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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.
@article{10_4007_annals_2019_190_2_1,
author = {Caucher Birkar},
title = {Anti-pluricanonical systems on {Fano} varieties},
journal = {Annals of mathematics},
pages = {345--463},
publisher = {mathdoc},
volume = {190},
number = {2},
year = {2019},
doi = {10.4007/annals.2019.190.2.1},
mrnumber = {3997127},
zbl = {07107180},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2019.190.2.1/}
}
TY - JOUR AU - Caucher Birkar TI - Anti-pluricanonical systems on Fano varieties JO - Annals of mathematics PY - 2019 SP - 345 EP - 463 VL - 190 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2019.190.2.1/ DO - 10.4007/annals.2019.190.2.1 LA - en ID - 10_4007_annals_2019_190_2_1 ER -
Caucher Birkar. Anti-pluricanonical systems on Fano varieties. Annals of mathematics, Tome 190 (2019) no. 2, pp. 345-463. doi: 10.4007/annals.2019.190.2.1
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