Generic vanishing on homogeneous spaces in arbitrary characteristic

2025/07/17 by Ankit Rai, K. V. Shuddhodan, Rai, Ankit +1
Mathematics · #Algebraic Geometry and Number Theory #Geometry and complex manifolds #Commutative Algebra and Its Applications

paper · pdf · doi:10.48550/arxiv.2507.13452

Abstract

Let X be a proper homogeneous space for a connected algebraic group G over an algebraically closed field. For locally closed smooth affine subvarieties W,Z⊂ X, we show that (-1)^{dim X-dim W+dim Z}χ(gW∩ Z)≥ 0 for generic g∈ G. This extends the characteristic-zero theorem of Schürmann--Simpson--Wang. Over finite fields, our methods give a trace-function identity on a dense open subset of G and a Lang--Weil estimate for the non-generic locus.

Citations

• A new generic vanishing theorem on homogeneous varieties and the positivity conjecture for triple intersections of Schubert cells
• Pieri and Murnaghan--Nakayama type Rules for Chern classes of Schubert Cells
• Conjectural positivity of Chern-Schwartz-MacPherson classes for Richardson cells
• Arithmetic Fourier transforms over finite fields: generic vanishing, convolution, and equidistribution
• Vanishing theorems for perverse sheaves on abelian varieties, revisited
• Holonomic D-modules on abelian varieties
• The perverse filtration and the Lefschetz hyperplane theorem, II

Related

• A new generic vanishing theorem on homogeneous varieties and the positivity conjecture for triple intersections of Schubert cells
• Generic vanishing in characteristic p>0 and the characterization of ordinary abelian varieties
• Generic vanishing fails for singular varieties and in characteristic p>0
• A derived category approach to generic vanishing
• Generic vanishing theory in positive characteristic