Seshadri positive submanifolds of polarized manifolds

TL;DR

This paper introduces and studies new positivity conditions called Seshadri A-bigness and A-ampleness for submanifolds within polarized complex manifolds, generalizing previous theories and motivated by numerous examples.

Contribution

It generalizes the theory of Seshadri positivity to higher-dimensional submanifolds, extending prior work on curves to broader cases.

Findings

Defines Seshadri A-bigness and A-ampleness for submanifolds

Establishes properties and implications of these positivity conditions

Provides a wide range of motivating examples

Abstract

Let $Y$ be a submanifold of dimension $y$ of a polarized complex manifold $(X,A)$ of dimension $k\geq 3$, with $1\leq y\leq k-1$. We define and study two positivity conditions on $Y$ in $(X,A)$, called Seshadri $A$-bigness and (a stronger one) Seshadri $A$-ampleness. In this way we get the natural generalization of the theory initiated by Paoletti in \cite{Pao} (which corresponds to the case $(k,y)=(3,1)$) and subsequently generalized and completed in \cite{BBF} (regarding curves in a polarized manifold of arbitrary dimension). The theory presented here, which is new even if $y=k-1$, is motivated by a reasonably large area of examples.