Anti-pluricanonical systems on Fano varieties

In this paper, we study the linear systems $\mathrm{\mid }-m{K}_X\mathrm{\mid }$ on Fano varieties $X$ with klt singularities. In a given dimension $d$, we prove $\mathrm{\mid }-m{K}_X\mathrm{\mid }$ is non-empty and contains an element with “good singularities” for some natural number $m$ depending only on $d$; if in addition $X$ is $ϵ$-lc for some $ϵ>0$, then we show that we can choose $m$ depending only on $d$ and $ϵ$ so that $\mathrm{\mid }-m{K}_X\mathrm{\mid }$ 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.