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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 Contemporary Mathematics Volume 647, 2015 http://dx.doi.org/10.1090/conm/647/12956 The stability manifolds of P1 and local P1 Aaron Bertram, Steffen Marcus, and Jie Wang Dedicated to Herb Clemens Abstract. In this expository paper, we review and compare the local home- omorphisms from the manifold of Bridgeland stability conditions to the space of central charges in the cases of both P1 and local P1. 1. Introduction The space of stability conditions on a triangulated category was introduced by Bridgeland in [3], following work on Π-stability in string theory due to Douglas [7]. A central result of [3] is that for an arbitrary triangulated category D the space Stab(D) of stability conditions is a topological manifold. Under nice conditions on D, the manifold Stab(D) is in fact a finite dimensional complex manifold. When working with an algebraic variety X, Bridgeland’s stability conditions allow one to abstract notions of slope stability for sheaves on projective varieties to the bounded derived category of coherent sheaves on X. Since [3], there have been many efforts towards computing explicit examples in order to better understand the theory and its relation to enumerative geometry and mirror symmetry. For a smooth curve C of genus at least one, there is a rather uniform and complete description of Stab(D(C)). Macr`ı[11] and Bridgeland [3]provedthat + they are all isomorphic to the universal cover GL (2, R)ofGL+(2, R). The case of C = P1 turns out to be more subtle and involved due to the existence of full exceptional collections in D(P1). There exists “pathological” regions in Stab(D(P1)) ∼ where line bundles are unstable. Okada [12] proved that Stab(D(P1)) = C2. In [2], Bridgeland describes a connected component of Stab(D(X)) when X is the minimal resolution of a Kleinian quotient singularity C2/G. In particular, for G = Z/2Z this includes the local case of P1 embedded as the zero section in the total space of its cotangent bundle. The aim of this expository note is to provide a thorough review and comparison of the two cases of P1 and local P1, with emphasis on the local homeomorphisms p from the space of stability conditions to the space of central charges. Through these explicit examples, the authors hope to illustrate some general methods to study stability manifolds. Some of these techniques can be applied to study more complicated examples. Bayer and Macr`ı[6] gave a fairly explicit description of the stability manifold of local P2. c 2015 American Mathematical Society 1 2 AARON BERTRAM, STEFFEN MARCUS, AND JIE WANG There is a natural free action of the additive group C on the stability manifold and nothing interesting happens to the semistable objects under this action (c.f.
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