Poisson Structures and Lie Algebroids in Complex Geometry by Brent Pym A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics University of Toronto c Copyright 2013 by Brent Pym Abstract Poisson Structures and Lie Algebroids in Complex Geometry Brent Pym Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2013 This thesis is devoted to the study of holomorphic Poisson structures and Lie algebroids, and their relationship with differential equations, singularity theory and noncommutative algebra. After reviewing and developing the basic theory of Lie algebroids in the framework of complex analytic and algebraic geometry, we focus on Lie algebroids over complex curves and their application to the study of meromorphic connections. We give concrete construc- tions of the corresponding Lie groupoids, using blowups and the uniformization theorem. These groupoids are complex surfaces that serve as the natural domains of definition for the fundamental solutions of ordinary differential equations with singularities. We explore the relationship between the convergent Taylor expansions of these fundamental solutions and the divergent asymptotic series that arise when one attempts to solve an ordinary differential equation at an irregular singular point. We then turn our attention to Poisson geometry. After discussing the basic structure of Poisson brackets and Poisson modules on analytic spaces, we study the geometry of the degeneracy loci|where the dimension of the symplectic leaves drops. We explain that Pois- son structures have natural residues along their degeneracy loci, analogous to the Poincar´e residue of a meromorphic volume form. We discuss the local structure of degeneracy loci that have small codimensions, and place strong constraints on the singularities of the degen- eracy hypersurfaces of log symplectic manifolds. We use these results to give new evidence for a conjecture of Bondal. Finally, we discuss the problem of quantization in noncommutative projective geometry. Using Cerveau and Lins Neto's classification of degree-two foliations of projective space, ii we give normal forms for unimodular quadratic Poisson structures in four dimensions, and describe the quantizations of these Poisson structures to noncommutative graded algebras. As a result, we obtain a (conjecturally complete) list of families of quantum deformations of projective three-space. Among these algebras is an \exceptional" one, associated with a twisted cubic curve. This algebra has a number of remarkable properties: for example, it supports a family of bimodules that serve as quantum analogues of the classical Schwarzen- berger bundles. iii Acknowledgements First and foremost, I wish to thank my thesis advisor and mentor, Marco Gualtieri. Marco's contagious passion for mathematics and his enthusiasm for teaching made our extended weekly meetings both exciting and memorable. I am deeply grateful for his countless efforts on my behalf, and for his apparently inexhaustible supply of advice and inspiration. I am grateful to several other faculty members in the Department of Mathematics, no- tably Lisa Jeffrey, Joel Kamnitzer, Yael Karshon, Boris Khesin, Eckhard Meinrenken and Michael Sigal, for their support and guidance. I am particularly indebted to Ragnar-Olaf Buchweitz, who patiently explained various aspects of algebra and singularity theory to me. I would also like to thank Michael Bailey, Alejandro Cabrera, Eleonore Faber, Jonathan Fisher, Eric Hart, Songhao Li, David Li-Bland, Brian Pike, Steven Rayan, Daniel Rowe and Jordan Watts for many hours of stimulating discussions. Special thanks are due to Nikita Nikolaev, who notified me of several typographical errors in an early draft of this thesis. The excellent administrative staff in the department have made it a very pleasant and efficient workplace. I am particularly thankful to Donna Birch, Ida Bulat, Jemima Mersica, and Patrina Seepersaud for their help on numerous occasions. Outside the University of Toronto, I have learned a great deal from conversations with many mathematicians, including Philip Boalch, Raf Bocklandt, William Graham, Nigel Hitchin, Jacques Hurtubise, Colin Ingalls, Jiang-Hua Lu, Alexander Odesskii, Alexander Polishchuk, Daniel Rogalski, Travis Schedler, Toby Stafford and Michel Van den Bergh. I am especially thankful to Colin Ingalls, who provided very helpful comments on Chapter 8, and Travis Schedler, who did the same for the paper [73]. Much of the work described in Chapter 8 occurred during visits to the Mathematical Sciences Research Institute (MSRI) for workshops associated with the program on Noncom- mutative Algebraic Geometry and Representation Theory. I would like to thank the MSRI and the organizers of the