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International Journal of Petrochemistry and Research
Adsorption of Chemically Prepared Cocoa Nibs Based Activated Carbon Onto Methylene Blue: Equilibrium and Kinetic Studies
KhairulAdli Nikman1,2, Fisal Ahmad3, MohdSukri Hassan1, Khairunisa Nikman4 and Mohd Azmier Ahmad5,6*
1Faculty of Science and Technology, Universiti Sains Islam Malaysia, 71800 Nilai, Negeri Sembilan, Malaysia
2Jabatan Perubatan Forensik, Hospital Kuala Lumpur, 50586 Kuala Lumpur, Malaysia
3Cocoa Innovative and Technology Centre, Malaysian Cocoa Board, Lot 12621, 71800 Nilai, Negeri Sembilan, Malaysia
4Academy of Language Studies, Universiti Teknologi MARA Pahang, Bandar Tun Abdul Razak, Jengka, Pahang, Malaysia
5School of Chemical Engineering, Engineering Campus, Universiti Sains Malaysia, 14300 NibongTebal, Penang, Malaysia
6Solid Waste Management Cluster, Science & Engineering Research Centre, Engineering Campus, Universiti Sains Malaysia, Nibong Tebal, Penang, Malaysia
*Corresponding author: Mohd Azmier Ahmad, Associate Professor, School of Chemical Engineering, Engineering Campus, Universiti Sains Malaysia, 14300 Nibong Tebal, Penang, Malaysia, Tel: +604 5996459, Fax: +604 5996908, E-mail: chazmier@usm.my
Received: March 23, 2017 Accepted: April 4, 2017 Published: April 10, 2017
Citation: Nikman KA, Ahmad F, Hassan MS, Nikman K, Ahmad MA. Adsorption of Chemically Prepared Cocoa Nibs Based Activated Carbon Onto Methylene Blue: Equilibrium and Kinetic Studies. Int J Petrochem Res. 2017; 1(1): 15-18. doi: 10.18689/ijpr-1000104
Copyright: © 2017 The Author(s). This work is licensed under a Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This study was aimed to prepare cocoa nibs based activated carbon (CNAC) via chemical activation for methylene blue (MB) dye adsorption from aqueous solution. The activation process was performed at 500°C under inert condition using K2CO3 as activation agent. The effect of contact time and initial concentration of adsorbate on the adsorption process were examined. Langmuir isotherm model fitted well the adsorption equilibrium data with monolayer adsorption capacity of 64.98 mg/g at 30°C. The adsorption kinetic was found to follow the pseudo-second-order kinetic model.
Keywords: Activated carbon, Chemical activation, Cocoa nibs, Methylene blue.
Water pollution has become a highlighted issue as some of the industrial effluent is directly discharged into the river and water bodies without proper treatment [1]. Dyes effluent treatments from textile industries are divided into biological, physical and chemical treatment processes. All these processes have different color removal capabilities, capital costs and operating rates [2]. Physical treatment via adsorption process using Agrowaste based activated carbon as adsorbent is among the most efficient and cheap method for dyes removal. Several studies on agrowastes have been employed to remove the contaminants such as methylene blue (MB) dye from the water bodies [3] [4]. Agrowaste is appreciated as economical, sustainable and ecologically friendly materials. In this study, an attempt was made to utilize cocoa nibs waste into activated carbon using chemical activation process. The performance of CNAC was studied in removing MB dye from aqueous solution.
All reagents and chemicals used in the research were analytical grade chemicals. For impregnation process, potassium carbonate (K2CO3) was used. Commercial methylene blue (C16H18ClN3S·3H2O) (MB) textile dye was obtained from Sigma-Aldrich (M) Sdn Bhd
Preparation of Stock and Test Solutions
Stock solution of MB with concentration of 500 mg/L was prepared by dissolving approximately 0.50 g of MB in 1000 mL distilled water. Test solution of MB ranging from 10 mg/L to 100 mg/L were prepared by subsequent dilution from stock solution.
Production of Pellet
The cocoa nibs waste was ground and sifted to a uniform size of 0.5 mm or smaller. The samples were shaped into pellets by compressing the mixture with a hydraulic press, in which the cocoa nibs were compacted into pellets with a specific density of 1.8 g/cm.
Preparation of Activated Carbon
The carbonization was performed by placing approximately 60 g of cocoa nibs pellet in a vertical furnace, which contained in a tubular stainless steel reactor. This step was carried outat 700°C for 1 hour under purified nitrogen (99.99%) with flowrate of 120 mL/min. Then, the furnace was allowed to cold to ambient temperature. Char yield was determined using following equation:
where is the mass of char after carbonization process and wi is the initial mass of cocoa nibs pellet. An amount of dried char was added with K2CO3 at various impregnation ratio (IR) in a 250 ml beaker. Both were mixed with deionized water to dissolve the salt. The IR was calculated as follows:
where wK2CO3 is the dry weight of potassium carbonate pellets and wchar is the dry weight of cocoa nibs char. The mixture was then dehydrated in an oven at 105°C for 24 hours. The activation step was done using similar reactor as in carbonization step but at final temperature of 500°C under nitrogen flowrate of 120 mL/min for 2 hours. The sample was then cooled to room temperature and washed with hot deionized water until the pH of the washed solution reached 6.5-7.
Removal of MB by Batch Adsorption Study
The batch equilibrium experiments of the adsorption capacity studies were conducted at 30 °C in a 250 mL conical flask in a water bath shaker. The stock solution (1000 mg/L) of MB was prepared by dissolving approximately weighed 1.0 gram of the MB in 1000 mL of distilled water in a 1500 mL volumetric flask. The stock solutions were stored in dark place to prevent direct sunlight. The sample solutions were withdrawn at equilibrium to determine the residual concentration. The concentrations of the filtrates were measured using UV-Visible Spectrophotometer (Model Agilent Cary 60, USA).
Effect of Initial Concentration
In order to study the effects of initial adsorbate concentration, 100 mL of adsorbate solutions with known initial concentration (10, 25, 50, 80, 100 mg/L of MB) were prepared in a series of 250 mL Erlenmeyer flasks. The amount of adsorbent that was added into each flask was fixed at 0.1 g. The flasks were placed in an isothermal water bath shaker at temperature and rotation speed of 30°C and 120 rpm, respectively until equilibrium point was reached.
Effect of Contact Time
In order to study the effects of contact time on the adsorption uptake, 100 mL of adsorbate solutions with known initial concentration (10-100 mg/L) were prepared in a series of 250 mL Erlenmeyer flasks. The amount of CNAC that was added into each flask containing MB solution was fixed at 0.1 g. The flasks were then placed in an isothermal water bath shaker of 30 °C with rotation speed of 120 rpm for 360 minutes. The percent removal of adsorbate was calculated as follows:
Where Ce is the concentration of adsorbate at equilibrium and Ci is the initial concentration of adsorbate. The MB uptake at equilibrium is calculated as follows:
where m is the mass of the adsorbent and V is the volume of the adsorbate.
Adsorption Isotherm
Adsorption isotherm study was carried out by fitting the equilibrium data to three isotherm model which are the Langmuir and Freundlich isotherm models, respectively represented by Equations (5),(6) and (7). The applicability and suitability of the isotherm equation to the equilibrium data was compared by judging the values of the correlation coefficients, R2. The Langmuir model is atypical model used to measure the amount of adsorbate on an adsorbent at equilibrium. The relation is expressed by following equation [5]:
where, qe is the amount adsorbed (mg/g), Ce is the equilibrium concentration of the metal ion (mg/L), qe is the maximum amount of adsorbed metal ion per unit mass of sorbent corresponding to complete coverage of the adsorptive sites(mg/g), KL is the Langmuir constant related to the energy of adsorption(L/mg)
The Freundlich adsorption isotherm or Freundlich equations are a relation between the concentrations of a solute on the surface of an adsorbent, to the concentration of the solute in the liquid with which it is in contact. The relationship is stated as follows [6]:
In this equation, qe (mg/g) is amount of adsorbed material in adsorbent surface k in arrangement are adsorption capacity and adsorption intensification.
Kinetic Model
The kinetics of adsorption describes the rate of adsorbate uptake on activated carbon prepared and it controls the equilibrium time. The kinetics of adsorbate uptake is required for selecting optimum operating conditions for the full-scale batch process. Therefore, models for liquid-phase adsorption such as pseudo-first-order and pseudo-second-order were used to analyze the adsorption kinetic data. The pseudo-first-order kinetic model equation of Largergren [7] is generally expressed as:
Where qe is the amount of adsorbate adsorbed at equilibrium (mg/g), qt is the amount of solute adsorb per unit weight of adsorbent at time (mg/g), k1 is the rate constant of pseudo-first order sorption (1/h). The pseudo-second-order equation is expressed as [8]:
Effect of contact time and MB initial concentration on adsorption equilibrium
Figure 2 and Table 1 show the effect of various initial concentrations on adsorption of MB by CNAC. An equilibrium time of 100 min was needed for MB dye solution with initial concentrations of 10-25 mg/l to reach equilibrium. However, for initial concentrations of 50-100 mg/l, longer equilibrium times of 22-24 hours were required for the system to reach equilibrium. Initially, adsorbate molecules have to first encounter the boundary layer effect. Then it has to diffuse from boundary layer film onto adsorbent surface and finally, it has to diffuse into the porous structure of the adsorbent [10]. The ratio of the initial number of dye molecules to the available surface area was low at lower initial concentration compared to higher initial concentration. Therefore, MB solution with higher initial concentration would take relatively longer contact time to attain equilibrium due to the higher amount of MB molecules.
Figure 2: Effects of contact time of MB onto CNAC
Table 1: MB percent removal by CNAC with IR 3:1
Effect of IR on adsorption equilibrium
Fig. 1 shows the influence of IR on CNAC in removing MB. Sample impregnated with 2:1 ratio has the lowest capability to adsorb the adsorbate where the value is 27.8 mg/g.
Figure 1: Effect of IR on MB uptake
The adsorption capacity of CNAC was decreased with increased in IR from 1:1 to 2:1 before increased at IR of 3:1. CNAC with 3:1 ratio managed to adsorb 30.9 mg/g of MB at equilibrium. Higher IR was favorable for enhancing the MB adsorption as more pores are developed on the sample surface [9].
The most appropriate correlation for the equilibrium curve needs to be established in order to understand the adsorption system. Therefore, the equilibrium adsorption data were analyzed using the Langmuir and Freundlich isotherms as shown in Figure 3 whereas Table 2 summarizes all the constants from both isoterm models.
Figure 3: Linear plots of (a) Langmuir and (b) Freundlich adsorption models
Table 2: Parameters of Langmuir and Freundlich adsorption isotherm for MB.
Langmuir model gave the highest R2 values which were greater than 0.96. Conformation of the experimental data into the Langmuir isotherm equation proved that the surface of CNAC for MB adsorption was made up of homogeneous adsorption patches with monolayer coverage of MB onto CNAC [9]. The monolayer saturation capacity of CNAC was found to 64.98 mg/g at 30 °C.
Adsorption Kinetics
All the experimental and calculated qe values obtained from the pseudo-first-order and pseudo-second-order kinetic model for adsorption of MB at 30 °C are tabulated in Table 3. Comparing the R2 values, pseudo-second-order model showed a significant agreement with adsorption mechanisms, which indicated the chemisorption with the heterogeneous active sites occurred on the surface of CNAC [11].
Table 3: Parameter values of the kinetic studies of the adsorption of MB onto CNAC.
CNAC was found to be suitable for the removal of MB from aqueous solution. Adsorption of MB was found to increase with increase in contact time and initial dye concentration. The Langmuir isotherm model was well described by the equilibrium data with maximum adsorption capacity of 64.98 mg/g. The pseudo-second-order kinetic model fits well with the kinetic data.
The authors are grateful for the funding from Solid Waste Management Cluster, USM (1001/CKT/870023) and research grant from Ministry of Science, Technology and Innovation (MOSTI), Malaysia.
Afifah N, Saleh R. Removal of methylene blue dye contaminant by combination of ultrasonic and visible light irradiation using perovskite LaMnO3 nanocatalyst. Materials Science Forum. 2016; 864: 99-105.
Amin N. Removal of direct blue-106 dye from aqueous solution using new activated carbons developed from pomegranate peel: Adsorption equilibrium and kinetics. Haz Mater J. 2009; 165(1-3): 52-62. doi: 10.1016/j.jhazmat.2008.09.067
Ghaedi M, GolestaniNasab A, Khodadoust S, Rajabi M, Azizian S. Application of activated carbon as adsorbents for efficient removal of methylene blue: Kinetics and equilibrium study. Ind Eng Chem J. 2014; 20(4): 2317-2324. doi: 10.1016/j.jiec.2013.10.007
Kumar A, Jena HM. Removal of methylene blue and phenol onto prepared activated carbon from Fox nutshell by chemical activation in batch and fixed-bed column. Cle Prod J. 2016; 137: 1246-1259. doi: 10.1016/j.jclepro.2016.07.177
Langmuir I. The adsorption of gases on plane surfaces of glass, mica and platinum. Journal of the American Chemistry Society. 1918; 40(9): 1361- 1403. doi: 10.1021/ja02242a004
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Langergren S, Svenska BK. Zurtheorie der sogenannten adsorption geloesterstoffe, Veternskapsakad. Handlingar. 1898; 24: 1-39.
Ho YS, McKay G. Sorption of dye from aqueous solution by peat. Chem Engg J. 1998; 70(2): 115-124. doi: 10.1016/S0923-0467(98)00076-1
Ahmad MA, Rahman NK. Equilibrium, kinetics and thermodynamic of Remazol Brilliant Orange 3R dye adsorption on coffee husk-based activated carbon. Chem Engg J. 2011; 170(1): 154-161. doi: 10.1016/j. cej.2011.03.045
Tan IAW, Ahmad AL, Hameed BH. Adsorption isotherms, kinetics, thermodynamics and desorption studies of 2,4,6-trichlorophenol on oil palm empty fruit bunch-based activated carbon. Haz Mater J. 2009; 164(2-3): 473-482. doi: 10.1016/j.jhazmat.2008.08.025
Ahmad F, Daud WMAW, Ahmad MA, Radzi R. Using cocoa (Theobroma cacao) shell-based activated carbon to remove 4-nitrophenol from aqueous solution: Kinetics and equilibrium studies. Chem Engg J. 2011; 178: 461-467. doi: 10.1016/j.cej.2011.10.044


