1,896 research outputs found
The pseudo-effective cone of a non-K\"ahlerian surface and applications
We describe the positive cone and the pseudo-effective cone of a
non-K\"ahlerian surface. We use these results for two types of applications:
- Describe the set of possible total Ricci scalars associated
with Gauduchon metrics of fixed volume 1 on a fixed non-K\"ahhlerian surface,
and decide whether the assignment is a deformation
invariant.
- Study the stability of the canonical extension
of a class VII surface
with positive . This extension plays an important role in our strategy to
prove the GSS conjecture using gauge theoretical methods.Comment: LaTeX, 25 pages; rv: minor correction in the proof of Remark 4.
A variation formula for the determinant line bundle. Compact subspaces of moduli spaces of stable bundles over class VII surfaces
This article deals with two topics: the first, which has a general character,
is a variation formula for the the determinant line bundle in non-K\"ahlerian
geometry. This formula, which is a consequence of the non-K\"ahlerian version
of the Grothendieck-Riemann Roch theorem proved recently by Bismut, gives the
variation of the determinant line bundle corresponding to a perturbation of a
Fourier-Mukai kernel on a product by a unitary flat line
bundle on the fiber . When this fiber is a complex surface and is
a holomorphic 2-bundle, the result can be interpreted as a Donaldson invariant.
The second topic concerns a geometric application of our variation formula,
namely we will study compact complex subspaces of the moduli spaces of stable
bundles considered in our program for proving existence of curves on minimal
class VII surfaces. Such a moduli space comes with a distinguished point
corresponding to the canonical extension of . The
compact subspaces Y\subset {\cal M}^\st containing this distinguished point
play an important role in our program. We will prove a non-existence result:
there exists no compact complex subspace of positive dimension Y\subset {\cal
M}^\st containing with an open neighborhood such that
consists only of non-filtrable bundles. In other words,
within any compact complex subspace of positive dimension Y\subset {\cal
M}^\st containing , the point can be approached by filtrable bundles.
Specializing to the case we obtain a new way to complete the proof of a
theorem in a previous article: any minimal class VII surface with has a
cycle of curves. Applications to class VII surfaces with higher will be
be discussed in a forthcoming article.Comment: 25 pages. Comments, suggestions are most welcome Revised version:
minor correction
On the torsion of the first direct image of a locally free sheaf
Let be a proper holomorphic submersion between complex manifolds
and a holomorphic bundle on . We study and describe explicitly
the torsion subsheaf of the first direct
image under the assumption .
We give two applications of our results. The first concerns the locus of points
in the base of a generically versal family of complex surfaces where the family
is non-versal. The second application is a vanishing result for
in a concrete situation related to
our program to prove the existence of curves on class VII surfaces.Comment: 27 pages. Comments, remarks, suggestions, bibliographic references
are most welcome. Revised version: minor correction
- …