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Moduli of K-stable Varieties

Author: Giulio Codogni

Publisher: Springer

Published: 2019-06-27

Total Pages: 181

ISBN-13: 3030131580

This volume is an outcome of the workshop "Moduli of K-stable Varieties", which was held in Rome, Italy in 2017. The content focuses on the existence problem for canonical Kähler metrics and links to the algebro-geometric notion of K-stability. The book includes both surveys on this problem, notably in the case of Fano varieties, and original contributions addressing this and related problems. The papers in the latter group develop the theory of K-stability; explore canonical metrics in the Kähler and almost-Kähler settings; offer new insights into the geometric significance of K-stability; and develop tropical aspects of the moduli space of curves, the singularity theory necessary for higher dimensional moduli theory, and the existence of minimal models. Reflecting the advances made in the area in recent years, the survey articles provide an essential overview of many of the most important findings. The book is intended for all advanced graduate students and researchers who want to learn about recent developments in the theory of moduli space, K-stability and Kähler-Einstein metrics.

Moduli of K-stable Varieties

Author: Giulio Codogni

Publisher:

Published: 2019

Total Pages: 181

ISBN-13: 9783030131593

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This volume is an outcome of the workshop "Moduli of K-stable Varieties", which was held in Rome, Italy in 2017. The content focuses on the existence problem for canonical Kèahler metrics and links to the algebro-geometric notion of K-stability. The book includes both surveys on this problem, notably in the case of Fano varieties, and original contributions addressing this and related problems. The papers in the latter group develop the theory of K-stability; explore canonical metrics in the Kèahler and almost-Kèahler settings; offer new insights into the geometric significance of K-stability; and develop tropical aspects of the moduli space of curves, the singularity theory necessary for higher dimensional moduli theory, and the existence of minimal models. Reflecting the advances made in the area in recent years, the survey articles provide an essential overview of many of the most important findings. The book is intended for all advanced graduate students and researchers who want to learn about recent developments in the theory of moduli space, K-stability and Kèahler-Einstein metrics.

Families of Varieties of General Type

Author: János Kollár

Publisher: Cambridge University Press

Published: 2023-04-30

Total Pages: 491

ISBN-13: 1009346105

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The first complete treatment of the moduli theory of varieties of general type, laying foundations for future research.

Birational Geometry, Kähler–Einstein Metrics and Degenerations

Author: Ivan Cheltsov

Publisher: Springer Nature

Published: 2023-05-23

Total Pages: 882

ISBN-13: 3031178599

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This book collects the proceedings of a series of conferences dedicated to birational geometry of Fano varieties held in Moscow, Shanghai and Pohang The conferences were focused on the following two related problems: • existence of Kähler–Einstein metrics on Fano varieties • degenerations of Fano varieties on which two famous conjectures were recently proved. The first is the famous Borisov–Alexeev–Borisov Conjecture on the boundedness of Fano varieties, proved by Caucher Birkar (for which he was awarded the Fields medal in 2018), and the second one is the (arguably even more famous) Tian–Yau–Donaldson Conjecture on the existence of Kähler–Einstein metrics on (smooth) Fano varieties and K-stability, which was proved by Xiuxiong Chen, Sir Simon Donaldson and Song Sun. The solutions for these longstanding conjectures have opened new directions in birational and Kähler geometries. These research directions generated new interesting mathematical problems, attracting the attention of mathematicians worldwide. These conferences brought together top researchers in both fields (birational geometry and complex geometry) to solve some of these problems and understand the relations between them. The result of this activity is collected in this book, which contains contributions by sixty nine mathematicians, who contributed forty three research and survey papers to this volume. Many of them were participants of the Moscow–Shanghai–Pohang conferences, while the others helped to expand the research breadth of the volume—the diversity of their contributions reflects the vitality of modern Algebraic Geometry.

Moduli Theory and Classification Theory of Algebraic Varieties

Author: H. Popp

Publisher: Springer

Published: 2006-11-15

Total Pages: 196

ISBN-13: 3540370315

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Modern Geometry: A Celebration of the Work of Simon Donaldson

Author: Vicente Muñoz

Publisher: American Mathematical Soc.

