Topics in Complex & Hypercomplex Geometry
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The IUM Report to the Simons Foundation, 2016
The IUM report to the Simons foundation, 2016 Contents 1 Introduction: list of awardees 4 2 Program: Simons stipends for students and graduate students 6 2.1 Research . 6 2.1.1 Elena Egorova . 6 2.1.2 Yakov Kononov . 7 2.1.3 Konstantin Loginov . 9 2.1.4 Maria Matushko . 9 2.1.5 Dmitry Oganesyan . 9 2.1.6 Vardan Oganesyan . 10 2.1.7 Pavel Sechin . 10 2.1.8 Danila Zaev . 11 2.1.9 Dmitry Zubov . 11 2.2 Scientific conferences and seminar talks . 12 2.2.1 Elena Egorova . 12 2.2.2 Yakov Kononov . 13 2.2.3 Konstantin Loginov . 13 2.2.4 Maria Matushko . 13 2.2.5 Dmitry Oganesyan . 14 2.2.6 Vardan Oganesyan . 14 2.2.7 Pavel Sechin . 14 2.2.8 Danila Zaev . 15 2.2.9 Dmitry Zubov . 15 3 Program: Simons IUM fellowships 16 3.1 Research . 16 3.1.1 Anton Aizenberg . 16 3.1.2 Alexander Belavin . 18 3.1.3 Alexei Elagin . 18 3.1.4 Alexei Gorodentsev . 19 1 3.1.5 Maxim Kazarian . 19 3.1.6 Anton Khoroshkin . 19 3.1.7 Iosif Krasilshchik . 20 3.1.8 Alexander Kuznetsov . 21 3.1.9 Maxim Leyenson . 24 3.1.10 Grigory Olshanski . 24 3.1.11 Alexei Penskoi . 25 3.1.12 Petr Pushkar' . 26 3.1.13 Sergey Rybakov . 26 3.1.14 George Shabat . 26 3.1.15 Arkady Skopenkov . 27 3.1.16 Mikhail Skopenkov . 31 3.1.17 Evgeni Smirnov . 32 3.1.18 Mikhail Verbitsky . -
Complex Geometry of Moment-Angle Manifolds
COMPLEX GEOMETRY OF MOMENT-ANGLE MANIFOLDS TARAS PANOV, YURY USTINOVSKIY, AND MISHA VERBITSKY Abstract. Moment-angle manifolds provide a wide class of examples of non- K¨ahler compact complex manifolds. A complex moment-angle manifold Z is constructed via certain combinatorial data, called a complete simplicial fan. In the case of rational fans, the manifold Z is the total space of a holomorphic bundle over a toric variety with fibres compact complex tori. In general, a complex moment-angle manifold Z is equipped with a canonical holomorphic foliation F which is equivariant with respect to the (C×)m-action. Examples of moment-angle manifolds include Hopf manifolds of Vaisman type, Calabi– Eckmann manifolds, and their deformations. We construct transversely K¨ahler metrics on moment-angle manifolds, un- der some restriction on the combinatorial data. We prove that any K¨ahler submanifold (or, more generally, a Fujiki class C subvariety) in such a moment- angle manifold is contained in a leaf of the foliation F. For a generic moment- angle manifold Z in its combinatorial class, we prove that all subvarieties are moment-angle manifolds of smaller dimension and there are only finitely many of them. This implies, in particular, that the algebraic dimension of Z is zero. Contents 1. Introduction 2 1.1. Non-K¨ahler geometry 2 1.2. Positive (1,1)-forms and currents on non-K¨ahler manifolds 3 1.3. Hopf manifolds and Vaisman manifolds 3 1.4. Moment-angle manifolds and their geometry 4 2. Basic constructions 6 3. Complex structures 10 4. Submanifolds, analytic subsets and meromorphic functions 12 4.1. -
The Holomorphic Σ-Model and Its Symmetries
The holomorphic s-model and its symmetries Brian R. Williams Contents 1 Introduction 3 1.1 Summary of the results...................................5 1.2 Overview...........................................5 2 Holomorphic quantum field theory7 2.1 The definition of a quantum field theory.........................9 2.1.1 Classical field theory................................9 2.1.2 Quantum field theory............................... 17 2.2 Holomorphic field theories................................ 21 2.2.1 The definition of a holomorphic theory..................... 22 2.2.2 Holomorphically translation invariant theories................. 29 2.3 Renormalization of holomorphic theories........................ 33 2.3.1 Holomorphic gauge fixing............................. 34 2.3.2 One-loop weights.................................. 36 2.3.3 A general result about chiral anomalies..................... 