188 research outputs found
Transformations of locally conformally K\"ahler manifolds
We consider several transformation groups of a locally conformally K\"ahler
manifold and discuss their inter-relations. Among other results, we prove that
all conformal vector fields on a compact Vaisman manifold which is neither
locally conformally hyperk\"ahler nor a diagonal Hopf manifold are Killing,
holomorphic and that all affine vector fields with respect to the minimal Weyl
connection of a locally conformally K\"ahler manifold which is neither
Weyl-reducible nor locally conformally hyperk\"ahler are holomorphic and
conformalComment: 8 page
Bi-HKT and bi-Kaehler supersymmetric sigma models
We study CKT (or bi-HKT) N = 4 supersymmetric quantum mechanical sigma
models. They are characterized by the usual and the mirror sectors displaying
each HKT geometry. When the metric involves isometries, a Hamiltonian reduction
is possible. The most natural such reduction with respect to a half of bosonic
target space coordinates produces an N = 4 model, related to the twisted
Kaehler model due to Gates, Hull and Rocek, but including certain extra F-terms
in the superfield action.Comment: 31 pages, minor corrections in the published versio
Potential one-forms for hyperk\"ahler structures with torsion
It is shown that an HKT-space with closed parallel potential 1-form has
-symmetry. Every locally conformally hyperk\"ahler manifold
generates this type of geometry. The HKT-spaces with closed parallel potential
1-form arising in this way are characterized by their symmetries and an
inhomogeneous cubic condition on their torsion.Comment: 16 pages, Latex, no figure
Vanishing Theorems and String Backgrounds
We show various vanishing theorems for the cohomology groups of compact
hermitian manifolds for which the Bismut connection has (restricted) holonomy
contained in SU(n) and classify all such manifolds of dimension four. In this
way we provide necessary conditions for the existence of such structures on
hermitian manifolds. Then we apply our results to solutions of the string
equations and show that such solutions admit various cohomological restrictions
like for example that under certain natural assumptions the plurigenera vanish.
We also find that under some assumptions the string equations are equivalent to
the condition that a certain vector is parallel with respect to the Bismut
connection.Comment: 25 pages, Late
Stable bundles on hypercomplex surfaces
A hypercomplex manifold is a manifold equipped with three complex structures
I, J, K satisfying the quaternionic relations. Let M be a 4-dimensional compact
smooth manifold equipped with a hypercomplex structure, and E be a vector
bundle on M. We show that the moduli space of anti-self-dual connections on E
is also hypercomplex, and admits a strong HKT metric. We also study manifolds
with (4,4)-supersymmetry, that is, Riemannian manifolds equipped with a pair of
strong HKT-structures that have opposite torsion. In the language of Hitchin's
and Gualtieri's generalized complex geometry, (4,4)-manifolds are called
``generalized hyperkaehler manifolds''. We show that the moduli space of
anti-self-dual connections on M is a (4,4)-manifold if M is equipped with a
(4,4)-structure.Comment: 17 pages. Version 3.0: reference adde
Compact Einstein-Weyl four-dimensional manifolds
We look for four dimensional Einstein-Weyl spaces equipped with a regular
Bianchi metric. Using the explicit 4-parameters expression of the distance
obtained in a previous work for non-conformally-Einstein Einstein-Weyl
structures, we show that only four 1-parameter families of regular metrics
exist on orientable manifolds : they are all of Bianchi type and
conformally K\"ahler ; moreover, in agreement with general results, they have a
positive definite conformal scalar curvature. In a Gauduchon's gauge, they are
compact and we obtain their topological invariants. Finally, we compare our
results to the general analyses of Madsen, Pedersen, Poon and Swann : our
simpler parametrisation allows us to correct some of their assertions.Comment: Latex file, 13 pages, an important reference added and a critical
discussion of its claims offered, others minor modification
The first coefficients of the asymptotic expansion of the Bergman kernel of the spin^c Dirac operator
We establish the existence of the asymptotic expansion of the Bergman kernel
associated to the spin-c Dirac operators acting on high tensor powers of line
bundles with non-degenerate mixed curvature (negative and positive eigenvalues)
by extending the paper " On the asymptotic expansion of Bergman kernel "
(math.DG/0404494) of Dai-Liu-Ma. We compute the second coefficient b_1 in the
asymptotic expansion using the method of our paper "Generalized Bergman kernels
on symplectic manifolds" (math.DG/0411559).Comment: 21 pages, to appear in Internat. J. Math. Precisions added in the
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Natural Connection with Totally Skew-Symmetric Torsion on Riemannian Almost Product Manifolds
On a Riemannian almost product manifold we consider a linear
connection preserving the almost product structure and the Riemannian
metric and having a totally skew-symmetric torsion. We determine the class
of the manifolds admitting such a connection and prove that this
connection is unique in terms of the covariant derivative of with respect
to the Levi-Civita connection. We find a necessary and sufficient condition the
curvature tensor of the considered connection to have similar properties like
the ones of the K\"ahler tensor in Hermitian geometry. We pay attention to the
case when the torsion of the connection is parallel. We consider this
connection on a Riemannian almost product manifold constructed by a
Lie group .Comment: 14 pages, a revised edition, an example is adde
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