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Register a new userHolomorphic projective connections on compact complex threefolds
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We prove that a holomorphic projective connection on a complex projective threefold is either flat, or it is a translation invariant holomorphic projective connection on an abelian threefold. In the second case, a generic translation invariant holomorphic affine connection on the abelian variety is not projectively flat. We also prove that a simply connected compact complex threefold with trivial canonical line bundle does not admit any holomorphic projective connection.
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We study holomorphic GL(2) and SL(2) geometries on compact complex manifolds. We show that a compact Kahler manifold of complex even dimension higher than two admitting a holomorphic GL(2)-geometry is covered by a compact complex torus. We classify compact Kahler-Einstein manifolds and Fano manifolds bearing holomorphic GL(2)-geometries. Among the compact Kahler-Einstein manifolds we prove that the only examples bearing holomorphic GL(2)-geometry are those covered by compact complex tori, the three dimensional quadric and those covered by the three dimensional Lie ball (the non compact dual of the quadric).
This is a survey paper dealing with holomorphic G-structures and holomorphic Cartan geometries on compact complex manifolds. Our emphasis is on the foliated case: holomorphic foliations with transverse (branched or generalized) holomorphic Cartan geometries.
We investigate stability of pullback bundles with respect to adiabatic classes on the total space of holomorphic submersions. We show that the pullback of a stable (resp. unstable) bundle remains stable (resp. unstable) for these classes. Assuming the graded object of a Jordan-Holder filtration to be locally free, we obtain a necessary and sufficient criterion for when the pullback of a strictly semistable vector bundle will be stable, in terms of intersection numbers on the base of the fibration. The arguments rely on adiabatic constructions of hermitian Yang-Mills connections together with the classical Donaldson-Uhlenbeck-Yau correspondence.
This paper is devoted to the study of holomorphic distributions of dimension and codimension one on smooth weighted projective complete intersection Fano manifolds threedimensional, with Picard number equal to one. We study the relations between algebro-geometric properties of the singular set of singular holomorphic distributions and their associated sheaves. We characterize either distributions whose tangent sheaf or conormal sheaf are arithmetically Cohen Macaulay (aCM) on smooth weighted projective complete intersection Fano manifolds threefold. We also prove that a codimension one locally free distribution with trivial canonical bundle on any Fano threefold, with Picard number equal to one, has a tangent sheaf which either splits or it is stable.
The contents of this article are now presented in the appendix of arXiv:0805.2195v2.
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