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Everyone knows that China is one of the most important and biodiverse countries on the planet. It is blessed with stunning wildlife, much of it found nowhere else in the world. China has, according to one measure, 7,516 species of vertebrates including 4,936 fish, 1,269 bird, 562 mammal, 403 reptile and 346 amphibian species. In terms of the number of species, China ranks third in the world in mammals, eighth in birds, seventh in reptiles and seventh in amphibians. In each category, China is the most biodiverse country outside of the tropics. Many species are endemic to China, including the country's most famous wildlife species, the Giant Panda. In all, about one-sixth of mammal species and two-thirds of amphibian species in China are endemic to the country.
However, not surprisingly, with rapid economic development and a human population of 1.3 billion, the environment in China is coming under huge pressure and, in addition to the obvious and well-publicised air pollution, China's water and soil are both in a desperate state, not to mention the ongoing destruction of valuable and biodiverse habitats, not least along the Yellow Sea coast where tidal mudflats – so important for millions of long-distance migratory shorebirds – are being lost at an alarming rate. In fact, at least 840 animal species are threatened, vulnerable or in danger of local extinction in China, due mainly to human activity such as habitat destruction, pollution and poaching for food, fur and ingredients for traditional Chinese medicine.
Despite the recent high-level political rhetoric about the importance of "ecological civilisation" and "green development", decision-making, particularly at the local level, is still not effectively taking into account the environmental cost. One reason for this disconnect is the low level of environmental awareness among the general population, in turn caused by an almost complete lack of environmental issues in the Chinese State Curriculum.
That is why education on the environment is so important and it's the main reason why EcoAction has developed an "Environmental Curriculum". The curriculum, focusing on migratory birds, has been piloted in two Beijing schools during the 2015-2016 academic year. Given the pressures on students in China, there was no room to fit in the lessons during normal school time, so these classes have been an optional extra for the participating students. It is testament to the thirst for knowledge of the children involved that they have committed to participate and seen it through to the end.
The curriculum has involved classroom-based lectures, field studies (including birding trips to Miyun Reservoir and Yeyahu Nature Reserve) and lectures by national and international experts, including leading ornithologist Professor Per Alström.
Professor Per Alström provides an introduction to taxonomy – the classification of birds.
Per's Chinese skills have come a long way!
The class is gripped by Per's lecture!
The students have also been encouraged to carry out an "investigation", for example visiting Beijing's wild bird markets to find out who are the buyers and sellers, where the birds come from and what can be done to accelerate their demise.
Students from Beijing's 94th Middle School receiving their certificates.
This week it was time for the participants to receive their certificates for completing the course.
Students from Beijing's 13th Middle School received their certificates this week.
The certificate. Big thanks to BirdLife and ZSL for supporting the programme.
Our hope is that we can expand the pilots to involve more schools in Beijing later this year and, if we can secure the resources, to train teachers to be able to deliver the course in other parts of China. Eventually, our aim is to do ourselves out of a job by having the Chinese government incorporate this course into the State Curriculum!
I'd like to pay tribute to EcoAction's Luo Peng for driving the development and delivery of the course and to BirdLife International and Zoological Society of London for their support. Can't wait for the 2016 course to begin!


One of the hot topics of the twenty teenies that was totally expected is the examination surrounding the ways in which AI or artificial intelligence will affect our lives and most specifically how AI will affect our jobs. For decades now, celebrated authors from Asimov to Zelazny have used robotics and AI as a key thread around which to weave a story.
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Q: Reset Error Messages JSF with Reset Button and Clear Fields I have a registration form with the following data:
First Name
Last Name
Age
Email address
Tel Number
password
re enter password.
Now if the passwords dont match or the age is a text error messages are show, i have a reset button to clear all the fields, but after the error comes up the resent button doesnt clear the messages or the fields.
Here is the code for the Button and whats in my java class:
<h:commandButton value="Reset" type="reset" action="#{javaBeanTest.reset()} />

In the Bean Class:
public void reset()
{
    this.foreName = null;
    this.surname = null;
    this.age = 0;
    this.password = null;
    this.confirmPass = null;
    this.email = null;
}

How would i go about this? The Fields gets set to blank but messages remain

A: Try to use render="@form" in your command button, it will update the form and errors in it. What are you using for showing errormessages?
Try this way:
<h:commandButton value="Reset" action="#{javaBeanTest.reset()} />
    <f:ajax render="@form"/>
</h:commandButton>

