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Okounkov bodies and geodesic rays in Kähler geometry · PDF file Okounkov bodies and geodesic rays in Kähler geometry David Witt Nyström ABSTRACT This thesis presents three papers

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  • THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

    Okounkov bodies and geodesic rays in Kähler geometry

    DAVID WITT NYSTRÖM

    Division of Mathematics

    Department of Mathematical Sciences

    CHALMERS UNIVERSITY OF TECHNOLOGY AND

    UNIVERSITY OF GOTHENBURG Göteborg, Sweden 2012

  • Okounkov bodies and geodesic rays in Kähler geometry

    David Witt Nyström

    ISBN 978-91-628-8478-9

    c© David Witt Nyström, 2012.

    Department of Mathematical Sciences Chalmers University of Technology and University of Gothenburg SE-412 96 GÖTEBORG, Sweden Phone: +46 (0)31-772 10 00

    Author e-mail: [email protected]

    Printed in Göteborg, Sweden, 2012

  • Okounkov bodies and geodesic rays in Kähler geometry David Witt Nyström

    ABSTRACT This thesis presents three papers dealing with questions in Kähler geometry.

    In the first paper we construct a transform, called the Chebyshev transform, which

    maps continuous hermitian metrics on a big line bundle to convex functions on the asso-

    ciated Okounkov body. We show that this generalizes the classical Legendre transform in

    convex and toric geometry, and also Chebyshev constants in pluripotential theory. Our

    main result is that the integral of the difference of two transforms over the Okounkov

    body is equal to the Monge-Ampère energy of the two metrics. The Monge-Ampère

    energy, sometimes also called the Aubin-Mabuchi energy or the Aubin-Yau functional,

    is a well-known functional in Kähler geometry; it is the primitive function to the Monge-

    Ampère operator. As a special case we get that the weighted transfinite diameter is equal

    to the mean over the unit simplex of the weighted directional Chebyshev constants. As

    an application we prove the differentiability of the Monge-Ampère on the ample cone,

    extending previous work by Berman-Boucksom.

    In the second paper we associate to a test configuration for a polarized variety a

    filtration of the section ring of the line bundle. Using the recent work of Boucksom-Chen

    we get a concave function on the Okounkov body whose law with respect to Lebesgue

    measure determines the asymptotic distribution of the weights of the test configuration.

    We show that this is a generalization of a well-known result in toric geometry.

    In the third paper, starting with the data of a curve of singularity types, we use the

    Legendre transform to construct weak geodesic rays in the space of positive singular

    metrics on an ample line bundle L. Using this we associate weak geodesics to suitable

    filtrations of the algebra of sections of L. In particular this works for the natural filtration

    coming from an algebraic test configuration, and we show how this in the non-trivial case

    recovers the weak geodesic ray of Phong-Sturm.

    Keywords: ample line bundles, Okounkov bodies, Monge-Ampère operator, Legendre

    transform, Chebyshev constants, test configurations, weak geodesic rays.

  • ii

  • Preface

    This thesis consists of an introduction and the following papers.

    • David Witt Nyström, Transforming metrics on a line bundle to the Okounkov body, submitted.

    • David Witt Nyström, Test configurations and Okounkov bodies, accepted for publication in Compositio Mathematica.

    • Julius Ross and David Witt Nyström, Analytic test configurations and geodesic rays, submitted.

    In order not to loose focus, the following paper is not included in this thesis.

    • Robert Berman, Sebastien Boucksom and David Witt Nyström, Fekete points and convergence towards equilibrium measure on

    complex manifolds, Acta Mathematica 207 (2011), no. 1, 1-27.

    iii

  • iv PREFACE

  • Acknowledgements

    First of all I would like to thank my thesis advisor Robert Berman. His en- thusiasm and love for mathematics has inspired my throughout this time. Also importantly, Robert has shown that you don’t have to sacrifice your family life in order to become a brilliant mathematician. I feel extremely grateful for ev- erything he has done for me, which is a lot.

    Secondly I would like to thank my co-advisor, Robert’s wingman, Bo Berndts- son. Bo was the reason I chose to specialize in complex analysis/geometry in the first place, and in hindsight it was the perfect choice. A very substantial part of my knowledge in this field I have him to thank for.

    My third big thank goes to my co-author Julius Ross in Cambridge. I know that we will continue writing papers together, in fact we are right now working on a new exciting thing, but that is a different story...

    I would also like to thank Sebastien Boucksom for his support and encour- agement.

    Thanks to all my friendly collegues at the department of mathematics in Gothenburg. In particular the other senior complex analysts: Elizabeth, Håkan och Mats, and my fellow graduate students: Ragnar, Martin, Johannes, Jacob, Rickard, Hossein, Oscar, Peter and Dawan to name a few. Thank you Erik for the kodak moments on the tennis court.

