Anilatmaja Aryasomayajula Indranil Biswas Archana S. Morye A.J. Parameswaran Editors Analytic and Algebraic Geometry Analytic and Algebraic Geometry Anilatmaja Aryasomayajula Indranil Biswas • Archana S. Morye A.J. Parameswaran Editors Analytic and Algebraic Geometry 123 Editors Anilatmaja Aryasomayajula Archana S. Morye Department of Mathematics School of Mathematics and Statistics IISER Tirupati University of Hyderabad Tirupati, Andhra Pradesh Hyderabad, Telangana India India Indranil Biswas A.J. Parameswaran School of Mathematics School of Mathematics Tata Institute of Fundamental Research Tata Institute of Fundamental Research Mumbai, Maharashtra Mumbai, Maharashtra India India ISBN 978-981-10-5648-2 (eBook) DOI 10.1007/978-981-10-5648-2 Library of Congress Control Number: 2017946031 This work is a co-publication with Hindustan Book Agency, New Delhi, licensed for sale in all countries in electronic form only. Sold and distributed in print across the world by Hindustan Book Agency, P-19 Green Park Extension, New Delhi 110016, India. ISBN: 978-93-86279-64-4 © Hindustan Book Agency 2017. © Springer Nature Singapore Pte Ltd. 2017 and Hindustan Book Agency 2017 This work is subject to copyright. All rights are reserved by the Publishers, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by Springer Nature The registered company is Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Preface For more than ten years, Indranil Biswas and A. J. Parameswaran of Tata In- stitute of Fundamental Research, Mumbai are organizing an annual international conference on the topics around Analytic and Algebraic Geometry. The first part of the conference is held at the Tata Institute of Fundamental Research and the second part in some other institute. These conferences are primarily in- tended to facilitate interactions, between mathematicians in India and in other countries, working in related areas. Another purpose of these conferences is to have series of lectures by experts on topics of interest to mathematicians in India. These annual conferences have initiated a large number of collaborative works in Mathematics. In 2015, a conference on Analytic and Algebraic Geometry was held at the Tata Institute of Fundamental Research and the University of Hyderabad. This volume is an outcome of that conference, although not all speakers submitted their lecture material. There are fifteen articles in this volume. The main purpose of the articles is to introduce recent and advanced techniques in Analytic and Algebraic Ge- ometry. We have attempted to give recent developments in the area to target mainly young researchers, who are new to this area. We have also added some research articles to give examples of how to use these techniques to prove new results. Our sincere thanks to the School of Mathematics, the University of Hyderabad for their generous financial and logistic support for this conference. Editors Contents Preface................................................................. v About the Editors ....... ..... ..... .............. ....... ... ix Marco Antei On the Bumpy Fundamental Group Scheme. ............................ 1 Anilatmaja Aryasomayajula HeatKernels,BergmanKernels,andCuspForms....................... 19 Usha N. Bhosle Ona ConjectureofButler.............................................. 29 Indranil Biswas and Mahan Mj A SurveyofLowDimensional(Quasi)ProjectiveGroups................ 49 Niels Borne Parabolic Sheaves and Logarithmic Geometry ........................... 67 Emre Coskun A Survey of Ulrich Bundles . ............................................ 85 Ananyo Dan Noether-Lefschetz Locus and a Special Case of the Variational HodgeConjecture:UsingElementaryTechniques..................... 107 A. El Mazouni and D.S. Nagaraj Tangent Bundle of P2 and Morphism from P2 to Gr(2, C4).............. 117 Michel Emsalem Twistingbya Torsor.................................................... 125 Viktoria Heu and Frank Loray HitchinHamiltoniansinGenus2........................................ 153 Inder Kaur Smoothness of Moduli Space of Stable Torsion-free Sheaves with Fixed DeterminantinMixedCharacteristic................................. 