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Jo Caulfield
Edfest Bouquets 2019
It's that time again when we award (virtual) bouquets to the best shows we saw at this year's EdFringe. From a plethora of performances over three weeks, here are our highlights. Congratulations to all concerned.
Endless Second – Theo Toksvig-Stewart/Madeleine Gray/Camilla Gurtler/ Cut the Cord
Who Cares? – Jessica Temple/Lizzie Mounter/Luke Grant/ Matt Woodhead/ LUNG & The Lowry
Shine – Olivier Leclair/Tiia-Mari Mäkinen/Hippana Theatre & From Start to Finnish
Ripped – Alex Gwyther/Max Lindsay/Robin Rayner Productions
On The Other Hand, We're Happy – Toyin Omari-Kinch/Charlotte Bate/Charlotte O'Leary/Daf James/Stef O'Driscoll/Paines Plough & Theatr Clwyd
Jo Caulfield: Voodoo Doll – The Stand Comedy Club
Daliso Chaponda: Blah Blah Blacklist – CKP and InterTalent Group
Showstopper! the Improvised Musical – The Showstoppers/Something for the Weekend
Fishbowl – SIT Productions with Le Fils Du Grand Réseau
Beep Boop – Richard Saudek/Crowded Outlet
Chris Dugdale – Down To One – Chris Dugdale Int Ents
Sexy Lamp – Katie Arnstein/Victoria Gagliano
Philip Caveney & Susan Singfield
Posted in Comedy, Theatre and tagged Alex Gwyther, Beep Boop, Camilla Gürtler, Charlotte Bate, Charlotte O'Leary, Chris Dugdale, CKP, Crowded Outlet, Cut the Cord, Daf James, Daliso Chaponda, Endless Second, Fishbowl, From Start to Finnish, Hippana Theatre, InterTalent Group, Jessica Temple, Jo Caulfield, Katie Arnstein, Le Fils DU Grand Réseau, Lizzie Mounter, Luke Grant, LUNG, Madeleine Gray, Matt Woodhead, Max Lindsay, Olivier Leclair, On The Other Hand, Paines Plough, Richard Saudek, Ripped, Robin Rayner Productions, Sexy Lamp, Shine, Showstopper: The Improvised Musical, SIT Productions, Something for the Weekend, Stef O'Driscoll, The Lowry, The Stand Comedy Club, Theatr Clwyd, The Showstoppers, Theo Toksvig-Stewart, Tiia-Mari Mäkinen, Toyin Omari-Kinch, Victoria Gagliano, We're Happy, Who Cares? on August 28, 2019 by Bouquets & Brickbats. Leave a comment
Jo Caulfield: Voodoo Doll
The Stand Comedy Club (Stand 3), Edinburgh
We weren't going to see Jo Caulfield this year. We always enjoy her shows, but we've reviewed her a few times already. We know her schtick; what can we say about her that we haven't said before? But, two-and-a-bit weeks into the Fringe, we find ourselves yearning for some guaranteed laughs, for quality comedy without a message or a solemn life-lesson.
So we're here. Again. And we're delighted with our decision. Because Caulfield is positively crackling with ire, her trademark causticity dialled right up to eleven. Voodoo Doll is loosely based on the idea of Jo trying to purge her anger by writing a list of all the things that annoy her. It's a long list. She's outraged – and outrageous. From her ageing mum to her hapless husband; from couples on holiday to millennial bar-tenders: no one is safe from her scathing putdowns. (Although the joke, usually, is more about her impotent fury than the ostensible target.)
Some of her subjects might seem worn: men vs women; young people nowadays. But every gag lands; every punchline takes us unawares. Clichés stop being clichés when surprises are revealed. The laughs keep coming, one sucker-punch after another, Caulfield never afraid to test the audience's boundaries, always at ease and in total command. The funniest Brexit gag I've ever heard is almost throwaway, delivered with a lightness of touch that stops the show from becoming serious or overtly political.
So do yourself a favour and see Voodoo Doll before the end of its Fringe run. You'll be hard-pressed to find a funnier hour.
Susan Singfield
Posted in Comedy and tagged Edinburgh 2019, Jo Caulfield, The Stand, The Stand Comedy Club on August 21, 2019 by Bouquets & Brickbats. Leave a comment
Pleasance Beyond, Edinburgh
The Pleasance Beyond is packed: the cast's pedigree or the subject matter or a combination of the two mean that the three-hundred-plus seats have all been sold. Which is great, obviously, but also… hot. It's a muggy day, so we thank our lucky stars we remembered to bring bottles of cold water, and hope the play is worth it.
It is. Hurrah! I'm not sure at first: there's an air conditioning unit running, and it's swallowing the sound a bit, so I have to strain to hear, and it's a wordy piece, so it matters; I need to catch the nuances. But I get used to it, and am soon drawn in, enjoying the intrigue and barbed repartee.
We're in the near future – a year or two hence – and Adam Masters (Timothy Bentinck) has just been elected as our new Prime Minister. He's inherited the Brexit stalemate, trying to tread a line between opposing factions in his cabinet, his main aim being to do nothing, to ride out the status quo. Adam's best friend and advisor, Paul Connell (Mike McShane), slyly suggests allocating key roles in the negotiations to arch rivals Simon Cavendish (Hal Cruttenden) and Diana Purdy (Pippa Evans), forcing them to work together, appeasing both the right and left wing commentariat. Chief EU negotiator Helena Brandt (Jo Caulfield) looks on in disbelief as the British government ties itself in knots, kiboshing every idea Adam presents with acerbic ripostes.
Adam's strategy – using his inaction to force others to act – is bound to end in disaster. And as the inevitable betrayal approaches, he becomes increasingly desperate.
Although Brexit is billed as a comedy – and there are plenty of laughs along the way – it's actually quite a serious piece. It's a smart move to cast comedians in the supporting roles – so that Adam is isolated, alone, facing an onslaught of expertly timed quips and snide putdowns. The performances are uniformly strong – Jo Caulfield is a real revelation, and we love her middle-European accent, which is subtle enough to avoid parody.
The staging is simple: a fixed set representing a series of offices, some neat cross-cutting highlighting the cut-throat nature of events. I feel for the actors in their three piece suits and formal dresses (especially Mike McShane, who seems to be wearing clothes he's borrowed from a much larger man – or perhaps they were his, several sizes ago); luckily, the characters are supposed to be stressed and sweaty, so their shiny faces don't seem out of place.
Sadly, the story is just too prescient; I can believe every word of it. It's Shakespearean in its exposure of human frailty and brutality – and sobering in the extreme. Still, it's definitely one to watch. Et tu, Boris?
Posted in Theatre and tagged Brexit, Hal Cruttenden, Jo Caulfield, Mike McShane, Pippa Evans, Pleasance, Pleasance Courtyard, Robert Khan, Timothy Bentinck, Tom Salinsky on August 7, 2018 by Bouquets & Brickbats. Leave a comment
Jo Caulfield: Killing Time
The Stand, Edinburgh
I don't go to see Jo Caulfield in order to be surprised. I've been a fan of her comedy for long enough to know what to expect – and I'm looking forward to another helping of her sly sarcasm. I'm not disappointed.
It's Saturday night and the room is packed; Caulfield's reputation means an audience is guaranteed, and she well deserves it. She makes it look effortless – her stage persona is all shrugs and don't-give-a-fuck – but it would be a mistake to underestimate the skill that makes this show. She's sharp, assessing her audience at the same time as engaging us, pushing boundaries with deceptive innocence.
In Killing Time Caulfield sticks to what she knows. 'There won't be a theme or a message to this show,' she says, 'If you want that, you'll need to go elsewhere. This'll just be me, talking about what I've done, what I've been thinking…' And it is, kind of – but it's so much more as well. It's observational comedy, sure, but a clear illustration of why that genre persists: in the right hands – in her hands – it's funny. She's outraged, regularly, by other people's behaviour, by their rudeness or their lack of awareness, by their sheer stupidity. She maintains a straight face throughout, a wide-eyed insouciance belying the audacity of some of what she says: she's the queen of bitchy put-downs but she keeps us on her side. It's an impressive tight-rope walk.
Okay, so there's quite a lot of men-do-this-and-women-do-that stuff, but she makes it work – it doesn't seem hack. The observations are fresh and precisely delivered, and the audience response is proof they hit their mark.
There's a real joy to be had in watching someone so confident and assured. And Jo Caulfield can be relied upon to deliver a great show.
Posted in Comedy and tagged Edinburgh, Edinburgh 2018, Jo Caulfield, Killing Time, The Stand, The Stand Comedy Club on August 5, 2018 by Bouquets & Brickbats. Leave a comment
Mervyn Stutter's Silver Jubilee Charity Gala
Assembly Theatre, George Square, Edinburgh
If the Edinburgh Fringe has anything resembling royalty then Mervyn Stutter is arguably first in line to the throne (now that Richard Herring seems to have abdicated). Amazingly, Mervyn has been running his Pick of the Fringe show for twenty-five years and this special gala has been arranged to celebrate the occasion and to raise money for charity. Think of an old-school variety show, offering short selections of comedy, theatre and music, and you've pretty much got the idea. As well as helping those in need, it's doubtless a great way of alerting the public to performers they might enjoy in more depth – fliers for all the featured acts are available on the way out.
On comes Mervyn in his trademark pink suit, and treats us to a few of his risqué ditties from down the years, before introducing the first act, Tom Binns, in his role as 'Hospital DJ,' Ivan Brackenbury. It's a great start to the show. Binns' comedy comes from the ineptness of his character, who makes a series of clumsy announcements and then compounds them with selection a toe-curlingly inappropriate songs. It's a great comic premise and Binns delivers on it, big time. You'll find his show at The Assembly George Square.
The next act, I Am Rhythm, features a lively troop of eight South African men who perform a couple of frenetic traditional dances from Soweto, which involves them slapping, clapping and chanting in a dazzling display of syncopation. Fans of this kind of music (and there are many) will find their show at Dance Base (Venue 22).
It's always a pleasure to see comedian Jo Caulfield, even if the short selection she offers here is already familiar to us from earlier shows. No matter, her tales of a louche Londoner adapting to a new life in Edinburgh are cannily observed and very, very funny. You'll find her at (where else?) The Stand Comedy Club.
Circa are difficult to describe. This four piece contemporary circus troop offer a short extract from their current show, Closer, which is all about relationships and combines incredible balancing with a series of bodily contortions that will astonish you. They are currently appearing at Edinburgh's infamous cow-shaped venue, Udderbelly.
Angel, a play by Henry Naylor is set in Kobane, Northern Syria and is the true-life tale of an acclaimed female sniper, credited with over one hundred kills. Lead actor Fillipa Braganca treats us to a short extract and I find myself wishing that she'd been granted more time. Naylor's last play, Echoes, won him the 2015 Fringe First Award and this one is also garnering great reviews. You'll find Angel at the Gilded Balloon, Teviot.
Next up is Sarah Louise Young (who we have already seen and loved in Royal Vauxhall). Here she's in her Cabaret Whore role as La Poule Plombée, a French chanteuse in the vein of Edith Piaf, but played (expertly) for laughs. She sings about baggage (pronounced to rhyme with Farage) and milks the song for every bit of humour. She's accompanied on the piano by Michael Roulston and, just in case you think she's not working hard enough, you can also catch the duo in their show Songs For Lovers (and Other Idiots) at Maggie's Chamber on Cowgate. La Poule Plombé, on the other hand, can be seen at The Voodoo Rooms.
In an extended set, the cast of Showstopper: The Improvised Musical stroll onstage and create songs inspired by suggestions shouted out randomly by members of the audience. They settle on a show set in 10 Downing Street, starring Theresa May and her cleaning lady. It's quite uncanny how they do it, but there they are, producing lyrics that actually make sense and rhyme, working in the style of Stephen Sondheim and Andrew Lloyd Webber. This Olivier award winning show is at The Pleasance Courtyard twice a day. Go along and marvel.
Rory Bremner is a name that surely needs no introduction, but he gets one anyway. He wanders onstage, tells us about his long relationship with Mr Stutter (they've been bumping into each other at the fringe for quarter of a century) and then he throws in a few of his brilliant impersonations. (His Barack Obama is quite something.) His show, Rory Bremner meets… is at The Gilded Balloon at the Museum.
Last but not least, we are treated to a couple of songs by The Jive Aces – purveyors of old time swing. Dressed to the nines in their yellow outfits, they soon have everybody clapping along to their song, before leading the crowd in a chorus of Happy Birthday, as somebody brings out a special cake to mark Mervyn's anniversary. The Jive Aces are at Assembly George Square Gardens – and Mervyn Stutter's Pick of the Fringe is at The Gilded Ballloon, Teviot every day at 13.00, with a fresh set of acts each time.
Posted in Comedy, Theatre and tagged Angel, Assembly George Square, Cabaret Whore, Circa, Closer, Edinburgh 2016, Fillipa Braganca, Henry Naylor, I Am Rhythm, Ivan Brackenbury, Jo Caulfield, La Poule Plombee, Mervyn Stutter, Michael Roulston, Rory Bremner, Sarah Louise Young, Showstopper: The Improvised Musical, The Jive Aces, Tom Binns on August 16, 2016 by Bouquets & Brickbats. Leave a comment
An Evening with Jo Caulfield & Friends
We missed Jo Caulfield at this year's Ed Fringe, so here's a golden opportunity to rectify the situation. It's an unusual gig – she's recording her stand up set for a new CD, Disappointed With You; and also an episode of the podcast gameshow The Good , The Bad and the Unexpected.' The UK's most iconic venue is packed with eager punters, ready for a good laugh, which is pretty much what they get tonight.
After a brief good-natured warm up from another Jo entirely – Jo Jo Sutherland, on comes Ms Caulfield, clearly in a mood to take no prisoners. A luckless Australian visitor, picked out in the intro session, is bluntly told that his country is of 'no interest at all.' 'I've looked into it,' she adds, 'but… no, nothing.' She then proceeds to take on a whole series of scattershot targets, knocking each one down with glee, revelling in the cantankerous middle class persona she habitually adopts. Subjects range from indecisive shoppers in queues, buying a fitted kitchen and the over friendly staff in Marks and Spencers – not the most promising list of subjects, I'll grant you and yet, her world-weary, bitchy demeanour manages to extract the maximum amount of laughter from each subject. Her occasional potty-mouthed utterances are delivered with perfect timing and one childhood memory concerning being disciplined by her Irish Catholic father results in a punchline so shocking, we feel guilty even as we laugh out loud. She finishes by thanking us for laughing 'for the recording' but it was ridiculously easy. She's a funny woman and I can't help but wonder why we don't see more of her on TV.
Next up, Jo introduces a short set by a young newcomer, Tony Sloan, looking like a chunkier Rick Moranis. He expertly plays the persona of the helpless loser, still living with his mother ('It's OK, she has the top bunk!') and looking for romance – ('You know what Tony is, backwards? Y-not?') One or two of his more obvious puns draw audible groans from the crowd, but there's an interesting idea to be developed here and I'm sure we'll be hearing more from him in the future.
After a short interval, on troop the four comedians taking part in the podcast – Keir McAllister, Stuart Murphy, Richard Melvin and Gareth Waugh, together with Jo acting as question master. Getting them all onto the tiny stage is quite an achievement in itself. Like so many of these podcasts, the show stands or falls by the quality of the comedians in any particular episode and it is well served tonight, particularly by Stuart Murphy, who must be one of the most quick-witted comics currently treading the boards. (Interested parties should check out his free Sunday lunchtime improv shows at The Stand, with partner Garry Dobson). He manages to reduce Jo to helpless laughter at several points so perhaps it's little wonder that his team win by a wide margin. There's a whole series of these shows which can be found, free of charge via iTunes.
Then it's all over and we troop off into a rainy Edinburgh night.
Jo Caulfield – 4.6 stars
The Good, the Bad and The Unexpected – 4 stars
Posted in Comedy and tagged Gareth Ward, Jo Caulfield, Keir McAllister, Richard Melvin, Stuart Murphy, The Stand, Tony Sloan on April 13, 2016 by Bouquets & Brickbats. Leave a comment
Jo Caulfield – Cancel My Subscription
The Stand Comedy Club, Edinburgh
In Cancel My Subscription, Jo Caulfield doesn't confound expectations so much as revel in them. She gives exactly what her audience expects: a caustic, bilious and gloriously profane account of the world as she sees it. The humour is largely narrative, and none the worse for it; Caulfield demonstrates with admirable aplomb how it is possible to push the boundaries of taste and decency without ever descending into "look-mum-no-hands-I-just-want-to-shock" territory. But shock she does: acutely timed references to polar opposites Jill Dando and Josef Fritzl leave the audience gasping, but with delight at her chutzpah more than anything else.
I really enjoyed this show. It didn't surprise me, but then I didn't go to be surprised. I went because I am a fan, and because I knew I would laugh myself silly.
And because I was born in the same hospital as her.
Posted in Comedy and tagged Cancel My Subscription, Comedy, Edinburgh 2014, Jo Caulfield, The Stand on August 12, 2014 by Bouquets & Brickbats. 1 Comment

redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction}
\label{sect-intro}

The Hartogs' extension theorem is one of the most distinctive results
in several complex variables. Let $\varOmega\subset \mathbb{C}^n$
{\small($n\geqslant 2$)} be a domain. Let $K\subset\subset \varOmega$ be
a compact subset such that $\varOmega\backslash K$ is connected. Denote
by $\mathscr{O}$ the sheaf of holomorphic functions on $\mathbb{C}^n$.

\begin{Thm}\label{Hart-1} {\em (Hartogs' extension theorem for holomorphic functions)} The restriction map
\begin{align*}
H^0(\varOmega,\mathscr{O})\longrightarrow H^0(\varOmega\backslash K,\mathscr{O})
\end{align*}
is bijective.
\end{Thm}

A proof using no $\overline{\partial}$ techniques
can be found in Merker-Porten's paper \cite{JM-P}.

