Proceedings of Symposia in PURE MATHEMATICS

Volume 99

Modern Geometry: A Celebration of the Work of Simon Donaldson

Vicente Munoz˜ Ivan Smith Richard P. Thomas Editors Volume 99

Modern Geometry: A Celebration of the Work of Simon Donaldson

Vicente Munoz˜ Ivan Smith Richard P. Thomas Editors Nathalie Wahl, 2017 Proceedings of Symposia in PURE MATHEMATICS

Volume 99

Modern Geometry: A Celebration of the Work of Simon Donaldson

Vicente Munoz˜ Ivan Smith Richard P. Thomas Editors 2010 Mathematics Subject Classification. Primary 32J25, 32L05, 53C07, 53C44, 53D35, 53D40, 53D50, 57R55, 57R57, 57R58.

Library of Congress Cataloging-in-Publication Data Names: Mu˜noz, V. (Vicente), 1971– editor. | Smith, Ivan, 1973– editor. | Thomas, Richard P., 1972– editor. Title: Modern geometry : a celebration of the work of Simon Donaldson / Vicente Mu˜noz, Ivan Smith, Richard P. Thomas, editors. Description: Providence, Rhode Island : American Mathematical Society, [2018] | Series: Pro- ceedings of symposia in pure mathematics ; volume 99 | Includes bibliographical references. Identifiers: LCCN 2017052437 | ISBN 9781470440947 (alk. paper) Subjects: LCSH: Donaldson, S. K. | Manifolds (Mathematics) | Four-manifolds (Topology) | Ge- ometry. | Topology. | AMS: Several complex variables and analytic spaces – Compact analytic spaces – Transcendental methods of algebraic geometry. msc | Several complex variables and analytic spaces – Holomorphic fiber spaces – Holomorphic bundles and generalizations. msc | Differential geometry – Global differential geometry – Special connections and metrics on vector bundles (Hermite-Einstein-Yang-Mills). msc | Differential geometry – Global differential geometry – Geometric evolution equations (mean curvature flow, Ricci flow, etc.). msc | Differential geome- try – Symplectic geometry, contact geometry – Global theory of symplectic and contact manifolds. msc | Differential geometry – Symplectic geometry, contact geometry – Floer homology and coho- mology, symplectic aspects. msc | Differential geometry – Symplectic geometry, contact geometry – Geometric quantization. msc | Manifolds and cell complexes – Differential topology – Differ- entiable structures. msc | Manifolds and cell complexes – Differential topology – Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants. msc | Manifolds and cell complexes – Differential topology – Floer homology. msc Classification: LCC QA613 .M6345 2018 | DDC 516/.07–dc23 LC record available at https://lccn.loc.gov/2017052437 DOI: http://dx.doi.org/10.1090/pspum/099

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Preface vii Graded linearisations Gergely Berczi,` Brent Doran, and Frances Kirwan 1 Atiyah-Floer conjecture: A formulation, a strategy of proof and generalizations Aliakbar Daemi and Kenji Fukaya 23 Weinstein manifolds revisited Yakov Eliashberg 59 Remarks on Nahm’s equations Nigel Hitchin 83

Conjectures on counting associative 3-folds in G2-manifolds Dominic Joyce 97 Toward an algebraic Donaldson-Floer theory Jun Li 161 Introduction to a provisional mathematical definition of Coulomb branches of 3-dimensional N = 4 gauge theories Hiraku Nakajima 193 An overview of knot Floer homology Peter Ozsvath´ and Zoltan´ Szabo´ 213 Descendants for stable pairs on 3-folds Rahul Pandharipande 251 The Dirichlet problem for the complex homogeneous Monge-Amp`ere equation Julius Ross and David Witt Nystrom¨ 289 K¨ahler-Einstein metrics Gabor´ Szekelyhidi´ 331 Donaldson theory in non-K¨ahlerian geometry Andrei Teleman 363 Two lectures on gauge theory and Khovanov homology Edward Witten 393

