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Vanishing theorems on wonderful varieties

Ruizhen Liu

TL;DR

The paper develops characteristic-free vanishing theorems for tautological bundles on wonderful varieties, tying the geometry of these compactifications to combinatorial invariants of matroids via the Orlik–Solomon algebra and logarithmic de Rham cohomology. It proves a characteristic-free analogue of Brieskorn’s arrangement-cohomology correspondence through pushforwards and degeneracy-locus techniques, and establishes comparison theorems that realize ${\operatorname{\,overline{OS}}}^*(L)$ inside global logarithmic forms on $W_L$. The work further extends vanishing results of Borel–Weil–Bott and Manivel type using Kempf–Weyman collapsings, and shows how cohomology of tautological bundles controls White’s conjecture for toric ideals of matroids, connecting geometric positivity to combinatorial syzygies and Tutte polynomials. Collectively, these results unify combinatorial, cohomological, and representation-theoretic perspectives, with implications for moduli spaces like $\\overline{M}_{0,m}$ and for questions about generators and syzygies of matroid toric ideals.

Abstract

We study vanishing theorems of tautological bundles in the sense of Berget--Eur--Spink--Tseng restricted to wonderful varieties. As an application, we prove a characteristic-independent analogue of Brieskorn's result on cohomology of arrangement complements, in addition to a comparison theorem between Orlik--Solomon algebra and the logarithmic de Rham cohomology of wonderful varieties. In a different direction, we extend a vanishing theorem of Borel--Weil--Bott type for tautological bundles. Finally, we reduce the weak version of White's basis conjecture to a problem about cohomology vanishing of tautological bundles.

Vanishing theorems on wonderful varieties

TL;DR

The paper develops characteristic-free vanishing theorems for tautological bundles on wonderful varieties, tying the geometry of these compactifications to combinatorial invariants of matroids via the Orlik–Solomon algebra and logarithmic de Rham cohomology. It proves a characteristic-free analogue of Brieskorn’s arrangement-cohomology correspondence through pushforwards and degeneracy-locus techniques, and establishes comparison theorems that realize inside global logarithmic forms on . The work further extends vanishing results of Borel–Weil–Bott and Manivel type using Kempf–Weyman collapsings, and shows how cohomology of tautological bundles controls White’s conjecture for toric ideals of matroids, connecting geometric positivity to combinatorial syzygies and Tutte polynomials. Collectively, these results unify combinatorial, cohomological, and representation-theoretic perspectives, with implications for moduli spaces like and for questions about generators and syzygies of matroid toric ideals.

Abstract

We study vanishing theorems of tautological bundles in the sense of Berget--Eur--Spink--Tseng restricted to wonderful varieties. As an application, we prove a characteristic-independent analogue of Brieskorn's result on cohomology of arrangement complements, in addition to a comparison theorem between Orlik--Solomon algebra and the logarithmic de Rham cohomology of wonderful varieties. In a different direction, we extend a vanishing theorem of Borel--Weil--Bott type for tautological bundles. Finally, we reduce the weak version of White's basis conjecture to a problem about cohomology vanishing of tautological bundles.

Paper Structure

This paper contains 22 sections, 34 theorems, 162 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The permutohedral variety
  4. Tautological bundles of linear spaces
  5. Matroidal operations and tautological bundles
  6. Characteristic-free vanishing results via pushforwards
  7. Tautological bundles on the projective line
  8. Vanishing results for higher direct images
  9. Running the induction
  10. Application to Tutte polynomials
  11. Vanishing theorems for hook-shaped Schur functors
  12. Logarithmic geometry and Orlik--Solomon algebras
  13. Degeneration of the Hodge--de Rham sequence
  14. Comparison theorems
  15. Other compactifications
  16. ...and 7 more sections

Key Result

Theorem A

Let $\Bbbk$ be a field and $L \subseteq \Bbbk^E$ be a linear subspace satisfying Condition eq:ll. Let $W_L$ be the de Concini--Procesi wonderful variety introduced above, and $D_L$ be its boundary divisor. Then,

Figures (2)

  • Figure 1: An illustration of the map $f:X_{0,1,2} \to X_{0,1}$; strata of the permutohedral varieties labelled by set partitions.
  • Figure 2: Fano plane $\check{\mathbb P}^2({\mathbb F}_2)$

Theorems & Definitions (85)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Conjecture 1.1: White
  • Conjecture 1.2: Conjecture \ref{['conj:white-deg3']}
  • Definition 2.1
  • Definition 2.2
  • Example 2.4
  • Proposition 2.5
  • ...and 75 more