Next, let $\mathscr{O}^*$ be the sheaf of invertible holomorphic functions
on $\mathbb{C}^n$. Arguing that a nowhere vanishing function $f \in
\varOmega \backslash K$ extends holomorphically to $\varOmega$ as well
as its inverse $g := \frac{1}{f}$, and that $f \, g \equiv 1$
transfers from $\varOmega \backslash K$ to $\varOmega$
by the uniqueness principle, one deduces the

\begin{Cor}\label{Hart-2} {\em (Extension of invertible holomorphic functions)} The restriction map
\begin{align}\label{res-inv}
H^0(\varOmega,\mathscr{O}^*)\longrightarrow H^0(\varOmega\backslash K,\mathscr{O}^*)
\end{align}
is bijective.\qed
\end{Cor}

Beyond functions, it is natural to ask whether
for holomorphic line bundles,
Hartogs' type extension holds from $\varOmega\backslash K$ to $\varOmega$.
If yes, is the extension unique modulo
isomorphism?

Recall that there is a bijection between the set of isomorphic classes
of holomorphic line bundles over $\varOmega$, and the Picard group
$H^1(\varOmega,\mathscr{O}^*)$, constructed in the following way. Any holomorphic
line bundle $\pi \colon L \longrightarrow \varOmega$ admits an open cover
$\{U_i\}$ of $\varOmega$ together with local trivialization maps
$\varphi_i \colon \pi^{-1}(U_i) \overset{\sim}{\longrightarrow} U_i
\times \mathbb{C}$ and transition maps $f_{ij}\in H^0(U_i\cap
U_j,\mathscr{O}^*)$. The data $\{f_{ij}\}$ is a \v{C}ech 1-cocycle
representing some element in $H^1(\{U_i\},\mathscr{O}^*)\hookrightarrow
H^1(\varOmega,\mathscr{O}^*).$ Reciprocally, any element in $H^1(\varOmega,\mathscr{O}^*)$
can be expressed by some \v{C}ech 1-cocycle $\{f_{ij}\}$ with respect
to some open cover $\{U_i\}$ of $\varOmega$ valued in $\mathscr{O}^*$. The data
($\{U_i\}$, $\{f_{ij}\}$) gives a holomorphic line bundle.

Using these notations, we may restate our question more
precisely.

\begin{Quest}\label{Quest-ext-L}
Given a holomorphic line bundle $L$ over $\varOmega\backslash K$, does there exist a holomorphic line bundle $\widetilde{L}$ over $\varOmega$ such that $\widetilde{L}|_{\varOmega\backslash K}\cong L$? Equivalently, is the restriction
map
\begin{align}\label{H-1-O-star-Omega-K}
H^1\big(
\varOmega,\mathscr{O}^*
\big)
\longrightarrow
H^1\big(
\varOmega\backslash K,\mathscr{O}^*
\big)
\end{align}
surjective? If yes, is it bijective?
\end{Quest}

A positive answer, under certain circumstances, was given by Fornaess-Sibony-Wold in \cite{FSW}.

\begin{Thm}\label{FSW} {\em (Extension across strictly pseudoconcave level sets)} Let $\varOmega\subset\mathbb{C}^n$ {\small($n\geqslant 3$)} be a pseudoconvex domain with a $\mathscr{C}^{\infty}$ strictly plurisubharmonic (psh) exhaustion function $\rho$, i.e. for each $a\in\mathbb{R}$, the sublevel set $K_a:=\rho^{-1}(-\infty,a]$ is compact in $\varOmega$. Then every holomorphic line bundle over $\varOmega\backslash K_a$ extends to $\varOmega$. The extension is unique modulo isomorphism.
\end{Thm}

Actually they proved a stronger version of this theorem, namely existence (resp. uniqueness) of an extension when the Levi form of $\rho$ has at least 3 (resp. 2) positive eigenvalues.

The proof of Theorem~\ref{FSW} uses (1) the exponential sequence and Cartan's theorem B (2) the extension of holomorphic functions across a totally real plane and (3) Andreotti-Grauert theory. We will present the first two ingredients in Section~\ref{sec-back} because we are going to use them later.

Now, let us come back to Question~\ref{Quest-ext-L}. For $n = 2$, Ivashkovich already presented in~\cite{Iva} a local counterexample (cex), but with $K \subset \varOmega$ not compact. In Section~\ref{sect-counter-dim-2}, we will briefly restate his construction, and by taking exponential, we will produce a domain $\varOmega \subset \mathbb{C}^2$ and a compact $K \subset\subset \varOmega$ through which some holomorphic line bundles do not extend.

\begin{Prop}
There exists a bounded pseudoconvex domain $\varOmega \subset
\mathbb{C}^2$ equipped with a $\mathscr{C}^\infty$ strictly
psh function
\[
\rho\colon\ \ \
\varOmega
\,\longrightarrow\,
[0,\infty)
\]
such that $K := \rho^{-1}(0) \cong S^1 \times S^1$ is a compact
totally real $2$-torus, and there exists a (nontrivial) holomorphic
line bundle $L$ on $\varOmega \backslash K$ having the property
that there exists {\em no} holomorphic line bundle
$\widetilde{L}$ on $\varOmega$ with $\widetilde{L}
\big\vert_{\varOmega \backslash K}\cong L$.
\end{Prop}

However, a similar construction in dimension $n \geqslant 3$, again
with a compact $K = \rho^{-1}(0) \cong (S^1)^n$ of the
same kind, would fall under
the {\em positive} (known) extension Theorem~\ref{FSW}.

Hence to really produce a cex to Hartogs' type extension
for holomorphic line bundles in {\em all} dimensions $n \geqslant
2$, the compact $K\subset\subset \varOmega$ should not be
of the shape $\{\rho\leqslant a\}$, i.e. a sublevel set of a
strictly psh exhaustion function.

In Section~\ref{sect-counter-dim-3}, we will perform
an alternative construction. In $\mathbb{C}^n$
{\small ($n \geqslant 2$)},
for $0 < \epsilon < n$, we introduce the domain:
\[
G_\epsilon
\,:=\,
\big\{
z\in\mathbb{C}^n
\colon\,
\sum_{j=1}^n\,
\big(\log\,\vert z_j\vert\big)^2
<
\epsilon
\big\},
\]
which contains the $n$-dimensional standard totally real torus:
\[
\mathbb{T}^n
\,=\,
\big\{
\vert z_1\vert
=\cdots=
\vert z_n\vert=1
\big\}
\,\cong\,
(S^1)^n.
\]
For $0 < \epsilon \ll n$ small, $G_\epsilon$ will appear to be
a thin Grauert tube around $\mathbb{T}^n$.
We will check that the domain $G_\epsilon$ is relatively
compact in the ball:
\[
\varOmega
\,:=\,
B
\big(
2\sqrt{n}\,e^{\sqrt{\epsilon}}
\big)
\]
centered at the origin and of radius $2\sqrt{n}\, e^{ \sqrt{\epsilon}}
$. Also, we will take a small open ball $U_p \subset
\mathbb{C}^n$ centered at the point:
\[
p
\,:=\,
\big(
e^{\sqrt{\epsilon/n}},
\dots,
e^{\sqrt{\epsilon/n}}
\big)
\,\in\,
\partial
G_\epsilon,
\]
as connecting the interior and the exterior of $\partial G_\varepsilon$ through a small hole at $p$. Our main result is the

\begin{Thm}
\label{main-counterexample}
With the compact:
\[
K
\,:=\,
\partial G_\epsilon
\big\backslash
U_p,
\]
the open set $\varOmega \backslash K$ is connected, and there
exists a (nontrivial) holomorphic
line bundle $L_{\text{cex}}$ on $\varOmega \backslash K$ having the property
that there exists {\em no} holomorphic line bundle
$\widetilde{L}$ on $\varOmega$ with $\widetilde{L}
\big\vert_{\varOmega \backslash K}\cong L_{\text{cex}}$. Here `cex' stands for `counterexample'
\end{Thm}

The way we construct this non-extendable $L_{\text{cex}}$ is by using the following gluing lemma.
\begin{Lem}\label{gluing-lemma}
Let $U,V\subset \mathbb{C}^n$ be two open subsets, $L_U,L_V$ be two holomorphic line bundles defined over $U$ and $V$ respectively. If $L_U|_{U\cap V}\cong L_V|_{V\cap U}$ are isomorphic as holomorphic line bundles, then there exists a holomorphic line bundle $L$ defined over $U\cup V$ such that $L|_U\cong L_U,L|_V\cong L_V$.
\end{Lem}

A more general version of this gluing lemma, for holomorphic vector bundles, is stated and proved in subsection \ref{subsection-gluing}.

Note that in this lemma, we assume no geometrical condition on $U,V$ and no triviality of $L_U,L_V$. The only condition is that $L_U|_{U\cap V}\cong L_V|_{V\cap U}$. In particular, when $H^1(U\cap V,\mathscr{O}^*)=0$, e.g. when $U\cap V$ is convex, this condition is always satisfied.

The picard group $H^1(G_\epsilon,\mathscr{O}^*)\cong \mathbb{Z}^{\binom{n}{2}}$ is nontrivial, which will be proved in Proposition \ref{prop:G}. So we can take a nontrivial holomorphic line bundle $L_{\text{nt}}$ over $G_\epsilon$. As a consequence of Proposition \ref{real-hessian} we show there exists a small ball $U_p$ centered at $p\in\partial G_\epsilon$ such that $U_p\cap G_\epsilon$ is convex. So in the gluing lemma, if we regard $U$ as $G_\epsilon$ and $V$ as $(\varOmega\backslash\bar{U})\cup U_p$, then $U\cap V=U_p\cap G_\epsilon$ is convex. Thus we can glue $L_{\text{nt}}$ with a trivial line bundle $L_{\text{triv}}$ over $V$ to obtain a line bundle $L_{\text{cex}}$ over $U\cup V=\varOmega\backslash(\partial G_\epsilon\backslash U_p)=\varOmega\backslash K$, which is connected by Proposition \ref{prop-omega}. Such $L_{\text{cex}}$ is nontrivial since $L|_U\cong L_{\text{nt}}$ is. It cannot be extended to $\varOmega$ since $H^1(\varOmega,\mathscr{O}^*)=0$.
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In dimension $n\geqslant 3$, the gluing lemma provides a way to extend holomorphic line bundles different from the method in Theorem \ref{FSW}.

