Birational boundedness for holomorphic symplectic varieties, Zarhin’s trick for $K3$ surfaces, and the Tate conjecture
Annals of mathematics, Tome 184 (2016) no. 2, pp. 487-526
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We investigate boundedness results for families of holomorphic symplectic varieties up to birational equivalence. We prove the analogue of Zarhin’s trick for $K3$ surfaces by constructing big line bundles of low degree on certain moduli spaces of stable sheaves, and proving birational versions of Matsusaka’s big theorem for holomorphic symplectic varieties.
As a consequence of these results, we give a new geometric proof of the Tate conjecture for $K3$ surfaces over finite fields of characteristic at least $5$, and a simple proof of the Tate conjecture for $K3$ surfaces with Picard number at least $2$ over arbitrary finite fields — including fields of characteristic $2$.
@article{10_4007_annals_2016_184_2_4,
author = {Fran\c{c}ois Charles},
title = {Birational boundedness for holomorphic symplectic varieties, {Zarhin{\textquoteright}s} trick for $K3$ surfaces, and the {Tate} conjecture},
journal = {Annals of mathematics},
pages = {487--526},
year = {2016},
volume = {184},
number = {2},
doi = {10.4007/annals.2016.184.2.4},
mrnumber = {3548531},
zbl = {06662218},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2016.184.2.4/}
}
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%0 Journal Article %A François Charles %T Birational boundedness for holomorphic symplectic varieties, Zarhin’s trick for $K3$ surfaces, and the Tate conjecture %J Annals of mathematics %D 2016 %P 487-526 %V 184 %N 2 %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2016.184.2.4/ %R 10.4007/annals.2016.184.2.4 %G en %F 10_4007_annals_2016_184_2_4
François Charles. Birational boundedness for holomorphic symplectic varieties, Zarhin’s trick for $K3$ surfaces, and the Tate conjecture. Annals of mathematics, Tome 184 (2016) no. 2, pp. 487-526. doi: 10.4007/annals.2016.184.2.4
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