Birational boundedness for holomorphic symplectic varieties, Zarhin’s trick for $K3$ surfaces, and the Tate conjecture

François Charles

doi:https://doi.org/10.4007/annals.2016.184.2.4

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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@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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@ARTICLE{Nikulin79, mrkey = {0525944}, number = {1}, author = {Nikulin, V. V.}, issn = {0373-2436}, mrclass = {10C05 (14G30 14J17 14J25 57M99 57R45 58C27)}, journal = {Izv. Akad. Nauk SSSR Ser. Mat.}, volume = {43}, mrnumber = {0525944}, fjournal = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya}, mrreviewer = {I. Dolgachev}, title = {Integer symmetric bilinear forms and some of their geometric applications}, pages = {111--177, 238}, year = {1979}, zblnumber = {0408.10011}, doi = {10.1070/IM1980v014n01ABEH001060}, }
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@INCOLLECTION{Tate95, mrkey = {1610977}, mrnumber = {1610977}, 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}, address = {Paris}, publisher = {Soc. Math. France}, pages = {Exp. No. 306, 415--440}, year = {1995}, zblnumber = {0199.55604}, }
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Birational boundedness for holomorphic symplectic varieties, Zarhin’s trick for $K3$ surfaces, and the Tate conjecture

Abstract

Authors