115 research outputs found
On the Kaehler rank of compact complex surfaces
Harvey and Lawson introduced the Kaehler rank and computed it in connection
to the cone of positive exact currents of bidimension (1,1) for many classes of
compact complex surfaces. In this paper we extend these computations to the
only further known class of surfaces not considered by them, that of Kato
surfaces. Our main tool is the reduction to the dynamics of associated
holomorphic contractions
A note on the cone of mobile curves
S. Boucksom, J.-P. Demailly, M. Paun and Th. Peternell proved that the cone
of mobile curves ME(X) of a projective complex manifold X is dual to the cone
generated by classes of effective divisors and conjectured an extension of this
duality in the Kaehler set-up. We show that their conjecture implies that ME(X)
coincides with the cone of integer classes represented by closed positive
smooth (n-1,n-1)-forms. Without assuming the validity of the conjecture we
prove that this equality of cones still holds at the level of degree functions
Moduli spaces of bundles over non-projective K3 surfaces
We study moduli spaces of sheaves over non-projective K3 surfaces. More
precisely, if is a Mukai vector on a K3 surface with
prime to and is a "generic" K\"ahler class on , we show that
the moduli space of stable sheaves on with associated
Mukai vector is an irreducible holomorphic symplectic manifold which is
deformation equivalent to a Hilbert scheme of points on a K3 surface. If
parametrizes only locally free sheaves, it is moreover hyperk\"ahler. Finally,
we show that there is an isometry between and
and that is projective if and only if is projective.Comment: 42 pages; major revisions; to appear in Kyoto J. Mat
Holomorphic vector bundles on non-algebraic surfaces
The existence problem for holomorphic structures on vector bundles over
non-algebraic surfaces is in general still open. We solve this problem in the
case of rank 2 vector bundles over K3 surfaces and in the case of vector
bundles of arbitrary rank over all known surfaces of class VII. Our methods,
which are based on Donaldson theory and deformation theory, can be used to
solve the existence problem of holomorphic vector bundles on further classes of
non-algebraic surfaces.Comment: LaTeX, 6 pages, to appear in Comptes Rendu
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