v

Preface

Simon Donaldson has been one of the central figures in modern geometry for thirty-five years, and remains as active today as ever. His work has revolutionised numerous fields; the breadth of the essays in this volume are testament to his profound influence across different areas of differential and algebraic geometry, and its connections to topology, to analysis and to theoretical physics. Simon Kirwan Donaldson was born on August 20th, 1957, in Cambridge, U.K. He attended secondary school at Sevenoaks in Kent, and was a mathematics under- graduate at Pembroke College, Cambridge, before going on to doctoral work under the joint supervision of Michael Atiyah and Nigel Hitchin at Oxford. After his DPhil degree, Donaldson became a Research Fellow at All Souls College, Oxford, and then (with a year at the Institute for Advanced Study in Princeton as inter- mission) the Wallis Professor at Oxford. He remained in Oxford until 1997, then spent one year at Stanford, California, before returning to the U.K. with a Chair at Imperial College, London. In 2014 he joined the Simons Center for Geometry and Physics at Stony Brook, and now divides his time between there and Imperial. Donaldson was an invited speaker at the 1982 ICM in Warsaw, and was awarded the Fields Medal at the 1986 ICM in Berkeley. Amongst his many other awards are the King Faisal International Prize (2006), the Nemmers Prize (2008), the Shaw Prize (2009, joint with Cliff Taubes), and the Breakthrough Prize (2015). He was knighted in the 2012 New Year Honours list for services to mathematics. Whilst still a graduate student, in 1982, Donaldson overturned the world of low-dimensional topology, bringing to bear methods from classical gauge theory and the Yang-Mills equations – ideas later recast by Witten in terms of quantum field theory – to prove new constraints on the topology of smooth four-dimensional manifolds, the nature of which have no analogue in either lower or higher di- mensions. Celebrated results in this period include: the diagonalisability theo- rem1 for the intersection forms of definite four-manifolds; the disproof of the four- dimensional s-cobordism conjecture and introduction of his polynomial invariants of four-manifolds; the Donaldson-Uhlenbeck-Yau (DUY) theorem describing the solutions of the Hermitian-Yang-Mills equations on K¨ahler manifolds; and his work on Nahm’s equations and monopoles. Whilst his work in low-dimensional topology dominated four-manifold theory from 1982–1994, Donaldson later made profound contributions to three quite differ- ent areas. In 1996 he introduced Lefschetz pencils into symplectic topology, proving the first general existence theorem for symplectic hypersurfaces. At the core of this

1The frontispiece to this volume, painted by Nathalie Wahl, merges Simon’s childhood passion for sailing with an abstracted version of the renowned image of the cobordism underlying the diagonalisability theorem. Readers might look for hints of other theorems hidden in the painting!

vii viii PREFACE work is an estimated transversality or quantitative Sard theorem, established via a novel h-principle based on analytical methods of approximately holomorphic geom- etry. In an attempt to fit Floer’s symplectic-geometric invariants into the formalism of topological quantum field theory, in analogy with the expected and known struc- tures for gauge-theoretic Floer homology, he introduced the triangle product in Lagrangian Floer cohomology, and the quantum category of a symplectic manifold – the cohomological version of which became the Fukaya category, central in mirror symmetry. At around the same time, Donaldson laid out a program in higher- dimensional gauge theory suggesting generalisations of both instanton theory and Lagrangian Floer theory to G2 and Spin(7)-manifolds, a program in rapid current development. In the mid 1990s, Donaldson began studying the existence question for con- stant scalar curvature K¨ahler metrics – the higher-dimensional analogue of the constant curvature metrics on Riemann surfaces provided by the uniformisation theorem. Over the following two decades, he introduced a huge array of new ideas into this part of complex differential geometry, partly based on intuitions derived from infinite-dimensional moment maps and ideas around geometric quantisation. He eventually successfully resolved (in 2013, with Xiuxiong Chen and Song Sun) the existence question for K¨ahler-Einstein metrics on Fano manifolds, as conjec- tured by Yau and Tian – a landmark achievement, once again binding together ideas from algebraic geometry and from infinite-dimensional analysis. Whilst the DUY theorem relied essentially on the link between stability of bundles and the existence of special-curvature connections, the results in complex geometry estab- lish a “more non-linear” analogue, reformulating the existence of K¨ahler-Einstein metrics in terms of the stability of the varieties themselves. Donaldson will give the opening lecture at the ICM in Rio in 2018, the 4th ICM which he will address. Donaldson’s influence on mathematics reaches very much further than his body of published results. He has had a huge number of graduate students (44 students and 132 descendents so far, according to the Mathematics Genealogy database). Our own extraordinarily priviliged experiences of being his students were that one was not just given a thesis problem, one was given a whole raft of problems, early entry to an intellectual landscape which other people had scarcely begun to think of populating. Donaldson suggested key examples which paved routes through these uncharted territories and made them familiar, generously leaving the impression one had surveyed and discovered the contours of the theory for oneself. Many people have worked on his suggestions without formally being his students or postdocs: he has always been incredibly generous with his ideas, and equally generous in stepping back from credit. His gentleness and kindness are renowned, and he has been a unique role model to generations of those who have learned from him, listened to his lectures and seminars2, or had the privilege of being party to one of his many informal asides, questions or car-ride reflections. The editors wish to thank Zak Turcinovic for help with the typesetting.

