Positivity on simple $G$-varieties

TL;DR

The paper addresses positivity for vector bundles on nonsingular simple $G$-projective varieties by reducing ampleness and nefness to restrictions on $B$-stable curves, and then computes precise Seshadri constants at the sink $x^-$ of such varieties. It establishes a nefness criterion for $ obreak \Gamma$-equivariant bundles via curve restrictions, and analyzes the geometry of the blow-up at the sink to describe nef and Mori cones explicitly. Concrete formulas are derived: for any ample line bundle $L=\\sum_i a_i D_i$, The Seshadri constant at $x^-$ is $\\varepsilon(L;x^-) = \\min_i a_i$, and for a $B$-equivariant nef vector bundle $E$, $\\varepsilon(E;x^-) = \\min_{i,C} a_i(C)$ where $E|_C$ splits on finitely many $B$-stable curves through $x^-$. These results yield a practical pathway to compute local positivity and Seshadri constants in the presence of group actions, with explicit instances for flag varieties and related simple $G$-varieties.

Abstract

Let $X$ be a normal projective variety equipped with an action of a semisimple algebraic group $G$, and assume that $X$ contains a unique closed orbit. Let $B$ be a Borel subgroup of $G$ and let $E$ be a $B$-equivariant vector bundle on $X$. In this article, we prove that $E$ is ample (respectively, nef) if and only if its restriction to the finite set of $B$-stable curves in $X$ is ample (respectively, nef). Moreover, we compute the nef cone of the blow-up of a nonsingular simple $G$-projective variety $X$ at a unique $B$-fixed point $x^-$, referred to as the sink of $X$. As an application, when $X$ is nonsingular, we calculate the Seshadri constants of any ample line bundle (not necessarily $G$-equivariant) at $x^-$. In addition, we compute the Seshadri constants of $B$-equivariant vector bundles at $x^{-}$.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Definition 1.3: Seshadri constant at a point
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1: Theorem 1, fmss
• proof : Proof of Theorem \ref{['theorem: nef']}
• Proposition 2.2
• proof
• proof : Proof of Theorem \ref{['theorem: ample']}