In \cite{FSW}, the strongly psh exhaustion function is modified to become a nice Morse exhaustion function, also denoted by $\rho$. For any $a\in\mathbb{R}$ and any holomorphic line bundle $L^a$, defined over the super level set $\varOmega^a:=\{\rho>a\}$, they proved that for any point $p$ in the level set $\Gamma^a:=\{\rho=a\}$, there exists a small neighborhood $U_p\subset\varOmega$ of $p$ such that $L^a|_{U_p\cap\varOmega^a}$ is trivial and $L^a$ can be extended trivially to $\varOmega^a\cup U_p$, no matter $p$ is a critical point of the Morse function $\rho$ or not. Since the level set $\Gamma^a$ is compact, after finitely many steps, $L^a$ extends as $L^b$ over $\varOmega^b$ with some $b<a$.

Keep extending $L^a$ until a local minimum $q$ of $\rho$ is reached. The minimum $q$ is an isolated point. There exists some small punctured ball $B_q^*$ centered at $q$ such that $H^1(B_q^*,\mathscr{O}^*)=0$ when $n\geqslant 3$, by a special case of Andreotti-Grauert theory, Proposition 12 in \cite{AG}, which is also proved in \cite{Scheja}. Thus one can extend any holomorphic line bundle trivially across any such local minimum. This proves Theorem \ref{FSW}.

The crucial point above is the following uniqueness result, which a consequence of \cite{FSW}. Let us call it `{\sl downward uniqueness}', since the isomorphism passes to a lower super level set.

\begin{Prop} {\em (Downward uniqueness)} If $L,L'$ are two holomorphic line bundles defined over $\varOmega^b$ that are isomorphic over $\varOmega^a\neq\emptyset$ with $b<a$, then they are isomorphic over $\varOmega^b$.
\end{Prop}

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\end{center}

However, by the gluing Lemma \ref{gluing-lemma}, we can lose uniqueness when we extend through compact sets having shapes different from $K_a$. In our cex constructed in Section \ref{sect-counter-dim-3}, the ball $\varOmega=B(2\sqrt{n}\, e^{\sqrt{\epsilon}})$ admits a strongly psh exhaustion function $\rho(z)=-\log(d(z,\partial\varOmega))$. Since $K$ is compact in $\varOmega$, there exists some $a\in \mathbb{R}$ such that $\varOmega^a=\{\rho>a\}\subset \varOmega\backslash K$. We could restrict the non-extendable holomorphic line bundle $L_{\text{cex}}$, mentioned above, to $\varOmega^a$, and extend $L_{\text{cex}}|_{\varOmega^a}$ to $\varOmega$ by Theorem \ref{FSW}. But in this way we will get a trivial line bundle, which does not agree with the initial bundle $L_{\text{cex}}$ over $\varOmega\backslash K$. In other words, we have the following commutative diagram of restriction maps that are group homomorphisms
\begin{diagram}
H^1(\varOmega,\mathscr{O}^*) & \rTo^{\res^4} & H^1(\varOmega\backslash K,\mathscr{O}^*)\\
& \rdTo^{\res^6} & \dTo^{\res^5}\\
& & H^1(\varOmega^a,\mathscr{O}^*).
\end{diagram}
By Theorem \ref{FSW}, the map $\res^6$ is bijective. But $\res^5$ is not injective since the nontrivial line bundle $L_{\text{cex}}$ and a trivial one $L_{\text{triv}}$ over $\varOmega\backslash K$ have the same restriction on $\varOmega^a$. Consequently $\res^4$ is {\em not} surjective.
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\end{center}

In conclusion, the map~{\thetag{\ref{H-1-O-star-Omega-K}}}
is {\em not} always surjective, in any dimension $n \geqslant 2$.

\medskip\noindent{\bf Acknowledgments.} The author adresses sincere thanks to Jo\"el Merker for driving him to this problem and for useful discussions. The author also thanks an anonymous referee for pointing out minor mistakes in the first version.

\section{Background}
\label{sec-back}

Now we present the ingredients (1) and (2) mentioned in the Introduction.

\begin{Thm} {\bf[Cartan's theorem B]} Let $X$ be a Stein manifold, $\mathscr{F}$ be a coherent analytic sheaf on $X$. Then
\[
H^r(X,\mathscr{F})=0
\quad{\scriptstyle (r\geqslant 1)}.
\]
\end{Thm}

A proof can be found in Cartan's original paper \cite{Cartan}. Recall the exponential sequence
\[0\longrightarrow\mathbb{Z}\xrightarrow{\times 2\pi i}\mathscr{O}\xrightarrow{\exp}\mathscr{O}^*\longrightarrow 0\] of sheaves over $X$ induces an exact sequence of cohomologies
\al{H^1(X,\mathscr{O})\longrightarrow H^1(X,\mathscr{O}^*)\longrightarrow H^2(X,\mathbb{Z}).}

When $X$ is stein, by Cartan's theorem B we know $H^1(X,\mathscr{O})=0$. Moreover, if $H^2(X,\mathbb{Z})=0$, for example when $X$ is contractible, then we get $H^1(X,\mathscr{O}^*)=0$. So we have the following criterion:

\begin{Cor}\label{crit-1} Every holomorphic line bundle over a Stein contractible manifold is trivial.
\end{Cor}

In particular, every convex domain in $\mathbb{C}^n$ {\small($n\geqslant 1$)} is Stein and contractible.

\begin{Cor}\label{crit-2} Every holomorphic line bundle over a convex domain in $\mathbb{C}^n$ {\small($n\geqslant 1$)} is trivial.
\end{Cor}

The next ingredient is the extension of holomorphic functions across a totally real plane.

\begin{Thm} Let $D\subset\mathbb{C}^n$ {\small($n\geqslant 2$)} be a domain, $K\subset \mathbb{C}^n$ be a totally real plane. Then the restriction map
\al{H^0(D,\mathscr{O})\longrightarrow H^0(D\backslash K,\mathscr{O})}
is bijective.
\end{Thm}

A proof can be found in the first Chapter of Siu's book \cite{Siu}. We can also apply the argument in the proof of Corollary \ref{Hart-2}.

\begin{Cor}\label{Hart-4} Under the same assumptions, the restriction map
\[
H^0(D,\mathscr{O}^*)\longrightarrow H^0(D\backslash K,\mathscr{O}^*)
\]
is bijective.
\end{Cor}

\section{Compactification of Ivashkovich's Counterexample}
\label{sect-counter-dim-2}

In this section we construct a cex in dimension 2. Let $z_1=x_1+iy_1$, $z_2=x_2+iy_2$ be the standard coordinates of $\mathbb{C}^2\cong\mathbb{R}^4$. For any $r>0$, let $D_r:=\{|x_1|<r,|x_2|<r,y_1^2+y_2^2<1\}$ be a convex bounded domain in $\mathbb{C}^2$. Let $K:=\{y_1=y_2=0\}$ be a totally real plane in $\mathbb{C}^2$.  Since $D_r\backslash K$ contracts to $C:=\{x_1=x_2=0,y_1^2+y_2^2=1/2\}\cong S^1$, we have $H^1(D_r\backslash K,\mathbb{Z})=\mathbb{Z}.$

We can represent a generator of this free $\mathbb{Z}$-module explicitly by using the \v{C}ech cohomology. Take an open cover $\mathscr{U}=\{U_1,U_2\}$ of $D_r\backslash K$ with $U_1:=\{y_2<|y_1|\}\cap D_r$ and $U_2:=\{y_2>-|y_1|\}\cap D_r$. Then $U_1\cap U_2$ has 2 components, $U_{12}^1:=\{y_1<-|y_2|\}\cap D_r$ and $U_{12}^2:=\{y_1>|y_2|\}\cap D_r$. Let $c=\{c_{12}\}\in Z^1(\mathscr{U},\mathbb{Z})$ be a \v{C}ech 1-cocycle defined by $c_{12}|_{U_{12}^1}=0$, $c_{12}|_{U_{12}^2}=1.$ Then $c$ is a nontrivial 1-cocycle representing a generator $[c]$ of the $\mathbb{Z}-$module $H^1(D_r\backslash K,\mathbb{Z})$.

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\end{center}

Recall that the exponential sequence
\al{0\longrightarrow\mathbb{Z}\xrightarrow{\times 2\pi i}\mathscr{O}\xrightarrow{\exp}\mathscr{O}^*\longrightarrow 0}
induces a long exact sequence
\begin{align}\label{long}
H^0(D_r\backslash K,\mathscr{O})\xrightarrow{\alpha} H^0(D_r\backslash K,\mathscr{O}^*)\xrightarrow{\beta} H^1(D_r\backslash K,\mathbb{Z})\xrightarrow{\gamma} H^1(D_r\backslash K,\mathscr{O})\xrightarrow{\delta} H^1(D_r\backslash K,\mathscr{O}^*).
\end{align}

By Corollary \ref{Hart-4}
\[
H^0(D_r\backslash K,\mathscr{O}^*)\cong H^0(D_r,\mathscr{O}^*).
\]
Since $D_r$ is simply connected, the map
\al{H^0(D_r,\mathscr{O})\xrightarrow{\exp}H^0(D_r,\mathscr{O}^*)}
is surjective, thus in the long exact sequence (\ref{long}), the map
\al{\alpha:H^0(D_r\backslash K,\mathscr{O})\longrightarrow H^0(D_r\backslash K,\mathscr{O}^*)}
is also surjective. We have $\im(\alpha)=\ker(\beta)=H^0(D_r\backslash K,\mathscr{O}^*)$ so $0=\im(\beta)=\ker(\gamma)$, i.e. $\gamma$ is injective. We know that the sequence
\al{0\longrightarrow H^1(D_r\backslash K,\mathbb{Z})\xrightarrow{\gamma} H^1(D_r\backslash K,\mathscr{O})\xrightarrow{\delta} H^1(D_r\backslash K,\mathscr{O}^*)}
is exact.

The generator $[c]\in H^1(D_r\backslash K,\mathbb{Z})$ maps to $\gamma([c])$ which is nontrivial since $\gamma$ is injective. Since $\frac{1}{2}\gamma([c])$ is not in the image of $\gamma$ we know $\delta\big(\frac{1}{2} \gamma([c])\big)$ represents a nontrivial holomorphic line bundle over $D_r\backslash K$, which cannot be extended to $D_r$ since $H^1(D_r,\mathscr{O}^*)=0$.

Using \v{C}ech cohomology, the element $\gamma([c])$ can be represented by $\{2\pi i c_{12}\}\in Z^1(\mathscr{U},\mathscr{O})$ and $\delta\big(\frac{1}{2}\gamma([c])\big)$ can be represented by $\{e^{\pi ic_{12}}\}\in Z^1(\mathscr{U},\mathscr{O}^*)$. Denote this 1-cocycle by $f=\{f_{12}\}$. We have $f_{12}|_{U_{12}^1}=1$, $f_{12}|_{U_{12}^2}=-1$. Let $L$ be a holomorphic line bundle defined on $D_r\backslash K$, trivial on $U_1$ and $U_2$ and the transition function is defined by $\{f_{12}\}$. Then $L$ is a nontrivial holomorphic line bundle. $L$ cannot be extended to $D_r$, since by Corollary \ref{crit-2} every holomorphic line bundle over $D_r$ is trivial.

