K-stability of Fano Varieties

Chenyang Xu

doi:10.1017/9781009538763

The K-stability of Fano varieties has been a major area of research over the last decade, ever since the Yau-Tian-Donaldson conjecture was resolved. This is the first book to give a comprehensive algebraic treatment of this emerging field. It introduces all the notions of K-stability that have been used over the development of the subject, proves their equivalence, and discusses newly developed theory, including several new proofs for existing theorems. Aiming to be as self-contained as possible, the text begins with a chapter covering essential background knowledge, and includes exercises throughout to test understanding. Written by an author at the forefront of developments in the area, it will be a source of inspiration for graduate students and researchers who work in algebraic geometry.

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Contents

Frontmatter
pp i-iv
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Contents
pp v-vi
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Preface
pp vii-viii
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Notation and Conventions
pp ix-xii
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Preliminaries
pp 1-11
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1 - Higher Dimensional Geometry Background
pp 12-53
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2 - K-stability via Test Configurations
pp 54-91
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3 - K-stability via Filtrations
pp 92-137
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4 - K-stability via Valuations
pp 138-206
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5 - Higher Rank Finite Generation
pp 207-237
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6 - Reduced Stability
pp 238-259
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7 - K-moduliStack
pp 260-287
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8 - K-moduli Space
pp 288-322
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9 - Positivity of the CM Line Bundle
pp 323-364
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Appendix A - Solutions to Exercises
pp 365-392
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Glossary
pp 393-398
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Bibliography
pp 399-408
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Index
pp 409-411
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K-stability of Fano Varieties

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Contents

Frontmatter
pp i-iv

Contents
pp v-vi

Preface
pp vii-viii

Notation and Conventions
pp ix-xii

Preliminaries
pp 1-11

1 - Higher Dimensional Geometry Background
pp 12-53

2 - K-stability via Test Configurations
pp 54-91

3 - K-stability via Filtrations
pp 92-137

4 - K-stability via Valuations
pp 138-206

5 - Higher Rank Finite Generation
pp 207-237

6 - Reduced Stability
pp 238-259

7 - K-moduliStack
pp 260-287

8 - K-moduli Space
pp 288-322

9 - Positivity of the CM Line Bundle
pp 323-364

Appendix A - Solutions to Exercises
pp 365-392

Glossary
pp 393-398

Bibliography
pp 399-408

Index
pp 409-411

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