Anti-pluricanonical systems on Fano varieties


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.


Caucher Birkar

DPMMS, Centre for Mathematical Sciences, University of Cambridge, Cambridge, UK