\medskip

Now we will construct the following objects:
\begin{itemize}
\item a bounded pseudoconvex domain $\varOmega\subset\mathbb{C}^2$ with a strongly $\mathscr{C}^\infty$ psh exhaustion function $\rho$;
\item some $a\in\mathbb{R}$ and some compact $K_a:=\rho^{-1}(-\infty,a]\subset\subset \varOmega$;
\item a holomorphic line bundle $L$ over $\varOmega\backslash K_a$ which can not be extended to $\varOmega$.
\end{itemize}

Recall $D_r=\{|x_1|,|x_2|<r,y_1^2+y_2^2<1\}$. Consider the map
\al{\varphi=\exp(i\cdot):\mathbb{C}^2&\longrightarrow \mathbb{C}^2,\\
(z_1,z_2)&\longmapsto (w_1,w_2):=(e^{iz_1},e^{iz_2}).}
This map $\varphi$ is locally biholomorphic. It is bijective, hence biholomorphic from $D_r$ onto $\varphi(D_r)$, when $r\leqslant\pi$. When $r>\pi$, the image is $\{(\log|w_1|)^2+(\log|w_2|)^2<1\}$. We define $\varOmega$ as this open set. Actually $\varOmega$ is a Grauert tube around the totally real torus $\{|w_1|=|w_2|=1\}$. It is a bounded pseudoconvex domain, as a special case of Proposition \ref{prop:G} with $n=2$ and $\epsilon=1$. The function
\al{\rho:\varOmega&\longrightarrow[0,+\infty)\\
(w_1,w_2)&\longmapsto-\log\big(1-(\log|w_1|)^2-(\log|w_2|)^2\big)}
is a strongly psh exhaustion function of $\varOmega$ and $\varphi(K\cap D_r)=\{|w_1|=|w_2|=1\}=\rho^{-1}(0)=K_0\subset\subset \varOmega$.

Recall the covering $\mathscr{U}$ of $D_r\backslash K$ and the \v{C}ech 1-cocycle $\{f_{12}\}$ above. Notice that $f_{12}$ is constant along the $(x_1,x_2)$-directions. In particular,
\al{f_{12}(x_1+2k_1\pi,x_2+2k_2\pi,y_1,y_2)=f_{12}(x_1,x_2,y_1,y_2)}
whenever $k_1$, $k_2\in\mathbb{Z}$, $(x_1,x_2,y_1,y_2)\in D_r$ and $(x_1+2k_1\pi,x_2+2k_2\pi,y_1,y_2)\in D_r$. So $f_{12}$ induces a function $\tilde{f}_{12}$ well defined on the disjoint union $\varphi(U_{12})=\varphi(U_{12}^1)\sqcup\varphi(U_{12}^2)$ with
\al{\tilde{f}_{12}|_{\varphi(U_{12}^1)}&=1,\\
\tilde{f}_{12}|_{\varphi(U_{12}^2)}&=-1.}
Here $\varphi(\mathscr{U}):=\{\varphi(U_1),\varphi(U_2)\}$ is an open cover of $\varOmega\backslash K_0$. This open cover, together with the transition function $\tilde{f}_{12}$, defines a nontrivial holomorphic line bundle $L$ over $\varOmega\backslash K_0$.

Suppose $L$ can be extended to a holomorphic line bundle $L'$ over $\varOmega$. Note that $\varphi=\exp(i\cdot)$ is a biholomorphism between $D_1=\{|x_1|,|x_2|<1,y_1^2+y_2^2<1\}$ and its image $\varphi(D_1)$. The pull-back $\varphi^*(L'|_{\varphi(D_1)})$ gives a holomorphic line bundle defined over $D_1$ which extends $\varphi^*(L|_{\varphi(D_1)\backslash K_0})$. Here $\varphi^*(L|_{\varphi(D_1)\backslash K_0})$ is a holomorphic line bundle defined over $D_1\backslash K$, since $\varphi^{-1}\big(\varphi(D_1)\backslash K_0\big)=D_1\backslash K$. However, due to our discussion in Section 2, such $\varphi^*(L|_{\varphi(D_1)\backslash K_0})$ is nontrivial hence cannot be extended across $K$. This contradiction shows that $L$ cannot be extended to $\varOmega$.

We can draw $\varOmega\subset \mathbb{R}^4$ as a movie of its 3d-sections (when $y_2$ is fixed) in $\mathbb{R}^3$.

\begin{figure}[h]
    \centering
    \includegraphics[width=0.7\textwidth]{dim2grauert.png}
  \caption{3d-sections of $\varOmega$, where the red surface is $\partial\varOmega$ and the blue curve is $K_0$}
\end{figure}

In fact, each 3d-section is obtained by rotating the 2d-section (when $y_1=0$) along the $x_2$-axis (the dashed line).

\begin{figure}[htbp]
    \centering
    \includegraphics[width=0.8\textwidth]{dim2grauert-covering21.png}
  \caption{2d-sections of $\varOmega$ when $y_1=0$}
\end{figure}

We can also draw $\varphi(U_1),\varphi(U_2),\varphi(U_{12}^1)$ and $\varphi(U_{12}^2)$ in this way.

\begin{figure}[htbp]
    \centering
    \includegraphics[width=0.8\textwidth]{dim2grauert-covering22.png}
  \caption{2d-sections of $\varphi(U_1)$}
\end{figure}

\begin{figure}[htbp]
    \centering
    \includegraphics[width=0.8\textwidth]{dim2grauert-covering23.png}
  \caption{2d-sections of $\varphi(U_2)$}
\end{figure}

\begin{figure}[htbp]
    \centering
    \includegraphics[width=0.8\textwidth]{dim2grauert-covering24.png}
  \caption{2d-sections of $\varphi(U_{12}^1)$ in green and of $\varphi(U_{12}^2)$ in yellow}
\end{figure}

\clearpage
\section{Counterexamples in general dimension}
\label{sect-counter-dim-3}

As announced in the Introduction, we will construct some `strange' bundles which cannot be extended from $\varOmega\backslash K$ to $\varOmega$. The key idea is a certain gluing lemma describing `flexibility' of holomorphic line bundles. Actually such lemma holds for holomorphic vector bundles.

\subsection{Gluing lemma}\label{subsection-gluing}
Roughly speaking, holomorphic vector bundles have more `flexibility' than holomorphic functions. Let $\varOmega\subset\mathbb{C}^n$ {\small($n\geqslant 2$)} be a domain and $U\subset \varOmega$ be a non-empty open subset. If two holomorphic functions $f$ and $g$ in $\varOmega$ are equal over $U$, then they are equal over $\varOmega$. However, if two holomorphic vector bundles $E$ and $F$ over $\varOmega$ are isomorphic over $U$, then they may not be isomorphic over $\varOmega$ in general. For example, let $E$ and $F$ be two non-isomorphic holomorphic line bundles over $\varOmega$. For any $x\in\varOmega$, there exists a neighborhood $U_E$ (resp. $U_F$) of $x$ in $\varOmega$ where $E$ (resp. $F$) is trivial. So $E$ and $F$ are trivial over $U:=U_E\cap U_F$, another neighborhood of $x$ in $\varOmega$. So they are isomorphic over a non-empty open subset $U$ of $\varOmega$.

\begin{Lem}\label{lem:gluing} {\em (Gluing lemma for holomorphic vector bundles)} Let $X$ be a complex manifold, let $U,V\subset X$ be two open subsets and let $W:=U\cap V$. For any integer $r\geqslant 1$, let $E_U$ be a holomorphic vector bundle of rank $r$ over $U$. Let $E_V$ be a holomorphic vector bundle of rank $r$ over $V$ such that $E_U|_W\cong E_V|_W$. Then there exists a holomorphic vector bundle $E$ of rank $r$ over $U\cup V$ such that $E|_U\cong E_U$ and $E|_V\cong E_V$.
\end{Lem}

\begin{Rmk} \em{Denote by $\mathcal{GL}_r(\co)$ the sheaf of invertible $r\times r$ matrices with coefficients in the sheaf $\mathscr{O}$ of holomorphic functions. In particular, $\mathcal{GL}_1(\mathscr{O})=\mathscr{O}^*$. Using the language of category theory, by the universal property of the fibre product, the following commutative diagram
\begin{diagram}
H^1\big(U\cup V,\mathcal{GL}_r(\co)\big) & \rTo^{\text{res}} & H^1\big(U,\mathcal{GL}_r(\co)\big)\\
\dTo^{\text{res}} & \circlearrowright& \dTo^{\text{res}}\\
H^1\big(V,\mathcal{GL}_r(\co)\big) & \rTo^{\text{res}} & H^1\big(U\cap V,\mathcal{GL}_r(\co)\big)
\end{diagram}
induces a canonical map
\al{q:H^1\big(U\cup V,\mathcal{GL}_r(\co)\big)\cong H^1\big(U,\mathcal{GL}_r(\co)\big)\times_{H^1(U\cap V,\mathcal{GL}_r(\co))}H^1\big(V,\mathcal{GL}_r(\co)\big).}
The gluing Lemma \ref{lem:gluing} states that $q$ is an epimorphism, by constructing a left inverse of $q$.}
\end{Rmk}

\begin{Rmk} \em{Note that we only have existence, but not uniqueness in general. That is to say, $E$ is not uniquely determined up to isomorphism by the information of $E_U$ and $E_V$. When $r=1$, a simple cex to uniqueness can be constructed by taking
\al{
&X=\mathbb{C}\mathbb{P}^1:=\{[Z_0:Z_1],(Z_0,Z_1)\in\mathbb{C}^2\backslash(0,0)\},\\
&U:=\{Z_0\neq 0\}=\{[1:{\textstyle{\frac{Z_1}{Z_0}}}],{\textstyle{\frac{Z_1}{Z_0}}}\in\mathbb{C}\}\cong \mathbb{C},\\
&V:=\{Z_1\neq 0\}=\{[{\textstyle{\frac{Z_0}{Z_1}}}:1],{\textstyle{\frac{Z_0}{Z_1}}}\in\mathbb{C}\}\cong\mathbb{C},\\
&W=U\cap V\cong \mathbb{C}^*
}
and $E_U:=U\times\mathbb{C}$ with coordinates $(z_1,s_1)$, $E_V:=V\times \mathbb{C}$ with coordinates $(z_0,s_0)$ being trivial line bundles with the identifications
\al{
(z_1,s_1)=({\textstyle{\frac{1}{z_0}}},{\textstyle{\frac{1}{z_0^n}}}s_0)
}
for each $n\in\mathbb{Z}$. This defines the holomorphic line bundle $O(n)$, trivial over $U$ and $V$. But $O(n)$ and $O(m)$ are not isomorphic whenever $n\neq m$.}
\end{Rmk}

\proof[Proof of Lemma~\ref{lem:gluing}] It suffices to consider the case where $U$, $V$, $W$ are non-empty. Denote the projection maps by $\pi_U:E_U\longrightarrow U$ and $\pi_V:E_V\longrightarrow V$. Let $\{(U_i,\varphi_{i})\}$ be a trivialization of $E_U$, $\{(V^j,\psi^j)\}$ be a trivialization of $E_V$. We will use the notations
\al{U_{i_1,\dots,i_s}:=\bigcap\limits_{x=1}^s U_{i_x},\\
V^{j_1,\dots,j_t}:=\bigcap\limits_{y=1}^t V^{j_y}.}
By trivializations of these vector bundles, we mean that $\{U_i\}$ is an open cover of $U$, $\{V^j\}$ is an open cover of $V$ and
\al{\varphi_{i}:\pi_U^{-1}(U_i)\longrightarrow U_i\times \mathbb{C}^r,\\
\psi^j:\pi_V^{-1}(V^j)\longrightarrow V^j\times \mathbb{C}^r,}
are homeomorphisms and the transition functions $f_{i_2,i_1}$, $g^{j_2,j_1}$ defined by
\al{\varphi_{i_2}\circ(\varphi_{i_1})^{-1}:U_{i_1,i_2}\times \mathbb{C}^r&\longrightarrow U_{i_1,i_2}\times \mathbb{C}^r\\
(z,l)&\longmapsto \big(z,f_{i_2,i_1}(z)\cdot l\big),\\
\psi^{j_2}\circ(\psi^{j_1})^{-1}:V^{j_1,j_2}\times \mathbb{C}^r&\longrightarrow V^{j_1,j_2}\times \mathbb{C}^r\\
(z,l)&\longmapsto \big(z,g^{j_2,j_1}(z)\cdot l\big),}
are holomorphic maps valued in $\mathcal{GL}_r(\co)(U_{i_1,i_2})$, $\mathcal{GL}_r(\co)(V^{j_1,j_2})$, satisfying the {\sl cocycle conditions}:
\begin{align}\label{cocy1}
f_{i_3,i_2}(z)\cdot f_{i_2,i_1} (z)=f_{i_3,i_1}(z)\quad {\scriptstyle(z\in U_{i_1,i_2,i_3})},\\
\label{cocy2}
g^{j_3,j_2}(z)\cdot g^{j_2,j_1}(z)=g^{j_3,j_1}(z)\quad {\scriptstyle(z\in V^{j_1,j_2,j_3})}.
\end{align}

