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Cohomology of the Moduli Space of Cubic Threefolds and Its Smooth Models
About this Title
Sebastian Casalaina-Martin, Samuel Grushevsky, Klaus Hulek and Radu Laza
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 282, Number 1395
ISBNs: 978-1-4704-6020-4 (print); 978-1-4704-7351-8 (online)
DOI: https://doi.org/10.1090/memo/1395
Published electronically: January 3, 2023
Table of Contents
Chapters
- 1. Introduction
- 2. Preliminaries
- 3. The cohomology of the Kirwan blowup, part I: equivariant cohomology of the semi-stable locus
- 4. The cohomology of the Kirwan blowup, part II
- 5. The intersection cohomology of the GIT moduli space $\mathcal {M}^{\operatorname {GIT}}$
- 6. The intersection cohomology of the ball quotient
- 7. The cohomology of the toroidal compactification
- A. Equivariant cohomology
- B. Stabilizers, normalizers, and fixed loci for cubic threefolds
- C. The moduli space of cubic surfaces
Abstract
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.- M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523–615. MR 702806, DOI 10.1098/rsta.1983.0017
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