155 research outputs found
Quasihomogeneous three-dimensional real analytic Lorentz metrics
We classify germs at the origin of real analytic Lorentz metrics on R^3 which
are quasihomogeneous, in the sense that they are locally homogeneous on an open
set containing the origin in its closure, but not locally homogeneous in the
neighborhood of the origin.Comment: 11 page
Quasihomogeneous three-dimensional real analytic Lorentz metrics do not exist
We show that a germ of a real analytic Lorentz metric on which is
locally homogeneous on an open set containing the origin in its closure is
necessarily locally homogeneous. We classifiy Lie algebras that can act
quasihomogeneously---meaning they act transitively on an open set admitting the
origin in its closure, but not at the origin---and isometrically for such a
metric. In the case that the isotropy at the origin of a quasihomogeneous
action is semisimple, we provide a complete set of normal forms of the metric
and the action.Comment: 23 pp. Took the place of "Quasihomogeneous three-dimensional real
analytic Lorentz metrics" (arXiv:1401.6272), which was withdrawn by the first
author. Revised version incorporates several minor corrections, including
those suggested by the refere
Branched holomorphic Cartan geometries and Calabi-Yau manifolds
We introduce the concept of a branched holomorphic Cartan geometry. It
generalizes to higher dimension the definition of branched (flat) complex
projective structure on a Riemann surface introduced by Mandelbaum. This new
framework is much more flexible than that of the usual holomorphic Cartan
geometries. We show that all compact complex projective manifolds admit
branched flat holomorphic projective structure. We also give an example of a
non-flat branched holomorphic normal projective structure on a compact complex
surface. It is known that no compact complex surface admits such a structure
with empty branching locus. We prove that non-projective compact simply
connected K\"ahler Calabi-Yau manifolds do not admit branched holomorphic
projective structures. The key ingredient of its proof is the following result
of independent interest: If E is a holomorphic vector bundle over a compact
simply connected K\"ahler Calabi-Yau manifold, and E admits a holomorphic
connection, then E is a trivial holomorphic vector bundle equipped with the
trivial connection.Comment: 24 pages; revised versio
Symmetries of holomorphic geometric structures on tori
We prove that any holomorphic locally homogeneous geometric structure on a
complex torus, modelled on a complex homogeneous surface, is translation
invariant. We conjecture that this result is true is any dimension. In higher
dimension we prove it here for nilpotent models. We also prove that in any
dimension the translation invariant -structures form a union of
connected components in the deformation space of -structures.Comment: 17 page
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