^{-k\varphi_{0}} \lesssim |\overline{\partial}\chi_{k}|^2_{\omega}e^{-k\varphi} + k^2|\chi_{k}\overline{\partial}\xi|^2_{\omega}e^{-k\varphi}.
\end{align*}
It follows that
\begin{align*}
& \int_{V} |\overline{\partial}\chi_{k}|^2_{\omega}e^{-k\varphi} \omega_{n} \lesssim \left(\frac{\sqrt{k}}{\log k}\right)^2 \int_{\frac{\log k}{2\sqrt{k}}< |y| < \frac{\log k}{\sqrt{k}}, (x, y) \in V} e^{-2k\pi|y|^2} dxdy \\
\lesssim & \left(\frac{\log k}{\sqrt{k}}\right)^{n-2}e^{-\pi (\log k)^2/2}
= O(k^{-\infty})
\end{align*}
and
\begin{align*}
k^2 \int_{V} |\chi_{k} \overline{\partial}\xi|^2_{\omega}e^{-k\varphi} \omega_{n}
\lesssim k^2 \int_{|y|< \frac{\log k}{\sqrt{k}}, (x, y) \in V} |y|^{2m} e^{-2k\pi|y|^2}dxdy = O(k^{2-m}).
\end{align*}
The last equality holds by the boundedness of $|\sqrt{k}y|^{2m}e^{-\pi|\sqrt{k}y|^2}$.
The lemma holds since $X$ is compact.
\end{proof}

Let $A$ be a finite sequence of points in $M \setminus X$ (possibly empty).
Assume that $A$ consists of $a_1, \ldots, a_N \in M \setminus X$ ($a_i \neq a_j$ if $i \neq j$) and $a_i$ occurs $l_i$ times in $A$.
Let $U_{i} \subset M$ be a small neighborhood of $a_i$.
We may assume that the support of $\chi_{k}$ does not intersect $U_{i}$ for any $i$ and $k$.
Let $0 \leq \kappa_i \leq 1$ be a smooth function on $M$ such that the support of $\kappa_i$ is contained in $U_i$ and $\kappa_i = 1$ on a neighborhood of $a_i$.
Let $z = (z_1, \ldots, z_n)$ be a holomorphic coordinate on $U_i$ such that $a_i$ corresponds to the origin.
Then we put $\tau_{i}(z) = \kappa_i (z) (n+l_i-1)\log |z|^2$.
By taking $U_i$ small, we have $\tau_{i} \leq 0$.
We define a singular Hermitian metric $h'_{k}$ of $L^k$ by $h'_{k} = h^k e^{-\sum_{i=1}^{N} \tau_{i}}$.
The Chern form induced by $h'_{k}$ is positive in the sense of currents if $k$ is sufficiently large.
Lemma~\ref{lemma:1} shows that $\| \nabla'' s_{k}\|^2_{h'_{k}, \omega} = O(k^{2-m})$.

Now we assume that $M$ satisfies one of the three conditions in Theorem~\ref{theorem:2}.
We regard $s_{k}$ as an $L^k \otimes \bigwedge^{n} T^{(1, 0)}M$-valued $(n, 0)$-form.
We show that there exists $t_{k} \in C^{\infty}(M, L^k)$ such that $\nabla'' t_{k} = \nabla'' s_{k}$ and $\|t_k\|^2_{h^k} \leq \|t_{k}\|^2_{h'_k} \lesssim \|\nabla'' s_{k}\|^2_{h'_k, \omega}$ for sufficiently large $k$ (cf.~\cite{Dem2}, ~\cite{Dem}).
If $M$ is a pseudoconvex domain in $\mathbb{C}^n$, $\bigwedge^n T^{(1, 0)}$ is trivial line bundle and there exists no problem.
If $M$ is a projective (resp.~Stein) manifold, we need to deal with the difference of the Chern form of $L^k$ and $L^k \otimes \bigwedge^n T^{(1, 0)} M$.
However the compactness (resp.~ the condition $Ric (\omega) \geq -C\omega$) gives the solution of $\overline{\partial}$-equation with the desired estimate for sufficiently large $k$.
We define a holomorphic section $\alpha_{k} = s_{k} - t_{k}$.
Since $\|\alpha_k\|^2_{h'_{k}, \omega} < + \infty$, we have $\alpha_k \in H^{0}_{(2), A} (M, L^k)$.
\begin{lemma}\label{lemma:2}
If $m > n/2 + 2$, we have
\[
\|\alpha_{k}\|^2_{h^k} \sim \frac{\mathrm{Vol}(X, \omega)}{(2k)^{n/2}} \quad (k \to +\infty).
\]
\end{lemma}
\begin{proof}
For any $\varepsilon > 0$,
it follows that
\[
(1-\varepsilon) \|s_{k}\|^2_{h^k} + \left(1-\frac{1}{\varepsilon}\right) \|t_{k}\|_{h^k}^2 \leq \|\alpha_{k}\|^2_{h^k} \leq (1+\varepsilon) \|s_{k}\|^2_{h^k} + \left(1+\frac{1}{\varepsilon}\right) \|t_{k}\|^2_{h^k}.
\]
Hence it is enough to show $\|s_{k}\|^2_{h^k} \sim \frac{\mathrm{Vol}(X, \omega)}{(2k)^{n/2}}$ ($k \to +\infty$) and we can prove Lemma~\ref{lemma:2} by the same argument as in Lemma~\ref{lemma:7}.
\end{proof}