    A huge thank you goes to Aron. It’s a privilege to get to work with such a good friend.

    I would also like to thank all my non-mathematical friends. Especially the Java gang with additions, who has been there for me always.

    v

  • vi ACKNOWLEDGEMENTS

    Thank you mum and dad for always encouraging me. A whole bunch of thank yous to Tomas, to my brothers Joel and Leonard,

    to Carina, William and Axel, to Moa, Morgan and Alice, and to Ann-Katrin, Henrik, Olof and Ylva, who taken all together make up my extended family.

    No words that I know of are able to express the intensity of gratitude and love I feel for my own little family, Johanna and Julian, so these have to suffice: I love you.

    David Witt Nyström Göteborg, March 2012

  • Contents

    Abstract i

    Preface iii

    Acknowledgements v

    I INTRODUCTION 1

    0 INTRODUCTION 3 0.1 Kähler geometry . . . . . . . . . . . . . . . . . . . . . . . . . 6

    0.1.1 Projective manifolds . . . . . . . . . . . . . . . . . . . 7

    0.1.2 Holomorphic functions . . . . . . . . . . . . . . . . . . 8

    0.1.3 Line bundles and sections . . . . . . . . . . . . . . . . 9

    0.1.4 Chern classes, self-intersection and volume . . . . . . . 10

    0.1.5 The Brunn-Minkowski inequality . . . . . . . . . . . . 12

    0.1.6 Okounkov bodies . . . . . . . . . . . . . . . . . . . . . 13

    0.1.7 Toric geometry and moment polytopes . . . . . . . . . . 15

    0.1.8 Symplectic geometry and moment maps . . . . . . . . . 17

    0.1.9 Plurisubharmonic functions . . . . . . . . . . . . . . . 18

    0.1.10 Hermitian metrics on line bundles . . . . . . . . . . . . 19

    0.1.11 The moment polytope revisited . . . . . . . . . . . . . 20

    0.1.12 The Monge-Ampère energy . . . . . . . . . . . . . . . 22

    0.1.13 The real Monge-Ampère operator . . . . . . . . . . . . 23

    vii

  • viii CONTENTS

    0.1.14 The Legendre transform . . . . . . . . . . . . . . . . . 24

    0.1.15 Symplectic potentials . . . . . . . . . . . . . . . . . . . 26

    0.2 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

    0.2.1 Capacity, transfinite diameter and Chebyshev constants . 29

    0.2.2 The Chebyshev transform . . . . . . . . . . . . . . . . 34

    0.2.3 Proof of main theorem . . . . . . . . . . . . . . . . . . 35

    0.2.4 Differentiability of Monge-Ampère energy . . . . . . . 37

    0.3 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

    0.3.1 The Yau-Tian-Donaldson conjecture . . . . . . . . . . . 38

    0.3.2 Test configurations . . . . . . . . . . . . . . . . . . . . 39

    0.3.3 Toric test configurations . . . . . . . . . . . . . . . . . 40

    0.3.4 Filtrations of the section ring . . . . . . . . . . . . . . . 42

    0.3.5 The concave transform of a test configuration . . . . . . 43

    0.3.6 Product test configurations and geodesic rays . . . . . . 45

    0.4 Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

    0.4.1 The Mabuchi K-energy . . . . . . . . . . . . . . . . . . 46

    0.4.2 Geodesic rays . . . . . . . . . . . . . . . . . . . . . . . 48

    0.4.3 Phong-Sturm rays . . . . . . . . . . . . . . . . . . . . 49

    0.4.4 The Legendre transform . . . . . . . . . . . . . . . . . 50

    0.4.5 Analytic test configurations . . . . . . . . . . . . . . . 52

    0.4.6 Maximal envelopes . . . . . . . . . . . . . . . . . . . . 54

    0.4.7 Proof of main theorem . . . . . . . . . . . . . . . . . . 55

    0.4.8 Connection to the work of Phong-Sturm . . . . . . . . . 56

    Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

    II PAPERS 61

    1 PAPER I 65 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

    1.1.1 Organization . . . . . . . . . . . . . . . . . . . . . . . 72

    1.1.2 Acknowledgement . . . . . . . . . . . . . . . . . . . . 73

    1.2 The Okounkov body of a semigroup . . . . . . . . . . . . . . . 74

  • CONTENTS ix

    1.3 Subadditive functions on semigroups . . . . . . . . . . . . . . . 75

    1.4 The Okounkov body of a line bundle . . . . . . . . . . . . . . . 83

    1.5 The Chebyshev transform . . . . . . . . . . . . . . . . . . . . . 85

    1.6 The Monge-Ampère energy of weights . . . . . . . . . . . . . . 90

    1.7

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