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Applications of Canonical Metrics on Berkovich Spaces Applications of Canonical Metrics on Berkovich Spaces by Matthew Stevenson A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in The University of Michigan 2019 Doctoral Committee: Professor Mattias Jonsson, Chair Professor Ratindranath Akhoury Postdoctoral Assistant Professor Eric Canton Professor Mircea Mustat, ˘a Professor Karen Smith Matthew Stevenson [email protected] ORCID iD: 0000-0003-0314-6518 c Matthew Stevenson 2019 All Rights Reserved To my family and teachers ii ACKNOWLEDGEMENTS First and foremost, I would like to thank my advisor Mattias Jonsson for his support and encouragement over the past five years. This thesis is immeasurably stronger because of his influence and suggestions, and it would not have been possible without his guidance. While teaching me the techniques at the core of this work, acting as a sounding board for my ideas, and being the voice of reason in the hills above Bogot´a,you have been an incredible mentor and it has been an absolute pleasure working with you. In addition, I would like to thank the members of my committee: Mircea Mustat, ˘a, Karen Smith, Ratindranath Akhoury, and Eric Canton. I have benefited greatly from classes and mathematical discussions with each of you, and you have been invaluable resources to me during my time at Michigan. Part of this thesis is based on joint work with Mirko Mauri and Enrica Mazzon. I am grateful for our fruitful collaboration. I have learned so much from both of you, and our work together was an exceptionally enjoyable and rewarding experience. Furthermore, this thesis benefitted greatly from conversations with Bhargav Bhatt, S´ebastienBoucksom, Antoine Ducros, Charles Favre, Kiran Kedlaya, Johannes Nicaise, J´er^omePoineau, Daniele Turchetti, Martin Ulirsch, Veronika Wanner, and Tony Yue Yu. Special thanks are due to Thibaud Lemanissier for our collaboration on hybrid analytifications, and to Michael Temkin for patiently explaining aspects of his work on canonical metrics. iii Throughout my time at Michigan, I benefited greatly from mathematical discus- sions with my fellow graduate students including (but not limited to) Harold Blum, Brandon Carter, Rankeya Datta, Yajnaseni Dutta, Jake Levinson, Devlin Mallory, Takumi Murayama, Ashwath Rabindranath, Emanuel Reinecke, and Harry Richman. Particular thanks are due to Takumi, Emanuel, and Enrica, as well as to Mathilde Gerbelli-Gauthier and Olivier Martin, all of whom had a profound impact on my mathematical career. Last but not least, I am thankful to have family, friends, and loved ones who supported and encouraged me throughout this process. I am especially grateful to Samantha for her constant support during the writing of this thesis. This dissertation is based upon work partially supported by NSF grants DMS- 1266207 and DMS-1600011, and ERC Starting Grant MOTZETA (project 306610). iv TABLE OF CONTENTS DEDICATION :::::::::::::::::::::::::::::::::::::::::: ii ACKNOWLEDGEMENTS :::::::::::::::::::::::::::::::::: iii LIST OF FIGURES :::::::::::::::::::::::::::::::::::::: vii ABSTRACT ::::::::::::::::::::::::::::::::::::::::::: viii CHAPTER I. Introduction .......................................1 1.1 Temkin's canonical metric and weight functions................4 1.2 Dual complexes and essential skeletons.....................6 1.2.1 Skeletons over a discretely-valued field................8 1.2.2 Skeletons over a trivially-valued field.................9 1.2.3 The closure of Kontsevich{Soibelman skeletons........... 12 1.3 A non-Archimedean Ohsawa{Takegoshi extension theorem.......... 14 1.3.1 A regularization theorem........................ 16 1.4 New evidence for the geometric P = W conjecture............... 19 II. Preliminaries ....................................... 23 2.1 Conventions.................................... 23 2.2 Berkovich analytifications............................. 24 2.3 Analytic generic fibres............................... 26 2.4 i-analytifications................................. 29 2.5 Models....................................... 29 2.6 Monomial and quasi-monomial points...................... 30 2.7 Gauss extensions.................................. 35 2.8 Logarithmic geometry............................... 36 2.9 Polyhedral complexes............................... 38 III. Canonical metrics on sheaves of differentials ................... 40 3.1 Metrics on non-Archimedean line bundles.................... 40 3.1.1 Definition of a metric on a line bundle................ 40 3.1.2 Metrics on analytifications of line bundles.............. 41 3.2 Weight metrics................................... 43 3.2.1 The weight metric over a discretely-valued field........... 44 3.2.2 The weight metric over a trivially-valued field............ 