A new generic vanishing theorem on homogeneous varieties and the positivity conjecture for triple intersections of Schubert cells

Schürmann, Jörg; Simpson, Connor; Wang, Botong

Forschungsartikel (Zeitschrift) | Peer reviewed

Zusammenfassung

In this paper we prove a new generic vanishing theorem for XX a complete homogeneous variety with respect to an action of a connected algebraic group. Let A,B0⊂XA,B0⊂X be locally closed affine subvarieties, and assume that B0B0 is smooth and pure dimensional. Let PP be a perverse sheaf on AA and let B=gB0B=gB0 be a generic translate of B0B0. Then our theorem implies (−1)codimBχ(A∩B,P|A∩B)≥0(−1)codim⁡Bχ(A∩B,P|A∩B)≥0. As an application, we prove in full generality a positivity conjecture about the signed Euler characteristic of generic triple intersections of Schubert cells. Such Euler characteristics are known to be the structure constants for the multiplication of the Segre–Schwartz–MacPherson classes of these Schubert cells.

Details zur Publikation

FachzeitschriftCompositio Mathematica (Compos. Math.)
Jahrgang / Bandnr. / Volume161
Ausgabe / Heftnr. / Issue1
Seitenbereich1-12
StatusVeröffentlicht
Veröffentlichungsjahr2025
DOI10.1112/S0010437X24007462
Stichwörter vanishing theorem; perverse sheaves; homogeneous variety, abelian variety; flag variety; Schubert cell; Chern–Schwartz–MacPherson class; Segre–Schwartz–MacPherson class; Euler characteristic; triple intersection; positivity conjecture

Autor*innen der Universität Münster

Schürmann, Jörg
Mathematisches Institut