Published: 2018-09-05

Total Pages: 416

ISBN-13: 1470440946

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This book contains a collection of survey articles of exciting new developments in geometry, written in tribute to Simon Donaldson to celebrate his 60th birthday. Reflecting the wide range of Donaldson's interests and influence, the papers range from algebraic geometry and topology through symplectic geometry and geometric analysis to mathematical physics. Their expository nature means the book acts as an invitation to the various topics described, while also giving a sense of the links between these different areas and the unity of modern geometry.

Moduli of Weighted Hyperplane Arrangements

Author: Valery Alexeev

Publisher: Birkhäuser

Published: 2015-05-18

Total Pages: 112

ISBN-13: 3034809158

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This book focuses on a large class of geometric objects in moduli theory and provides explicit computations to investigate their families. Concrete examples are developed that take advantage of the intricate interplay between Algebraic Geometry and Combinatorics. Compactifications of moduli spaces play a crucial role in Number Theory, String Theory, and Quantum Field Theory – to mention just a few. In particular, the notion of compactification of moduli spaces has been crucial for solving various open problems and long-standing conjectures. Further, the book reports on compactification techniques for moduli spaces in a large class where computations are possible, namely that of weighted stable hyperplane arrangements (shas).

Complex and Symplectic Geometry

Author: Daniele Angella

Publisher: Springer

Published: 2017-10-12

Total Pages: 262

ISBN-13: 331962914X

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This book arises from the INdAM Meeting "Complex and Symplectic Geometry", which was held in Cortona in June 2016. Several leading specialists, including young researchers, in the field of complex and symplectic geometry, present the state of the art of their research on topics such as the cohomology of complex manifolds; analytic techniques in Kähler and non-Kähler geometry; almost-complex and symplectic structures; special structures on complex manifolds; and deformations of complex objects. The work is intended for researchers in these areas.

Stacks Project Expository Collection

Author: Pieter Belmans

Publisher: Cambridge University Press

Published: 2022-09-30

Total Pages: 308

ISBN-13: 1009063286

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The Stacks Project Expository Collection (SPEC) compiles expository articles in advanced algebraic geometry, intended to bring graduate students and researchers up to speed on recent developments in the geometry of algebraic spaces and algebraic stacks. The articles in the text make explicit in modern language many results, proofs, and examples that were previously only implicit, incomplete, or expressed in classical terms in the literature. Where applicable this is done by explicitly referring to the Stacks project for preliminary results. Topics include the construction and properties of important moduli problems in algebraic geometry (such as the Deligne–Mumford compactification of the moduli of curves, the Picard functor, or moduli of semistable vector bundles and sheaves), and arithmetic questions for fields and algebraic spaces.

Algebraic Geometry: Salt Lake City 2015 (Part 1)

Author: Tommaso de Fernex

Publisher: American Mathematical Soc.

Published: 2018-06-01

Total Pages: 655

ISBN-13: 1470435772

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This is Part 1 of a two-volume set. Since Oscar Zariski organized a meeting in 1954, there has been a major algebraic geometry meeting every decade: Woods Hole (1964), Arcata (1974), Bowdoin (1985), Santa Cruz (1995), and Seattle (2005). The American Mathematical Society has supported these summer institutes for over 50 years. Their proceedings volumes have been extremely influential, summarizing the state of algebraic geometry at the time and pointing to future developments. The most recent Summer Institute in Algebraic Geometry was held July 2015 at the University of Utah in Salt Lake City, sponsored by the AMS with the collaboration of the Clay Mathematics Institute. This volume includes surveys growing out of plenary lectures and seminar talks during the meeting. Some present a broad overview of their topics, while others develop a distinctive perspective on an emerging topic. Topics span both complex algebraic geometry and arithmetic questions, specifically, analytic techniques, enumerative geometry, moduli theory, derived categories, birational geometry, tropical geometry, Diophantine questions, geometric representation theory, characteristic and -adic tools, etc. The resulting articles will be important references in these areas for years to come.