program for supporting my visits financially and for providing a highly invigorating working environment. My research was made possible, in part, by fellowships from the University of Toronto and the Natural Sciences and Engineering Research Council of Canada. I am very grateful for their generosity. Finally, I wish to thank my family and friends for their love and support|particularly my partner, Natalie Symchych, whose unwavering patience and encouragement made the completion of this thesis possible. iv Contents 1 Introduction1 1.1 Motivation.....................................1 1.2 The Example....................................2 1.3 Guiding principles.................................5 1.4 Summary of the thesis...............................6 2 Lie algebroids in complex geometry8 2.1 Preliminaries....................................8 2.1.1 Analytic spaces...............................8 2.1.2 Holomorphic vector bundles and sheaves................. 11 2.1.3 Calculus on analytic spaces........................ 14 2.2 Lie algebroids.................................... 15 2.3 Lie algebroids associated with hypersurfaces................... 18 2.4 Lie algebroid modules............................... 21 2.5 Lie algebroid cohomology............................. 24 2.5.1 The Picard group.............................. 26 2.6 The universal envelope, jets and higher order connections........... 28 2.7 Holomorphic Lie groupoids............................ 30 2.7.1 Integration and source-simply connected groupoids........... 34 2.8 Groupoids in analytic spaces........................... 35 2.9 The question of algebraicity............................ 37 3 Lie theory on curves and meromorphic connections 40 3.1 Invitation: divergent series and the Stokes phenomenon............ 40 3.2 Lie algebroids on curves.............................. 44 3.2.1 Basic properties.............................. 44 3.2.2 Higher order connections and singular differential equations...... 46 3.2.3 Meromorphic projective structures.................... 49 3.3 Lie groupoids on curves.............................. 50 v 3.3.1 Motivation: integration of representations................ 50 3.3.2 Blowing up: the adjoint groupoids.................... 51 3.3.3 Examples on P1 ............................... 52 3.3.4 Uniformization............................... 57 3.4 Local normal forms................................. 64 3.4.1 Local normal form for twisted pair groupoids.............. 64 3.4.2 Source-simply connected case....................... 66 3.5 Summation of divergent series........................... 68 4 Poisson structures in complex geometry 73 4.1 Multiderivations.................................. 73 4.2 Poisson brackets.................................. 75 4.3 Poisson subspaces.................................. 78 4.4 Poisson hypersurfaces and log symplectic structures.............. 82 4.5 Lie algebroids and symplectic leaves....................... 84 5 Geometry of Poisson modules 87 5.1 Poisson modules.................................. 87 5.2 Pushforward of Poisson modules......................... 89 5.3 Restriction to subspaces: Higgs fields and adaptedness............. 90 5.4 Lie algebroids associated with Poisson modules................. 94 5.5 Residues of Poisson line bundles......................... 95 5.6 Modular residues.................................. 99 6 Degeneracy loci 102 6.1 Motivation: Bondal's conjecture.......................... 102 6.2 Degeneracy loci in algebraic geometry...................... 104 6.3 Degeneracy loci of Lie algebroids......................... 106 6.4 Degeneracy loci of Poisson structures....................... 107 6.5 Degeneracy loci of Poisson modules........................ 110 6.6 Structural results in small codimension...................... 112 6.6.1 Codimension one: log symplectic singularities.............. 112 6.6.2 Codimension two: degeneracy in odd dimension............. 114 6.6.3 Codimension three: submaximal degeneracy in even dimension.... 115 6.7 Non-emptiness via Chern classes......................... 115 6.8 Fano manifolds................................... 119 vi 7 Poisson structures on projective spaces 122 7.1 Review of Poisson cohomology.......................... 122 7.2 Multivector fields in homogeneous coordinates.................. 123 7.2.1 Helmholtz decomposition on vector spaces................ 124 7.2.2 Multivector fields on projective space.................. 125 7.2.3 A comparison theorem for quadratic Poisson structures........ 128 7.3 Projective embedding............................... 130 7.4 Poisson structures on Pn admitting a normal crossings anticanonical divisor. 132 7.5 Poisson structures associated with linear free divisors.............