In fact, we can use the notation $H^{0,0}_{\bar{\partial}}(U_{I},E_U)$ for the set of holomorphic sections of $E_U$ over $U_{I}$ where $I=\{i_1,\dots,i_s\}$. It is indeed a free $H^0(U_I,\mathscr{O})$-module of rank $r$. If we use $e_1,\dots,e_r$ as the standard basis of the $\mathbb{C}$-vector space $\mathbb{C}^r$, then $\{(\varphi_{i})^{-1}(z,e_1),$ $\dots$ $,(\varphi_{i})^{-1}(z,e_r)\}$ is a set of nowhere vanishing holomorphic sections of $E_U|_{U_i}$ which generate $H^{0,0}_{\bar{\partial}}(U_i,E_U)$. So actually $f_{i_2,i_1}$ is a $H^0(U_{i_1,i_2},\mathscr{O})$-coefficients invertible linear map such that for $\alpha=1,\dots,r$,
\al{(\varphi_{i_2})^{-1}(z,e_\alpha)=\sum\limits_{\beta=1}^r\big(f_{i_2,i_1}(z)\big)_{\alpha,\beta}(\varphi_{i_1})^{-1}(z,e_\beta)\quad {\scriptstyle(\beta=1,\dots,r;~z\in U_{i_1,i_2})},}
and the cocycle conditions are automatically satisfied.

In the language of \v{C}ech cohomology we would say that $E_U$ is represented by the open cover $\{U_i\}$ of $U$ and the 1-cocycle $\{f_{i_2,i_1}\}\in Z^1\big(\{U_i\},\mathcal{GL}_r(\co)\big)$. We have similar statements for $\big(E_V,\{V^j\},\{g^{j_2,j_1}\}\big)$.

Then $\{W_i^j:=U_i\cap V^j\}$ is an open cover of $W$ trivializing $E_U$ and $E_V$ simultaneously, i.e. $H^{0,0}_{\bar{\partial}}(W_i^j,E_U)$ \big(resp. $H^{0,0}_{\bar{\partial}}(W_i^j,E_V)$\big) is a rank $r$ free $H^0(W_i^j,\mathscr{O})$-module generated by $\{(\varphi_{i})^{-1}(z,e_1),$ $\dots$ $,(\varphi_{i})^{-1}(z,e_r)\}$ \big(resp. $\{(\psi^{j})^{-1}(z,e_1),$ $\dots$ $,(\psi^{j})^{-1}(z,e_r)\}$\big). The isomorphism $h:E_U|_W\cong E_V|_W$ induces an isomorphism between the rank $r$ free $H^0(W_i^j,\mathscr{O})$-modules $H^{0,0}_{\bar{\partial}}(W_i^j,E_U)$ and $H^{0,0}_{\bar{\partial}}(W_i^j,E_V)$. It is determined by $h^j_{~i}\in H^0\big(W_i^j,\mathcal{GL}_r(\co)\big)$ such that for $\alpha=1,\dots,r$,
\al{(\psi^{j})^{-1}(z,e_\alpha)=\sum\limits_{\beta=1}^r\big(h^j_{~i}(z)\big)_{\alpha,\beta}(\varphi_{i})^{-1}(z,e_\beta)\quad{\scriptstyle(\beta=1,\dots,r;~ z\in W^j_i)}.
}

Use the notation $W_{i_1,\dots, i_s}^{j_1,\dots, j_t}:=(\bigcap\limits_{x=1}^sU_{i_x})\cap(\bigcap\limits_{y=1}^t V^{j_y})$. For any indices $i_1$, $i_2$, $j_1$, $j_2$ we get the {\sl transition equations}
\begin{align}\label{liso}
f_{i_2,i_1}(z)=\big(h_{~~i_2}^{j_2}(z)\big)^{-1}\cdot g^{j_2,j_1}(z)\cdot h_{~~i_1}^{j_1}(z)\quad{\scriptstyle(z\in W_{i_1,i_2}^{j_1,j_2})}.
\end{align}

Now we define $E$, a holomorphic vector bundle over $U\cup V$. Note that $\{U_i,V^j\}$ is actually an open cover of  $U\cup V$. We let $E$ to be trivial on each $U_i$ and $V^j$ and define transition functions $l\in Z^1\big(\{U_i,V^j\},\mathcal{GL}_r(\co)\big)$ by
\al{l_{i_2,i_1} &:=f_{i_2,i_1},\\
l^{j_2,j_1} &:=g^{j_2,j_1},\\
l_{~i}^j &:=h_{~i}^j.}
The cocycle conditions among ($U_{i_1}$, $U_{i_2}$, $U_{i_3}$) and ($V^{j_1}$, $V^{j_2}$, $V^{j_3}$) are satisfied because of (\ref{cocy1}) and (\ref{cocy2}). We only need to check the cocycle conditions among ($U_{i_1}$, $U_{i_2}$, $V^j$) and ($U_i$, $V^{j_1}$, $V^{j_2}$), i.e.
\al{h_{~i_2}^j(z)\cdot f_{i_2,i_1}(z) &=h_{~i_1}^j(z)\quad{\scriptstyle(z\in W_{i_1,i_2}^j)},\\
g^{j_2,j_1}(z)\cdot h_{~~i}^{j_1}(z) &=h_{~~i}^{j_2}(z)\quad{\scriptstyle(z\in W_i^{j_1,j_2})}.}
But this can be achieved by taking $j_1=j_2=j$ or $i_1=i_2=i$ in (\ref{liso}) and using $f_{i,i}=id$, $g^{j,j}=id$. We have $E|_U\cong E_U$, because both bundles are given by the same transition functions $\{l_{i_2,i_1}=f_{i_2,i_1}\}$ with respect to the same open covering $\{U_i\}$ of $U$. For the same reason $E|_V\cong E_V$. \endproof

\subsection{A Stein manifold with a nontrivial holomorphic line bundle}

Take $\rho:(\mathbb{C}^*)^n\longrightarrow[0,+\infty)$, $(z_1,\dots,z_n)\mapsto\sum\limits_{j=1}^n(\log|z_j|)^2$. For any $\epsilon>0$, define $G_\epsilon:=\rho^{-1}[0,\epsilon)$. Denote by $B(r):=\{\sum\limits_{j=1}^n|z_j|^2<r^2\}$ the open ball centered at the origin of radius $r>0$ in $\mathbb{C}^n$.
\begin{Prop}\label{prop:G} For every $\epsilon>0$,
\begin{enumerate}[{\bf (i)}]
\item $G_\epsilon\subset B(\sqrt{n}e^{\sqrt{\epsilon}})$ is bounded;
\item $G_\epsilon$ is pseudoconvex with smooth boundary;
\item $G_\epsilon$ is connected. In fact $G_\epsilon$ contracts to a $n$-dimensional torus. Its Picard group is $H^1(G_\epsilon,\mathscr{O}^*)\cong\mathbb{Z}^{\binom{n}{2}}$. In particular, $G_\epsilon$ carries a nontrivial holomorphic line bundle.
\end{enumerate}
\end{Prop}
\proof {\bf (i)} Since
\begin{align*}
  \sum\limits_{j=1}^n(\log|z_j|)^2 <\epsilon
  ~\Longrightarrow~ &(\log|z_j|)^2 <\epsilon\cond{j=1,\dots,n}\\
  \Longrightarrow~ &e^{-\sqrt{\epsilon}}<|z_j| <e^{\sqrt{\epsilon}}\cond{j=1,\dots,n}\\
  \Longrightarrow~ &\sum\limits_{j=1}^n|z_j|^2 <ne^{2\sqrt{\epsilon}},
\end{align*}
we see that $G_\epsilon\subset B(\sqrt{n}e^{\sqrt{\epsilon}})$.

{\bf (ii)} Actually the boundary $\partial G_\epsilon=\rho^{-1}(\epsilon)$. We know that $\rho$ is smooth on $(\mathbb{C}^*)^n$ and $d\rho=\sum\limits_{j=1}^n\log|z_j|({\textstyle{\frac{dz_j}{z_j}}}+{\textstyle{\frac{d\bar{z}_j}{\bar{z}_j}}})$.
\al{&d\rho=0\\
\Longleftrightarrow~ &|z_j| =1\cond{j=1,\dots,n}\\
\Longleftrightarrow~ &\rho=0.}
So $\epsilon>0$ is a regular value of $\rho$, hence $\partial G_\epsilon$ is smooth.

To prove that $G_\epsilon$ is pseudoconvex, we check the Levi-condition. For any $z\in\partial G_\epsilon$ and $(w_1,\dots,w_n)\in T_z(\mathbb{C}^*)^n=T_z\mathbb{C}^n$ satisfying $\sum\limits_{j=1}^n\frac{\partial\rho(z)}{\partial z_j}w_j=0$, we have
\al{\sum\limits_{j,k=1}^n\frac{\partial^2\rho(z)}{\partial z_j\bar{\partial}z_k}w_j\bar{w}_k =\sum\limits_{j=1}^n\frac{|w_j|^2}{2|z_j|^2}\geqslant \frac{1}{2e^{2\sqrt{\epsilon}}}\sum\limits_{j=1}^n|w_j|^2.}
Thus $G_\epsilon$ is pseudoconvex.

{\bf (iii)} We claim that $G_\epsilon$ contracts to $\rho^{-1}(0)=\{|z_j|=1,j=1,\dots,n\}\cong \underbrace{S^1\times \ldots \times S^1}_{n \text{ times}}=T^n$. To show this, we construct a contraction map
\al{H:G_\epsilon\times [0,1] &\longrightarrow \mathbb{C}^n,\\
(z,t) &\longmapsto (|z_1|^{\lambda(z,t)}z_1,\dots,|z_n|^{\lambda(z,t)}z_n),}
where $\lambda(z,t):G_\epsilon\times[0,1]\longrightarrow \mathbb{R}$ is a continuous map. Note that
\al{\rho\big(H(z,t)\big)=\big(\lambda(z,t)+1\big)^2\rho(z).}

Then $H$ is a contraction map if for any $z\in G_\epsilon$ and any $t\in[0,1]$ we have
\begin{equation}
\left\{
\begin{aligned}
H(z,0) =z &\Longrightarrow \lambda(z,0)=0,\\
\im(H) \in G_\epsilon &\Longrightarrow \big(\lambda(z,t)+1\big)^2\rho(z)<\epsilon.
\end{aligned}
\right.
\end{equation}

If we take $\lambda(z,t)=-t$, then these conditions are satisfied and $\im\big(H(z,1)\big)=\rho^{-1}(0)=\{|z_j|=1,j=1,\dots,n\}\cong T^n$. We see $G_\epsilon$ is connected since $T^n$ is.