\begin{lemma}\label{lemma:3}
We have
\[
|t_{k}|^2_{h^k} = O(k^{2+2n-m})
\]
uniformly on $X$.
\end{lemma}

\begin{proof}
Let $k \in \mathbb{N}$ be a sufficiently large number.
Let $p \in X$ and
$(V, (z_{1}, \ldots, z_{n}))$ ($p \in V \subset M$) be a holomorphic local coordinate system.
Let $v_{k} \in C^{\infty}(V)$ such that $t_{k} = v_{k} \xi^k s^k_{0}$.
By Lemma~15.1.8 of \cite{Hor},
we have
\[
|v_{k}\xi^k(p)|^2 \lesssim k^{-2} \sup_{B(p, k^{-1})} |\overline{\partial}_{z}(v_{k}\xi^k)|^2 + k^{2n} \int_{B(p, k^{-1})} |v_{k}\xi^k|^2 d\mu_{\mathrm{Leb}}.
\]
Here $B(p, r)$ is the Euclidean ball of center $p$ and radius $r$, and $d\mu_{\mathrm{Leb}}$ is the Lebesgue measure on $V$.
We have $\overline{\partial}_{z} (v_{k}\xi^k) = \overline{\partial}_{z} (\chi_{k}\xi^k)$.
Let $\eta = \sup_{B(p, k^{-1})} |\varphi_{0}(p)-\varphi_{0}(z)|$.
We have $\eta = O(k^{-1})$.
Then
\begin{align*}
|t_{k}(p)|^2_{h^k} & \lesssim k^{-2} e^{k\eta} \sup_{B(p, k^{-1})} (|\overline{\partial}\chi_{k} \xi^k|^2 + k^2|\chi_{k}\xi^{k-1}\overline{\partial}\xi|^2)e^{-k\varphi_{0}} + k^{2n}e^{k\eta} \int_{B(p, k^{-1})} |t_{k}|^2_{h^k} \omega_n \\
& \lesssim
k^{-2}  \sup_{B(p, k^{-1})} (|\overline{\partial}\chi_{k}|^2 + k^2|\chi_{k}\overline{\partial}\xi|^2)e^{-k\varphi} +  O(k^{2+2n-m}).
\end{align*}
It follows that
\[
k^{-2} \sup_{B(p,k^{-1})} |\overline{\partial} \chi_{k}|^2e^{-k\varphi} \lesssim \frac{1}{k(\log k)^2} \sup_{B(p, k^{-1})} \large|\chi'(\frac{\sqrt{k}d_{X}}{\log k})\large|^2 e^{-k\pi (d_{X})^2}
= O(k^{-\infty}),
\]
\[
|\chi_{k}\overline{\partial}\xi|^2 e^{-k\varphi} \lesssim \chi_{k} (d_{X})^{2m}e^{-k\pi (d_{X})^2} = O(k^{-m}).
\]
Since $X$ is compact, the above estimates do not depend on $p$ and the lemma is proved.
\end{proof}

If we take $m > 2 + 2n$, we have $\lim_{k \to +\infty}|\alpha_{k}|^2_{h^k} = 1$ uniformly on $X$.
Hence we obtain
\[
\liminf_{k \to +\infty} \left(\frac{\mathrm{Vol}(X, \omega)}{(2k)^{n/2}} \sup_{f \in H^0_{(2), A}(M, L^k), f \neq 0}\frac{\inf_{x \in X} |f(x)|^2_{h^k}}{\|f\|^2_{h^k}}\right) \geq 1.
\]
This completes the proof of Theorem~\ref{theorem:3}.

redpajama_set_name: RedPajamaArXiv


Issue At times, system administrators need to retrieve the RSA Authentication Manager license file to rebuild the system, but the license file is misplaced or lost after RSA Authentication Manager installation or perhaps the system administrator cannot download the license from Download Central.
Resolution The RSA Authentication Manager server stores the license file in the file system so administrators can retrieve it from the server, either via SSH or by a KVM that is directly attached to the primary.
In RSA Authentication Manager 8.x, the license file is located in the /tmp/ directory.
drwxr-xr-x  2 root root  4096 Jan 14 10:01 .
drwxrwxrwt 45 root root  4096 Jan 14 15:00 ..
The license files can only be retrieved from the RSA Authentication Manager 8.x primary server.
Please contact RSA Customer Support to generate a replacement license file if you still cannot locate the license file.


Red magic sparkles are 100% edible and ideal for adding extra glamour to your cupcakes. These red magic sparkles are great for decorating Christmas and valentines cupcakes.
Sprinkle on dry icing or stick to icing by sprinkling on confectioners glaze or edible glaze.