Anti-pluricanonical systems on Fano varieties

TL;DR

This paper investigates the properties of anti-pluricanonical systems on Fano varieties with klt singularities, establishing non-emptiness, existence of well-behaved elements, and birational maps depending on dimension and singularity parameters.

Contribution

It proves the non-emptiness of $|-mK_X|$, existence of elements with good singularities, and the birationality of the map for Fano varieties with controlled singularities, also confirming Shokurov's boundedness conjecture.

Findings

$|-mK_X|$ is non-empty for some $m$ depending on dimension.

Existence of elements with good singularities in $|-mK_X|$.

For $ ext{epsilon}$-lc Fano varieties, $|-mK_X|$ defines a birational map.

Abstract

In this paper, we study the linear systems $|-mK_X|$ on Fano varieties $X$ with klt singularities. In a given dimension $d$, we prove $|-mK_X|$ is non-empty and contains an element with "good singularities" for some natural number $m$ depending only on $d$; if in addition $X$ is $\epsilon$-lc for some $\epsilon>0$, then we show that we can choose $m$ depending only on $d$ and $\epsilon$ so that $|-mK_X|$ defines a birational map. Further, we prove Shokurov's conjecture on boundedness of complements, and show that certain classes of Fano varieties form bounded families.