97 research outputs found
Kodaira-Saito vanishing and applications
The first part of the paper contains a detailed proof of M. Saito's
generalization of the Kodaira vanishing theorem, following the original
argument and with ample background, based on a lecture given at a Clay workshop
on mixed Hodge modules. The second part contains some recent applications, and
a Kawamata-Viehweg-type statement in the setting of mixed Hodge modules.Comment: 33 pages; final version, with expository improvements, updates, and
substantial additions in the background section
Stable maps and Quot schemes
In this paper we study the relationship between two different
compactifications of the space of vector bundle quotients of an arbitrary
vector bundle on a curve. One is Grothendieck's Quot scheme, while the other is
a moduli space of stable maps to the relative Grassmannian.
We establish an essentially optimal upper bound on the dimension of the two
compactifications. Based on that, we prove that for an arbitrary vector bundle,
the Quot schemes of quotients of large degree are irreducible and generically
smooth. We precisely describe all the vector bundles for which the same thing
holds in the case of the moduli spaces of stable maps. We show that there are
in general no natural morphisms between the two compactifications. Finally, as
an application, we obtain new cases of a conjecture on effective base point
freeness for pluritheta linear series on moduli spaces of vector bundles.Comment: 39 pages, 1 figure; final version with a few expository changes
suggested by the refere
Viehweg's hyperbolicity conjecture for families with maximal variation
We use the theory of Hodge modules to construct Viehweg-Zuo sheaves on the
base spaces of families with maximal variation and fibers of general type, or
more generally whose geometric generic fiber has a good minimal model. We
deduce Viehweg's hyperbolicity conjecture in this context, namely the fact that
the base spaces of such families are of log general type. This is approached as
part of a general problem of identifying what spaces can support Hodge
theoretic objects with certain positivity properties.Comment: 28 pages; final version with a few expository improvements, to appear
in Invent. Mat
Generic vanishing and minimal cohomology classes on abelian varieties
We establish a, and conjecture further, relationship between the existence of
subvarieties representing minimal cohomology classes on principally polarized
abelian varieties, and the generic vanishing of certain sheaf cohomology. The
main ingredient is the Generic Vanishing criterion of math.AG/0608127, based on
the Fourier-Mukai transform.Comment: 12 pages; final version, to appear in Math. Annalen; contains
expository changes and a proof in the case of abelian varieties of the
generic vanishing criterion we use, according to suggestions from the refere
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