Now we calculate the Picard group $H^1(G_\epsilon,\mathscr{O}^*)$. By {\bf(ii)}, $G_\epsilon$ is Stein and by Cartan's theorem B, we have $H^1(G_\epsilon,\mathscr{O})=H^2(G_\epsilon,\mathscr{O})=0$.
Recall that the exponential exact sequence
\al{0\longrightarrow \mathbb{Z}\xrightarrow{\times 2\pi i}\mathscr{O}\xrightarrow{exp}\mathscr{O}^*\longrightarrow 0}
induces a long exact sequence
\al{H^1(G_\epsilon,\mathscr{O})\longrightarrow H^1(G_\epsilon,\mathscr{O}^*)\longrightarrow H^2(G_\epsilon,\mathbb{Z})\longrightarrow H^2(G_\epsilon,\mathscr{O}).}

Since the first and the last term vanish, we have
\al{H^1(G_\epsilon,\mathscr{O}^*)\cong H^2(G_\epsilon,\mathbb{Z})\cong H^2(T^n,\mathbb{Z}).}

Recall that by using a Mayer-Vietoris sequence we have $H^k(S^1\times M,\mathbb{Z})\cong H^{k-1}(M,\mathbb{Z})\oplus H^k(M,\mathbb{Z})$ for any simplicial complex $M$ and any $k\in\mathbb{Z}_+$. Hence $H^k(T^n,\mathbb{Z})\cong\mathbb{Z}^{\binom{n}{k}}$. In particular $H^1(G_\epsilon,\mathscr{O}^*)\cong H^2(T^n,\mathbb{Z})\cong\mathbb{Z}^{\binom{n}{2}}$ is nontrivial since $n\geqslant 2$. \endproof

\subsection{Gluing process}

Now we take $0<\epsilon<n$, for example $\epsilon=n/2$. The boundary $\partial G_\epsilon$ is then given by the equation
\[
\rho(z)=\sum\limits_{j=1}^n(\log|z_j|)^2=\epsilon.
\]

\begin{Prop}\label{real-hessian}
The real Hessian $H\rho$ is positive definite at the point $p=(e^{\sqrt{\epsilon/n}},\dots,e^{\sqrt{\epsilon/n}})\in\partial G_\epsilon$.
\end{Prop}

\proof For $j=1,\dots,n$, we have
\al{
\frac{\partial^2 \rho}{\partial x_j^2} &=\frac{2}{|z_j|^4}\big((y_j^2-x_j^2)\log|z_j|+x_j^2\big),\\
\frac{\partial^2 \rho}{\partial x_j\partial y_j} &=\frac{2x_jy_j}{|z_j|^4}(1-2\log|z_j|),\\
\frac{\partial^2 \rho}{\partial y_j^2} &=\frac{2}{|z_j|^4}\big((x_j^2-y_j^2)\log|z_j|+y_j^2\big).
}

So the real Hessian is
\[
H\rho=
\begin{bmatrix}
    \frac{2}{|z_1|^4}H^{2\times 2}_1 & 0  & \dots  & 0 \\
    0 & \frac{2}{|z_2|^4}H^{2\times 2}_2 &  \dots  & 0 \\
    \vdots & \vdots & \ddots & \vdots \\
    0 & 0 & \dots  & \frac{2}{|z_n|^4}H^{2\times 2}_n
\end{bmatrix},
\]
where
\[
H^{2\times 2}_j=
\begin{bmatrix}
    (y_j^2-x_j^2)\log|z_j|+x_j^2 & x_jy_j(1-2\log|z_j|) \\
    x_jy_j(1-2\log|z_j|) & (x_j^2-y_j^2)\log|z_j|+y_j^2
\end{bmatrix}\cond{j=1,\dots,n}.
\]

The real Hessian $H\rho$ is positive definite if and only if for all $j$, $H^{2\times 2}_j$ is positive definite. That is equivalent to
\begin{equation*}
\left\{
\begin{aligned}
\text{tr} (H^{2\times 2}_j)>0,\\
\det(H^{2\times 2}_j)>0,
\end{aligned}
\right.
\end{equation*}
for all $j$. The first inequality is achieved since $\text{tr}(H^{2\times 2}_j)=x_j^2+y_j^2>0$ for all $j$ and all $z\in \partial G_\epsilon\subset (\mathbb{C}^*)^n$. For the second inequality, we calculate
\[
\det(H^{2\times 2}_j) = (x_j^2+y_j^2)^2(\log|z_j|-(\log|z_j|)^2),
\]
hence
\al{
\det(H^{2\times 2}_j) >0 &\Longleftrightarrow~ \log|z_j|-(\log|z_j|)^2>0\\
&\Longleftrightarrow~ \log|z_j|\in(0,1)\\
&\Longleftrightarrow~ |z_j|\in(1,e).
}
So at $p=(e^{\sqrt{\epsilon/n}},\dots,e^{\sqrt{\epsilon/n}})\in\partial G_\epsilon$ we have $\det\big(H^{2\times 2}_j(p)\big)>0$ for each $j=1,\dots,n$. We conclude that $H\rho(p)$ is positive definite. \qed

\begin{Rmk} \em{The condition $\epsilon<n$ is necessary and sufficient for the existence of some point in $\partial G_\epsilon$ where the real Hessian $H\rho$ is positive definite. This is because when $\epsilon\geq n$, for any $p'=(z_1,\dots,z_n)\in \partial G_\epsilon$ there exists at least one $j=1,\dots,n$ such that $(\log|z_j|)^2\geq 1$. Thus $\det\big(H^{2\times 2}_j(p')\big)\leq 0$, hence $H\rho(p')$ is not positive definite.}
\end{Rmk}

At the point $p=(e^{\sqrt{\epsilon/n}},\dots,e^{\sqrt{\epsilon/n}})\in\partial G_\epsilon$, since $H\rho(p)$ is positive definite and $\rho$ is smooth, there exists some open convex neighborhood $U_p$ of $p$ in $(\mathbb{C}^*)^n$ (e.g. a sufficiently small open ball centered at $p$), such that $H\rho(z)$ is positive definite for all $z\in U_p$. Thus $\rho$ is strictly convex in $U_p$, hence for any $p_1,p_2\in G_\epsilon\cap U_p$ and any $t\in(0,1)$ we have
\[
\rho\big(tp_1+(1-t)p_2\big)<t\rho(p_1)+(1-t)\rho(p_2)<\epsilon.
\]
So $tp_1+(1-t)p_2\in G_\epsilon$ and is also contained in $U_p$ since $U_p$ is convex. So $tp_1+(1-t)p_2\in G_\epsilon\cap U_p$. We proved that $G_\epsilon\cap U_p$ is convex.

Now we define $\varOmega:=B(2\sqrt{n}e^{\sqrt{\epsilon}})$, $\varOmega':=(\varOmega\backslash \overline{G_\epsilon})\cup U_p$ and $K:=\varOmega\backslash (G_\epsilon\cup \varOmega')$. In fact we have $K=\partial G_\epsilon\backslash U_p\subset\subset \varOmega$.

\begin{Prop}\label{prop-omega} The open set $\varOmega'\cup G_\epsilon=\varOmega\backslash K$ is connected.
\end{Prop}

To prove the proposition, we will use the language in Range's book \cite{Range} Chap 3.7. We call a compact set $A\subset \mathbb{C}^n$ a {\sl Stein compactum} if it has a neighborhood basis of Stein domains.

\begin{Lem}\label{lem:Steincompactum} For any bounded Stein compactum $A\subset \mathbb{C}^n$ $(n\geqslant 2)$, the complement $A^c:=\mathbb{C}^n\backslash A$ is connected.
\end{Lem}

\proof Since $A$ is bounded, there exists some $R>0$ such that $A\subset B(R)\subset \mathbb{C}^n$. Thus the open set $A^c$ has an unbounded component containing $B(R)^c$.

Suppose $A^c$ is not connected and $W$ is another component, then $W\subset B(R)$ is bounded. Also $\partial W\subset\partial A$. Take $p\in W\subset A^c$. We have $p\notin A$. Thus $\mathbb{C}^n\backslash\{p\}$ is an open neighborhood of $A$. Since $A$ is a Stein compactum, there exists an open Stein neighborhood $V$ of $A$ contained in $\mathbb{C}^n\backslash\{p\}$. The set $\partial W\subset \partial A\subset A$ is relatively compact in $V$. The holomorphically convex hull of $\partial W$ with respect to $V$
\al{
\widehat{\partial W}_V:=\{z\in V||f(z)|\leqslant \sup\limits_{w\in \partial W}|f(w)|,\forall f\in\mathscr{O}(V)\},
}
on one hand, should be relatively compact in $V$ since $V$ is Stein.

But on the other hand $W\backslash V\subset\subset V\cup W$ and $(V\cup W)\backslash (W\backslash V)=V$ is connected. Thus by Hartogs' extension theorem for holomorphic functions we have $\mathscr{O}(V)=\mathscr{O}(V\cup W)$ and by maximal principle we have
\al{
|f(z)|\leqslant\sup\limits_{w\in\partial W}|f(w)|,\cond{z\in W, f\in \mathscr{O}(V\cup W)\subset\mathscr{O}(W)}.
}
Hence $V\cap W\subset\widehat{\partial W}_V$. If $\widehat{\partial W}_V$ is relatively compact in $V$, so is $W\cap V$. But since $W$ is a connected domain, it is pathly connected. We take a point $q\in W\cap V$ and a path $\gamma$ in $W$ connecting $q$ and $p\in W\backslash V$. We get a subpath in $W\cap V$ approaching $\partial V$. Hence $W\cap V$ is not relatively compact in $V$, a contradiction.
\qed

\proof[Proof of Proposition \ref{prop-omega}] In fact $U_p$ is a domain meeting both $G_\epsilon$ and $\varOmega\backslash\overline{G_\epsilon}$. By Proposition \ref{prop:G} {\bf(iii)} we know $G_\epsilon$ is connected. So it suffices to show that $\varOmega\backslash\overline{G_\epsilon}$ is connected. Note that the bounded compact set $\overline{G_\epsilon}$ has a neighborhood basis of Stein domains $\{G_{\epsilon+\delta},\delta>0\}$. By Lemma \ref{lem:Steincompactum} we know $\overline{G_{\epsilon}}^c$ is connected. Since $\overline{G_{\epsilon}}\subset\subset\varOmega=B(2\sqrt{n})$, we know $\varOmega\backslash\overline{G_\epsilon}$ is connected.\qed

\proof [End of the proof of Theorem \ref{main-counterexample}]
Take the trivial holomorphic line bundle $L_{\text{triv}}$ over $\varOmega'$, take $L_{\text{nt}}$ a nontrivial holomorphic line bundle over $G_\epsilon$. Since $G_\epsilon\cap U_p$ is convex, by Corollary \ref{crit-2} we know the restrictions of $L_{\text{nt}}$ and $L_{\text{triv}}$ to $G_\epsilon\cap U_p$ are trivial, hence isomorphic. By the gluing Lemma \ref{lem:gluing} we get a holomorphic line bundle $L_{\text{cex}}$ over $G_\epsilon\cup \varOmega'=\varOmega\backslash K$, which is nontrivial since $L_{\text{cex}}|_{G_\epsilon}\cong L_{\text{nt}}$ is. Thus $H^1(\varOmega\backslash K,\mathscr{O}^*)\neq 0$. However, $H^1(\varOmega,\mathscr{O}^*)=0$ since $\varOmega$ is convex. Thus the restriction map
\al{H^1(\varOmega,\mathscr{O}^*)\longrightarrow H^1(\varOmega\backslash K,\mathscr{O}^*)}
cannot be surjective. In particular, $L_{\text{cex}}$ cannot be extended to $\varOmega$.
\endproof

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