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Hodge Theory and Classical Algebraic Geometry 647 Hodge Theory and Classical Algebraic Geometry Conference on Hodge Theory and Classical Algebraic Geometry May 13–15, 2013 The Ohio State University, Columbus, Ohio Gary Kennedy Mirel Caibar˘ Ana-Maria Castravet Emanuele Macrì Editors American Mathematical Society Hodge Theory and Classical Algebraic Geometry Conference on Hodge Theory and Classical Algebraic Geometry May 13–15, 2013 The Ohio State University, Columbus, Ohio Gary Kennedy Mirel Caibar˘ Ana-Maria Castravet Emanuele Macrì Editors 647 Hodge Theory and Classical Algebraic Geometry Conference on Hodge Theory and Classical Algebraic Geometry May 13–15, 2013 The Ohio State University, Columbus, Ohio Gary Kennedy Mirel Caibar˘ Ana-Maria Castravet Emanuele Macrì Editors American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash Misra Martin J. Strauss 2010 Mathematics Subject Classification. Primary 14C30, 14D07, 32G20, 58A14. Library of Congress Cataloging-in-Publication Data Hodge theory and classical algebraic geometry : a conference on Hodge theory and classical algebraic geometry : May 13-15, 2013, the Ohio State University, Columbus, Ohio / Gary Kennedy, Mirel Caib˘ar, Ana-Maria Castravet, Emanuele Macr`ı, editors. pages cm. – (Contemporary mathematics ; volume 647) Includes bibliographical references. ISBN 978-1-4704-0990-6 (alk. paper) 1. Geometry, Algebraic–Congresses. 2. Hodge theory–Congresses. I. Kennedy, Gary, 1950- II. Caib˘ar, Mirel, 1967- III. Castravet, Ana-Maria, 1980- IV. Macr`ı, Emanuele. QA564.H55 2015 514.74–dc23 2015006623 Contemporary Mathematics ISSN: 0271-4132 (print); ISSN: 1098-3627 (online) DOI: http://dx.doi.org/10.1090/conm/647 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2015 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 201918171615 Contents Preface vii The stability manifolds of P1 and local P1 Aaron Bertram, Steffen Marcus, and Jie Wang 1 Reduced limit period mappings and orbits in Mumford-Tate varieties Mark Green and Phillip Griffiths 19 The primitive cohomology of theta divisors Elham Izadi and Jie Wang 79 Neighborhoods of subvarieties in homogeneous spaces Janos´ Kollar´ 91 Unconditional noncommutative motivic Galois groups Matilde Marcolli and Gonc¸alo Tabuada 109 Differential equations in Hilbert-Mumford Calculus Ziv Ran 117 Weak positivity via mixed Hodge modules Christian Schnell 129 v Preface This volume contains a selection of papers stemming from the conference Hodge Theory and Classical Algebraic Geometry, held on the campus of The Ohio State University in Columbus, Ohio, from May 13 to 15, 2013. The conference web page is still accessible at http://go.osu.edu/hodge. Most of the conference talks were captured on video, which may be viewed by following the appropriate links. A program and abstracts are also available there. In some instances the paper in this volume closely adheres to the conference lecture; in other instances there is a great difference. The idea of the conference was to offer young researchers a global view of recent developments and to have the speakers share their vision of the future. The papers in this proceeding follow essentially a similar idea; there are a few survey papers while others contain original research. The topics range from more classical aspects of Hodge theory to modern developments in compactifications of period domains, applications of Saito’s theory of mixed Hodge modules, and connections with derived category theory and non-commutative motives. The reader may note an odd feature: although there is no dedication on the title page, each paper in our volume is dedicated to Professor Herb Clemens. This is because our conference inadvertently used the venerable sales technique known as “bait and switch,” with Clemens as our bait. That is to say: we announced that he was retiring, and that we were organizing a conference in his honor. But it turned out after all that he was not in fact retiring, but instead beginning yet another chapter in his distinguished career. In view of this history, we tried