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Rational cohomology of toric diagrams

Grigory Solomadin

TL;DR

The note develops a unified framework to compute rational Betti numbers of homotopy colimits of toric diagrams and their classifying spaces by translating T-actions into toric diagrams over finite posets and applying BKSS, EMSS, and Zeeman–McCrory dualities. It proves BKSS collapse over $\mathbb{Q}$ for toric diagrams, relates ordinary and equivariant cohomology to sheaf/co-sheaf data on orbit posets, and establishes a Cohen–Macaulay criterion for torus actions on hocolims. The results yield explicit bigraded Betti numbers for skeletons of toric manifolds and nonsingular toric varieties, with precise formulas for $T$-characteristic diagrams and consistency checks against known CP^{m-1} decompositions. Together, these findings provide practical tools to compute cohomology and Betti numbers in a broad class of toric-geometry/topological settings, including equivariant formality and CM actions.

Abstract

In this note, (rational) Betti numbers of homotopy colimits for toric diagrams and their classifying spaces are described in terms of sheaf cohomology under CW poset assumption on opposite to the respective indexing category. Split $T$-CW complexes with a CW orbit poset have such a decomposition (up to an equivariant homeomorphism). Cohen-Macaulayness (over $\mathbb{Q}$) of the $T$-action on $hocolim\ D$ is equivalent to acyclicity for a certain sheaf provided that $D$ is a $T$-diagram over $C$, and $C^{op}$ is a homology manifold. The ordinary and bigraded Betti numbers are computed for skeletons of equivariantly formal spaces from this class (in particular, of compact nonsingular toric varieties).

Rational cohomology of toric diagrams

TL;DR

The note develops a unified framework to compute rational Betti numbers of homotopy colimits of toric diagrams and their classifying spaces by translating T-actions into toric diagrams over finite posets and applying BKSS, EMSS, and Zeeman–McCrory dualities. It proves BKSS collapse over for toric diagrams, relates ordinary and equivariant cohomology to sheaf/co-sheaf data on orbit posets, and establishes a Cohen–Macaulay criterion for torus actions on hocolims. The results yield explicit bigraded Betti numbers for skeletons of toric manifolds and nonsingular toric varieties, with precise formulas for -characteristic diagrams and consistency checks against known CP^{m-1} decompositions. Together, these findings provide practical tools to compute cohomology and Betti numbers in a broad class of toric-geometry/topological settings, including equivariant formality and CM actions.

Abstract

In this note, (rational) Betti numbers of homotopy colimits for toric diagrams and their classifying spaces are described in terms of sheaf cohomology under CW poset assumption on opposite to the respective indexing category. Split -CW complexes with a CW orbit poset have such a decomposition (up to an equivariant homeomorphism). Cohen-Macaulayness (over ) of the -action on is equivalent to acyclicity for a certain sheaf provided that is a -diagram over , and is a homology manifold. The ordinary and bigraded Betti numbers are computed for skeletons of equivariantly formal spaces from this class (in particular, of compact nonsingular toric varieties).
Paper Structure (15 sections, 32 theorems, 115 equations)

This paper contains 15 sections, 32 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Spectral sequence of a diagram and its collapse
  3. (Co)homology of (co)sheaves on posets
  4. Zeeman-McCrory spectral sequence for homology manifolds
  5. Torus actions arising from $T$-diagrams
  6. Faces and orbits
  7. Orbit spectral sequence collapse
  8. Criterion for homotopy decomposition of a $T$-action
  9. Borel space of a homotopy colimit
  10. EMSS and BKSS
  11. Cohen-Macaulay torus actions on toric diagrams
  12. Skeletons of compact nonsingular toric varieties
  13. Betti numbers for equivariantly formal $T$-diagrams
  14. Betti numbers for skeletons of $T$-characteristic diagrams
  15. Skeletons of the complex projective space

Key Result

Theorem 1

For any complete fan $\Sigma$ the orbit spectral sequence (over $\mathbb Q$) of $X_{\Sigma}$ collapses at the second page. There is the isomorphism

Theorems & Definitions (94)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: bo-ka-72bo-87
  • Definition 2.4
  • Remark 2.5
  • Remark 2.6
  • Definition 2.7: we-zi-zi-99
  • ...and 84 more