Complete intersections in spherical varieties

TL;DR

This paper investigates the geometry of complete intersections in spherical varieties, providing formulas for their invariants and criteria for nonemptiness, generalizing classical results from toric geometry.

Contribution

It introduces a comprehensive framework for analyzing complete intersections in spherical varieties, linking their properties to moment and Newton-Okounkov polytopes, and extends criteria for nonemptiness beyond toric cases.

Findings

Computed arithmetic genus and h^{p,0} numbers in terms of polytopes.

Established a criterion for nonemptiness of intersections.

Unified results with classical toric geometry when specialized.

Abstract

Let G be a complex reductive algebraic group. We study complete intersections in a spherical homogeneous space G/H defined by a generic collection of sections from G-invariant linear systems. Whenever nonempty, all such complete intersections are smooth varieties. We compute their arithmetic genus as well as some of their h^{p,0} numbers. The answers are given in terms of the moment polytopes and Newton-Okounkov polytopes associated to G-invariant linear systems. We also give a necessary and sufficient condition on a collection of linear systems so that the corresponding generic complete intersection is nonempty. This criterion applies to arbitrary quasi-projective varieties (i.e. not necessarily spherical homogeneous spaces). When the spherical homogeneous space under consideration is a complex torus (C^*)^n, our results specialize to well-known results from the Newton polyhedra theory and toric varieties.