1,293 research outputs found
Birational boundedness for holomorphic symplectic varieties, Zarhin's trick for surfaces, and the Tate conjecture
We investigate boundedness results for families of holomorphic symplectic
varieties up to birational equivalence. We prove the analogue of Zarhin's trick
by for 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 surfaces over finite fields of characteristic at least ,
and a simple proof of the Tate conjecture for surfaces with Picard number
at least over arbitrary finite fields -- including characteristic .Comment: 27 pages. Zarhin's trick for K3 surfaces is now stated for arbitrary
fields, and the proof of Theorem 3.3 has been fixed. Minor typos fixe
Rational Lagrangian fibrations on punctual Hilbert schemes of K3 surfaces
A rational Lagrangian fibration f on an irreducible symplecitc variety V is a
rational map which is birationally equivalent to a regular surjective morphism
with Lagrangian fibers. By analogy with K3 surfaces, it is natural to expect
that a rational Lagrangian fibration exists if and only if V has a divisor D
with Bogomolov--Beauville square 0. This conjecture is proved in the case when
V is the punctual Hilbert scheme of a generic algebraic K3 surface S. The
construction of f uses a twisted Fourier--Mukai transform which induces a
birational isomorphism of V with a certain moduli space of twisted sheaves on
another K3 surface M, obtained from S as its Fourier--Mukai partner.Comment: Final version; minor change
Deformation of canonical morphisms and the moduli of surfaces of general type
In this article we study the deformation of finite maps and show how to use
this deformation theory to construct varieties with given invariants in a
projective space. Among other things, we prove a criterion that determines when
a finite map can be deformed to a one--to--one map. We use this criterion to
construct new simple canonical surfaces with different and . Our
general results enable us to describe some new components of the moduli of
surfaces of general type. We also find infinitely many moduli spaces having one component whose general point corresponds to a
canonically embedded surface and another component whose general point
corresponds to a surface whose canonical map is a degree 2 morphism.Comment: 32 pages. Final version with some simplifications and clarifications
in the exposition. To appear in Invent. Math. (the final publication is
available at springerlink.com
Moduli spaces of sheaves on K3 surfaces
In this survey article we describe moduli spaces of simple, stable, and
semistable sheaves on K3 surfaces, following the work of Mukai, O'Grady,
Huybrechts, Yoshioka, and others. We also describe some recent developments,
including applications to the study of Chow rings of K3 surfaces, determination
of the ample and nef cones of irreducible holomorphic symplectic manifolds, and
moduli spaces of Bridgeland stable complexes of sheaves.Comment: 24 pages, to appear in the Journal of Geometry and Physics special
issue: proceedings of the "Workshop on Instanton Counting: Moduli Spaces,
Representation Theory and Integrable Systems" (Leiden, 16-20 June 2014
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