Complete intersections in spherical varieties

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 hp,0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h^{p,0}$$\end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathbb {C}^*)^n$$\end{document}, our results specialize to well-known results from the Newton polyhedra theory and toric varieties.