to enforce the following rule at our confer- ence: you are forbidden to say anything nice about Herb. Of course the rule was skirted repeatedly. One speaker tried to evade it by praising not Herb but rather his basement, where he had stayed as a houseguest while learning to ski in the mountains of Utah. Others broke the rule quite brazenly. For example, it would be violating this rule to remark how much energy he has brought to the Ohio State Mathematics Department and in particular to those who work in algebraic geom- etry. And again it would be against the rule to note the awe with which we seem to observe at least two or three different people doing full-time jobs, each one of them named Herb Clemens. One of us once had the opportunity to speak to Robert Moses, a civil rights pioneer and the founder of the Algebra Project, who told us of his admiration for Herb’s work in mathematics education, and Moses seemed genuinely astonished to learn that Herb was also famous for a conjecture named after him in pure mathematics. We are very glad to report that his efforts on behalf of mathematics research, education, and infrastructure continue unceasingly. vii viii PREFACE The conference was supported in part by National Science Foundation Grant No. 1302880, with additional funding provided by The Ohio State University De- partment of Mathematics, including its Mathematics Research Institute. We used the lecture room and other facilities of the Mathematical Biosciences Institute. The conference organizers were the editors of the present volume, together with Elham Izadi and Christian Schnell, who likewise helped us in preparing the conference grant proposal. Paul Nylander created the striking image used for our poster and conference web pages. Gary Kennedy Mirel Caib˘ar Ana-Maria Castravet Emanuele Macr`ı SELECTED PUBLISHED TITLES IN THIS SERIES 647 Gary Kennedy, Mirel Caib˘ar, Ana-Maria Castravet, and Emanuele Macr`ı, Editors, Hodge Theory and Classical Algebraic Geometry, 2015 641 Maria Basterra, Kristine Bauer, Kathryn Hess, and Brenda Johnson, Editors, Women in Topology, 2015 640 Gregory Eskin, Leonid Friedlander, and John Garnett, Editors, Spectral Theory and Partial Differential Equations, 2015 639 C. S. Aravinda, William M. Goldman, Krishnendu Gongopadhyay, Alexander Lubotzky, Mahan Mj, and Anthony Weaver, Editors, Geometry, Groups and Dynamics, 2015 638 Javad Mashreghi, Emmanuel Fricain, and William Ross, Editors, Invariant Subspaces of the Shift Operator, 2015 637 St´ephane Ballet, Marc Perret, and Alexey Zaytsev, Editors, Algorithmic Arithmetic, Geometry, and Coding Theory, 2015 636 Simeon Reich and Alexander J. Zaslavski, Editors, Infinite Products of Operators and Their Applications, 2015 635 Christopher W. Curtis, Anton Dzhamay, Willy A. Hereman, and Barbara Prinari, Editors, Nonlinear Wave Equations, 2015 634 Steven Dougherty, Alberto Facchini, Andr´e Leroy, Edmund Puczylowski, and Patrick Sol´e, Editors, Noncommutative Rings and Their Applications, 2015 633 Delaram Kahrobaei and Vladimir Shpilrain, Editors, Algorithmic Problems of Group Theory, Their Complexity, and Applications to Cryptography, 2015 632 Gohar Kyureghyan, Gary L. Mullen, and Alexander Pott, Editors, Topics in Finite Fields, 2015 631 Siddhartha Bhattacharya, Tarun Das, Anish Ghosh, and Riddhi Shah, Editors, Recent Trends in Ergodic Theory and Dynamical Systems, 2015 630 Pierre Albin, Dmitry Jakobson, and Fr´ed´eric Rochon, Editors, Geometric and Spectral Analysis, 2014 629 Milagros Izquierdo, S. Allen Broughton, Antonio F. Costa, and Rub´ıE. Rodr´ıguez, Editors, Riemann and Klein Surfaces, Automorphisms, Symmetries and Moduli Spaces, 2014 628 Anita T. Layton and Sarah D. Olson, Editors, Biological Fluid Dynamics: Modeling, Computations, and Applications, 2014 627 Krishnaswami Alladi, Frank Garvan, and Ae Ja Yee, Editors, Ramanujan 125, 2014 626 Veronika Furst, Keri A. Kornelson, and Eric S. Weber, Editors, Operator Methods in Wavelets, Tilings, and Frames, 2014 625 Alexander Barg and Oleg R. Musin, Editors, Discrete Geometry and
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