Abstract
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 $2$.
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number = {1},
issn = {0020-9910},
author = {Mukai, Shigeru},
mrclass = {14D20 (14F05)},
doi = {10.1007/BF01389137},
journal = {Invent. Math.},
zblnumber = {0565.14002},
volume = {77},
mrnumber = {0751133},
fjournal = {Inventiones Mathematicae},
mrreviewer = {Fran{ç}ois Cossec},
coden = {INVMBH},
title = {Symplectic structure of the moduli space of sheaves on an abelian or {$K3$} surface},
year = {1984},
pages = {101--116},
} -
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@ARTICLE{Mukai87b, mrkey = {0922020},
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mrclass = {32L10 (14J28 32G13 32J25 53C57)},
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volume = {39},
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title = {Moduli of vector bundles on {$K3$} surfaces and symplectic manifolds},
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@INCOLLECTION{Mukai87, mrkey = {0893604},
author = {Mukai, Shigeru},
mrclass = {14J28 (14D22 14F05)},
series = {Tata Inst. Fund. Res. Stud. Math.},
address = {Bombay},
publisher = {Tata Inst. Fund. Res.},
zblnumber = {0674.14023},
volume = {11},
mrnumber = {0893604},
booktitle = {Vector Bundles on Algebraic Varieties},
mrreviewer = {Mei Chu Chang},
venue = {{B}ombay, 1984},
title = {On the moduli space of bundles on {$K3$} surfaces. {I}},
pages = {341--413},
year = {1987},
} -
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[Tate66]
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author = {Tate, John},
mrclass = {11G40 (14G10)},
booktitle = {Séminaire {B}ourbaki, {V}ol. 9},
title = {On the conjectures of {B}irch and {S}winnerton-{D}yer and a geometric analog},
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[Yoshioka01]
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[Zarhin74]
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}