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Cohomology of the moduli space of cubic threefolds and its smooth models

Sebastian Casalaina-Martin, Samuel Grushevsky, Klaus Hulek, Radu Laza

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 of cubic surfaces is discussed in an appendix.

Cohomology of the moduli space of cubic threefolds and its smooth models

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 of cubic surfaces is discussed in an appendix.

Paper Structure

This paper contains 80 sections, 48 theorems, 256 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and conventions
  4. The general setting
  5. The case of hypersurfaces
  6. The case of cubic threefolds
  7. Strictly polystable points
  8. Moduli space of cubic threefolds and its standard compactifications $\mathcal{M}^{\operatorname{GIT}}$ and $(\mathcal{B}/\Gamma)^*$
  9. The GIT compactification $\mathcal{M}^{\operatorname{GIT}}$
  10. The ball quotient model $(\mathcal{B}/\Gamma)^*$
  11. The Kirwan blowup $\mathcal{M}^{\operatorname{K}}$ of the moduli space of cubic threefolds
  12. Introduction
  13. The Kirwan blowup in general
  14. The Kirwan blowup of the moduli space of cubic threefolds
  15. The toroidal compactification
  16. ...and 65 more sections

Key Result

Theorem 1.1

The Betti numbers of $\mathcal{M}^{\operatorname{K}}$ and $\overline{\mathcal{B}/\Gamma}$, and the intersection Betti numbers of $\mathcal{M}^{\operatorname{GIT}}$, $\widehat{\mathcal{M}}$, and $(\mathcal{B}/\Gamma)^*$ are as follows: while all the odd degree (intersection) cohomology vanishes.

Figures (1)

  • Figure 1: A sample codimension $4$ element is given above in blue as $\textcolor{blue}{\beta'}= \frac{1}{2}(-\frac{2}{3},\frac{1}{3},\frac{1}{3})\in \mathcal{B}$. Another sample codimension $4$ element is given in green as $\textcolor{green}{\beta'}= \frac{1}{2}(\frac{1}{3},-\frac{2}{3},\frac{1}{3})\in \mathcal{B}$. A sample codimension $5$ element is given in red as $\textcolor{red}{\beta'}= \frac{1}{7}(2,1,-3)\in \mathcal{B}$. We have dropped the last two $0$-coordinates for brevity.

Theorems & Definitions (118)

  • Theorem 1.1
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 2.1
  • Theorem 2.2: GIT compactification for cubic threefolds, allcock
  • Remark 2.3
  • Theorem 2.4: The ball quotient model, act and ls
  • Theorem 2.5: GIT to ball quotient comparison, act and ls
  • Remark 2.6
  • ...and 108 more