Memoirs of the American Mathematical Society Series ; Volume 282 [More in this series]
Summary note
"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.
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Includes bibliographical references.
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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.
ISBN
1-4704-7351-8
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