41 2.4 Equivariant BV quantization................................ 46 2.4.1 Classical equivariance............................... 48 2.4.2 Quantum equivariance............................... 48 2.4.3 The case of a local Lie algebra........................... 49 3 The holomorphic s-model 52 3.1 Gelfand-Kazhdan formal geometry............................ 55 3.1.1 A Harish-Chandra pair for the formal disk................... 55 3.1.2 Formal coordinates................................. 57 3.1.3 The category of formal vector bundles...................... 61 3.1.4 Gelfand-Kazhdan descent............................. 63 3.1.5 Formal characteristic classes........................... -
The Holomorphic Σ-Model and Its Symmetries
NORTHWESTERN UNIVERSITY The Holomorphic σ-model and its Symmetries A DISSERTATION SUBMITTED TO THE GRADUATE SCHOOL IN PARTIAL FULFILLMENT OF THE REQUIREMENTS for the degree DOCTOR OF PHILOSOPHY Field of Mathematics By Brian R. Williams EVANSTON, ILLINOIS June 2018 2 c Copyright by Brian R. Williams 2018 All Rights Reserved 3 ABSTRACT The Holomorphic σ-model and its Symmetries Brian R. Williams The holomorphic σ-model is a field theory that exists in any complex dimension that describes the moduli space of holomorphic maps from one complex manifold to another. We introduce the general notion of a holomorphic field theory, which is one that is sensi- tive to the underlying complex structure of the spacetime manifold (and potentially other geometric input data). Throughout, we rigorously study perturbative quantum field the- ory in a way that combines the Batalin-Vilkovisky formalism and the effective approach to renormalization. In addition to computing the one-loop anomaly of the holomorphic σ-model, we study the local operators of the theory using factorization algebras and com- pute the local index. The final part of this thesis investigates the symmetries present in a general holomorphic theories. We characterize the local central extensions of the two fundamental symmetry algebras, which provide higher dimensional generalizations of the Kac-Moody and Virasoro vertex algebras. 4 Acknowledgements My journey to pursue mathematics started well before entering the Ph.D. program at Northwestern. Throughout this, I've relied on the constant support of my family, especially my parents. I thank my mother for being a great listener and for giving me the freedom to follow my passions. -
Complex Geometry (Pdf)
Complex Geometry George Torres Fall 2020 Prof. Bernd Siebert 1 Introduction 3 2 Local Theory 4 2.1 Holomorphic functions of several variables . .4 2.2 Complex Vector Spaces . .9 n 2.3 The local ring OC ;0 .............................................9 2.4 Inverse and Implicit Function Theorems . 11 2.5 Analytic Sets: Local Finite Mappings and Dimension . 12 3 Complex Manifolds 15 3.1 Projective Space . 15 3.2 Non Algebraic Manifolds . 16 3.3 Complex Tori . 18 3.4 Holomorphic Group Actions . 20 3.5 Calculus on Complex Manifolds . 23 4 Vector Bundles and Sheaves 24 4.1 Holomorphic Vector Bundles . 24 4.2 Sheaves . 26 4.3 Sheaf Cohomology . 28 4.4 Divisors and Line Bundles . 29 4.5 Linear Systems and Maps to Pn ...................................... 30 5 Kähler Geometry 31 5.1 Hermitian Manifolds . 31 5.2 Kähler Manifolds . 33 5.3 The Kähler Identities . 35 5.4 Hodge Theory . 36 5.5 The @@¯-Lemma . 39 5.6 The Hard Lefschetz Theorem . 39 5.7 Formality of Kähler Manifolds . 41 6 Chern-Weil Theory 43 6.1 Connections and Curvature . 43 6.2 Chern Classes . 45 6.3 Interpretations of c1 on line bundles . 45 6.4 The Hirzebruch-Riemann-Roch Formula . 45 Last updated December 13, 2020 1 These are lecture notes from the Fall 2020 course M392C Complex Geometry at the University of Texas at Austin taught by Prof. Bernd Seibert. The prerequisites for following these notes is single variable complex analysis, manifold theory, (e.g. from Guillemin and Pollack), and some familiarity with commutative alge- bra, algebraic geometry, and sheaf theory.