6,282 research outputs found
Prime exceptional divisors on holomorphic symplectic varieties and monodromy-reflections
Let X be a projective irreducible holomorphic symplectic manifold. The second
integral cohomology of X is a lattice with respect to the Beauville-Bogomolov
pairing. A divisor E on X is called a prime exceptional divisor, if E is
reduced and irreducible and of negative Beauville-Bogomolov degree.
Let E be a prime exceptional divisor on X. We first observe that associated
to E is a monodromy involution of the integral cohomology of X, which acts on
the second cohomology lattice as the reflection by the cohomology class of E
(Theorem 1.1).
We then specialize to the case that X is deformation equivalent to the
Hilbert scheme of length n zero-dimensional subschemes of a K3 surface. We
determine the set of classes of exceptional divisors on X (Theorem 1.11). This
leads to a determination of the closure of the movable cone of X.Comment: v2: 53 pages, Latex. The main Conjecture 1.11 is now Theorem 1.11.
Final version. To appear in KJM, Maruyama memorial volum
Integral generators for the cohomology ring of moduli spaces of sheaves over Poisson surfaces
Let M be a smooth and compact moduli space of stable coherent sheaves on a
projective surface S with an effective (or trivial) anti-canonical line bundle.
We find generators for the cohomology ring of M, with integral coefficients.
When S is simply connected and a universal sheaf E exists over SxM, then its
class [E] admits a Kunneth decomposition as a class in the tensor product of
the topological K-rings K(S) and K(M). The generators are the Chern classes of
the Kunneth factors of [E] in K(M). The general case is similarComment: v3: Latex, 27 pages. Final version, to appear in Advances in Math.
The proof of Lemma 21 is corrected and several other minor changes have been
made. v2: Latex, 26 pages. The paper was split. The new version is a rewrite
of the first three sections of version 1. The omitted results, about the
monodromy of Hilbert schemes of point on a K3 surface, constitute part of the
new paper arXiv:math.AG/0601304. v1: Latex, 53 page
Generators of the cohomology ring of moduli spaces of sheaves on symplectic surfaces
Let M be a moduli space of stable sheaves on a K3 or Abelian surface S. We
express the class of the diagonal in the cartesian square of M in terms of the
Chern classes of a universal sheaf. Consequently, we obtain generators of the
cohomology ring of M. When S is a K3 and M is the Hilbert scheme of length n
subschemes, this set of generators is sufficiently small in the sense that
there aren't any relations among them in the stable cohomology ring.
When S is the cotangent bundle of a Riemann surface, we recover the result of
T. Hausel and M. Thaddeus: The cohomology ring of the moduli spaces of Higgs
bundles is generated by the universal classes.Comment: Latex, 23 pages. The introduction is expanded, the coefficient in
part 3 of Theorem 1 is corrected, plus several other minor change
The Standard Conjectures for holomorphic symplectic varieties deformation equivalent to Hilbert schemes of K3 surfaces
We prove the standard conjectures for complex projective varieties that are
deformations of the Hilbert scheme of points on a K3 surface. The proof
involves Verbitsky's theory of hyperholomorphic sheaves and a study of the
cohomology algebra of Hilbert schemes of K3 surfaces.Comment: 15 pages, minor change
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