Vicente Mu˜noz, Ivan Smith, Richard Thomas

2In the early 1990s, at his “Geometry and Analysis” seminar at Oxford, instead of inviting a speaker, Donaldson would sometimes talk about a result which excited him, outlining the proof he imagined the author had given. Often this had no resemblance to the actual work, and opened up an entirely new perspective. PUBLISHED TITLES IN THIS SERIES

99 Vicente Mu˜noz, Ivan Smith, and Richard P. Thomas, Editors, Modern Geometry, 2018 96 Si Li, Bong H. Lian, Wei Song, and Shing-Tung Yau, Editors, String-Math 2015, 2017 95 Izzet Coskun, Tommaso de Fernex, and Angela Gibney, Editors, Surveys on Recent Developments in Algebraic Geometry, 2017 94 Mahir Bilen Can, Editor, Algebraic Groups: Structure and Actions, 2017 93 Vincent Bouchard, Charles Doran, Stefan M´endez-Diez, and Callum Quigley, Editors, String-Math 2014, 2016 92 Kailash C. Misra, Daniel K. Nakano, and Brian J. Parshall, Editors, Lie Algebras, Lie Superalgebras, Vertex Algebras and Related Topics, 2016 91 V. Sidoravicius and S. Smirnov, Editors, Probability and Statistical Physics in St. Petersburg, 2016 90 Ron Donagi, Sheldon Katz, Albrecht Klemm, and David R. Morrison, Editors, String-Math 2012, 2015 89 D. Dolgopyat, Y. Pesin, M. Pollicott, and L. Stoyanov, Editors, Hyperbolic Dynamics, Fluctuations and Large Deviations, 2015 88 Ron Donagi, Michael R. Douglas, Ljudmila Kamenova, and Martin Rocek, Editors, String-Math 2013, 2014 87 Helge Holden, Barry Simon, and Gerald Teschl, Editors, Spectral Analysis, Differential Equations and Mathematical Physics: A Festschrift in Honor of Fritz Gesztesy’s 60th Birthday, 2013 86 Kailash C. Misra, Daniel K. Nakano, and Brian J. Parshall, Editors, Recent Developments in Lie Algebras, Groups and Representation Theory, 2012 85 Jonathan Block, Jacques Distler, Ron Donagi, and Eric Sharpe, Editors, String-Math 2011, 2012 84 Alex H. Barnett, Carolyn S. Gordon, Peter A. Perry, and Alejandro Uribe, Editors, Spectral Geometry, 2012 83 Hisham Sati and Urs Schreiber, Editors, Mathematical Foundations of Quantum Field Theory and Perturbative String Theory, 2011 82 MichaelUsher,Editor, Low-dimensional and Symplectic Topology, 2011 81 Robert S. Doran, Greg Friedman, and Jonathan Rosenberg, Editors, Superstrings, Geometry, Topology, and C∗-algebras, 2010 80 D. Abramovich, A. Bertram, L. Katzarkov, R. Pandharipande, and M. Thaddeus, Editors, Algebraic Geometry, 2009 79 Dorina Mitrea and Marius Mitrea, Editors, Perspectives in Partial Differential Equations, Harmonic Analysis and Applications, 2008 78 Ron Y. Donagi and Katrin Wendland, Editors, From Hodge Theory to Integrability and TQFT, 2008 77 Pavel Exner, Jonathan P. Keating, Peter Kuchment, Toshikazu Sunada, and Alexander Teplyaev, Editors, Analysis on Graphs and Its Applications, 2008

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/pspumseries/. PSPUM Modern Geometry • Munoz˜ et al., Editors 99 AMS