173 Johan Martens GroupCompactificationsandModuliSpaces............................ 187 viii Contents Archana S. Morye TheSerre-SwanTheoremforRingedSpaces............................. 207 Georg Schumacher An Extension Theorem for Hermitian Line Bundles . .................... 225 Tathagata Sengupta Elliptic Fibrations on Supersingular K3 Surface with Artin Invariant 1 in Characteristic3...................................................... 239 About the Editors Anilatmaja Aryasomayajula is assistant professor at Indian Institute of Science Education and Research (IISER), Tirupati, India. He did his Ph.D. from the Humboldt University of Berlin. A DST inspire fellow, his areas of interest are automorphic forms, Hilbert modular forms, Arakelov theory and Diophantine geometry. Indranil Biswas is professor of mathematics at the Tata Institute of Fundamental Research (TIFR), Mumbai, India. He works in algebraic geometry, differential geometry, several complex variables and analytic space, and deformation quantization. He has more than 400 publications to his credit with many collaborators across the world. In 2006, the government of India awarded him the Shanti Swarup Bhatnagar Prize in Mathematical Sciences for his contributions to algebraic geometry, more precisely moduli problems of vector bundles. He is a fellow of Indian Academy of Sciences and Indian National Science Academy (INSA). Archana S. Morye is a faculty in the University of Hyderabad, India. She did her Ph.D. from Harish-Chandra Research Institute, Allahabad, India. Her research area is algebraic geometry more specifically in vector bundles. A.J. Parameswaran is professor of mathematics at the Tata Institute of Fundamental Research (TIFR), Mumbai, India. He was the director of Kerala School of Mathematics (KSOM), Kozhikode, India, f or a short period of time. He works in algebraic geometry, several complex variables and analytic space. He is a fellow of Indian Academy of Sciences. On the Bumpy Fundamental Group Scheme Marco Antei∗,† Abstract In this short paper we first recall the definition and the construction of the fundamental group scheme of a scheme X in the known cases: when it is defined over a field and when it is defined over a Dedekind scheme. It classifies all the finite (or quasi-finite) fpqc torsors over X.WhenX is defined over a noetherian regular scheme S of any dimension we do not know if such an object can be constructed. This is why we introduce a new category, containing the fpqc torsors, whose objects are torsors for a new topology. We prove that this new category is cofiltered thus generating a fundamental group scheme over S,said bumpy as it may not be flat in general. We prove that it is flat when S is a Dedekind scheme, thus coinciding with the classical one. Mathematics Subject Classification (2000). 14G99, 14L20, 14L30 1. Introduction In his famous [SGA1] Alexander Grothendieck constructs the ´etale fundamental group π´et(X, x)ofaschemeX, endowed with a geometric point x : Spec(Ω) → X. It is a pro-finite group classifying the ´etale covers over X. Grothendieck elaborates for π´et(X, x) a specialization theory ([SGA1], Chapitre X) but it seems he is not entirely satisfied of it. Indeed at the end of the aforementioned tenth chapter he claims (here we freely translate from French1)that ∗The author thanks the project TOFIGROU (ANR-13-PDOC-0015-01) †Laboratoire J.A.Dieudonn´e, UMR CNRS-UNS No 7351 Universit´e de Nice Sophia- Antipolis, Parc Valrose, 06108 NICE Cedex 2, France. E-mail: [email protected] 1Une th´eorie satisfaisante de la sp´ecialisation du groupe fondamental doit tenir compte de la “composante continue” du “vrai” groupe fondamental, correspondant `a la classification des revtements principaux de groupe structural des groupes infinit´esimaux; moyennant quoi © Springer Nature Singapore Pte Ltd. 2017 and Hindustan Book Agency 2017 A. Aryasomayajula et al. (eds.), Analytic and Algebraic Geometry, DOI 10.1007/978-981-10-5648-2_1 2 Marco Antei a satisfactory specialization theory of the fundamental group should consider the “continuous component” of the “true” fundamental group corresponding to the classification of principal homogeneous