Cohomology of the Moduli Space of Cubic Threefolds and Its Smooth Models / Sebastian Casalaina-Martin, [and three others].

Format:

Book

Author/Creator:

Casalaina-Martin, Sebastian, author.

Series:

Memoirs of the American Mathematical Society ; Volume 282.

Memoirs of the American Mathematical Society Series ; Volume 282

Language:

English

Subjects (All):

Cohomology operations.

Moduli theory.

Threefolds (Algebraic geometry).

Physical Description:

1 online resource (112 pages)

Edition:

First edition.

Place of Publication:

Providence, RI : American Mathematical Society, [2023]

Summary:

"We compute and compare the (intersection) cohomology of various natural geometric compactifications of the moduli space of cubic threefolds: the GIT compactification and its Kirwan blowup, as well as the Baily-Borel and toroidal compactifications of the ball quotient model, due to Allcock-Carlson-Toledo. Our starting point is Kirwan's method. We then follow by investigating the behavior of the cohomology under the birational maps relating the various models, using the decomposition theorem in different ways, and via a detailed study of the boundary of the ball quotient model. As an easy illustration of our methods, the simpler case of the moduli space of cubic surfaces is discussed in an appendix"-- Provided by publisher.

Contents:

Cover

Title page

Chapter 1. Introduction

Acknowledgments

Chapter 2. Preliminaries

2.1. Notation and conventions

2.2. Moduli space of cubic threefolds and its standard compactifications \GIT and \BG

2.3. The Kirwan blowup \MK of the moduli space of cubic threefolds

2.4. The toroidal compactification

Chapter 3. The cohomology of the Kirwan blowup, part I: equivariant cohomology of the semi-stable locus

3.1. The equivariantly perfect stratification and the equivariant cohomology of the semi-stable locus in general

3.2. The equivariant cohomology of the locus of semi-stable cubic threefolds

Chapter 4. The cohomology of the Kirwan blowup, part II

4.1. The correction terms in general

4.2. The main correction terms for cubic threefolds

4.3. The extra correction terms for cubic threefolds

4.4. Putting the terms together to compute the cohomology of \calM^{ }

Chapter 5. The intersection cohomology of the GIT moduli space \GIT

5.1. Obtaining the intersection cohomology of the GIT quotient from the cohomology of the Kirwan blowup, in general

5.2. The intersection cohomology of the GIT quotient for cubic threefolds

5.3. Putting the terms together to compute the cohomology of \GIT

5.4. The intersection cohomology of ̂\calM

Chapter 6. The intersection cohomology of the ball quotient

6.1. A special case of the decomposition theorem

6.2. The intersection cohomology of the ball quotient

Chapter 7. The cohomology of the toroidal compactification

7.1. The arithmetic of the two cusps of \calB/Γ

7.2. The cohomology of the toroidal boundary divisors

7.3. The cohomology of the toroidal compactification

Appendix A. Equivariant cohomology

A.1. Review of Atiyah-Bott

A.2. Compact and complex Lie groups

A.3. Kirwan's result for compact groups acting on symplectic manifolds.

A.4. Fibrations

Appendix B. Stabilizers, normalizers, and fixed loci for cubic threefolds

B.1. Connected component \CC*

B.2. Connected component \PGL(2,\CC)

B.3. Connected component (\CC*)²

Appendix C. The moduli space of cubic surfaces

C.1. The moduli space of cubic curves

C.2. The moduli space of cubic surfaces

C.3. The proof of Theorem C.1

C.4. The cohomology of the Naruki compactification

Bibliography

Back Cover.

Notes:

Includes bibliographical references.

Description based on print version record.

Other Format:

Print version: Casalaina-Martin, Sebastian Cohomology of the Moduli Space of Cubic Threefolds and Its Smooth Models

ISBN:

9781470473518

1470473518