47 3.2.3 Alternative expressions for the weight function........... 48 3.3 Temkin's metrization of pluricanonical sheaves................. 50 3.3.1 Seminorms on modules of K¨ahlerdifferentials............ 51 v 3.3.2 Temkin's metric............................. 53 3.3.3 Temkin's metric on divisorial points................. 55 3.4 Comparison theorems with Temkin's metric.................. 59 3.4.1 Temkin's comparison theorem with the weight metric........ 59 3.4.2 Divisorial points under Gauss extensions............... 66 3.4.3 Proof of TheoremA.......................... 72 IV. Essential skeletons of pairs .............................. 75 4.1 Skeletons over a discretely-valued field..................... 75 4.2 Skeletons over a trivially-valued field...................... 77 4.2.1 The faces of the skeleton of a log-regular scheme........... 77 4.2.2 The skeleton of a log-regular scheme.................. 80 4.2.3 The retraction to the skeleton..................... 81 4.2.4 Functoriality of the skeleton....................... 84 4.2.5 Comparison with the dual complex.................. 84 4.2.6 The skeleton of a product....................... 86 4.2.7 The Kontsevich{Soibelman and essential skeletons......... 89 4.2.8 Comparison of the trivially-valued and discretely-valued settings. 92 4.3 Closure of the skeleton of a log-regular pair................... 98 4.3.1 The decomposition of the closure of the skeleton.......... 100 4.3.2 The case of the toric varieties..................... 103 4.3.3 The closure of a Kontsevich{Soibelman skeleton........... 106 V. A non-Archimedean Ohsawa{Takegoshi extension theorem ......... 114 5.1 The structure of the Berkovich unit disc.................... 114 5.1.1 The Berkovich unit disc........................ 114 5.1.2 Temkin's metric on the Berkovich unit disc............. 119 5.1.3 Metric structure on the Berkovich unit disc............. 120 5.1.4 Quasisubharmonic functions on the Berkovich unit disc....... 121 5.2 An Ohsawa{Takegoshi-type extension theorem................. 124 5.2.1 Proof of Theorem 5.2.0.1........................ 126 5.3 A non-Archimedean Demailly approximation.................. 136 5.3.1 Construction of the non-Archimedean Demailly approximation.. 138 5.3.2 A regularization theorem........................ 143 5.3.3 Non-Archimedean multiplier ideals.................. 145 VI. On the geometric P=W conjecture ......................... 150 6.1 The geometric P = W conjecture......................... 150 6.2 Dual boundary complex of GLn-character varieties of a genus one surface.. 153 6.2.1 Dlt modifications and dual complexes................. 153 6.2.2 Hilbert scheme of n points of a toric surface............. 155 6.2.3 A proof of Theorem 6.2.0.1 for n = 2.................. 157 6.2.4 The essential skeleton of a logCY pair................ 158 6.2.5 Proof of Theorem 6.2.0.1........................ 163 6.2.6 An alternative proof of Theorem 6.2.0.1................ 165 6.3 Dual boundary complex of SLn-character varieties of a genus one surface.. 170 6.3.1 An alternative proof of Theorem 6.3.0.1............... 173 6.4 Local computations on the Tate curve...................... 176 BIBLIOGRAPHY :::::::::::::::::::::::::::::::::::::::: 182 vi LIST OF FIGURES Figure 3.1 A commutative diagram describing the model associated to the Gauss extension.. 68 3.2 A comparison between the constructions of [BHJ17] and x3.4.2............ 72 4.1 An illustration of Proposition 4.2.8.3 for a Tate elliptic curve............. 97 4.2 A comparison of the decomposition of the extended fan and the closure of the skeleton for a model of P2................................. 106 4.3 An example illustrating that the inclusion of Proposition 4.3.3.1 may be strict... 112 5.1 The Berkovich unit disc over an algebraically closed, spherically complete field, with the radius function shown on the vertical axis................... 116 5.2 A possible configuration for Γ';n in Lemma 5.2.1.5................... 132 vii ABSTRACT This thesis examines the nature of Temkin's canonical metrics on the sheaves of differentials of Berkovich spaces, and discusses 3 applications thereof. First, we show a comparison theorem between Temkin's metric on the i-analytification of a smooth variety over a trivially-valued field of characteristic zero, and a weight metric defined in terms of log discrepancies. This result is the trivially-valued counterpart to a comparison theorem of Temkin between his metric and the weight metric of Mustat, ˘a{Nicaisein the discretely-valued setting. These weight metrics are used to define an essential skeleton of a pair over a trivially-valued field; this is done following the approach of Brown{Mazzon in the discretely-valued case, and we show a compatibility result between the essential skele-
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