Second Chern class of Fano manifolds and anti-canonical geometry

TL;DR

This paper establishes bounds on the second Chern class of Fano manifolds with Picard number one and explores the geometric properties of certain Fano manifolds, including basepoint freeness and birationality of linear systems.

Contribution

It provides new lower bounds for the second Chern class and analyzes the anti-canonical linear systems of Fano manifolds, including singular weak Fano varieties.

Findings

Lower bound for second Chern class in terms of index and degree

Linear system  mH is basepoint free for m   7

The map defined by  mH is birational for m   5

Abstract

Let $X$ be a Fano manifold of Picard number one. We establish a lower bound for the second Chern class of $X$ in terms of its index and degree. As an application, if $Y$ is a $n$-dimensional Fano manifold with $-K_Y=(n-3)H$ for some ample divisor $H$, we prove that $h^0(Y,H)\geq n-2$. Moreover, we show that the rational map defined by $\vert mH\vert$ is birational for $m\geq 5$, and the linear system $\vert mH\vert$ is basepoint free for $m\geq 7$. As a by-product, the pluri-anti-canonical systems of singular weak Fano varieties of dimension at most $4$ are also investigated.