Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userHolomorphic G-structures and foliated Cartan geometries on compact complex manifolds
114
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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).
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.
An n-dimensional complex manifold is a manifold by biholomorphic mappings between open sets of the finite direct product of the complex number field. On the other hand, when A is a commutative Banach algebra, Lorch gave a definition that an A-valued function on an open set of A is holomorphic. The definition of a holomorphic function by Lorch can be straightforwardly generalized to an A-valued function on an open set of the finite direct product of A. Therefore, a manifold modeled on the finite direct product of A (an n-dimensional A-manifold) is easily defined. However, in my opinion, it seems that so many nontrivial examples were not known (including the case of n=1, that is, Riemann surfaces). By the way, if X is a compact Hausdorff space, then the algebra C(X) of all complex valued continuous functions on X is the most basic example of a commutative Banach algebra (furthermore, a commutative C*-algebra). In this note, we see that if the set of all continuous cross sections of a continuous family M of compact complex manifolds (a topological deformation M of compact complex analytic structures) on X is denoted by G(M), then the structure of a C(X)-manifold modeled on the C(X)-modules of all continuous cross sections of complex vector bundles on X is introduced into G(M). Therefore, especially, if X is contractible, then G(M) is a finite-dimensional C(X)-manifold.
This is the second part of a series of two papers dedicated to a systematic study of holomorphic Jacobi structures. In the first part, we introduced and study the concept of a holomorphic Jacobi manifold in a very natural way as well as various tools. In the present paper, we solve the integration problem for holomorphic Jacobi manifolds by proving that they integrate to complex contact groupoids. A crucial tool in our proof is what we call the homogenization scheme, which allows us to identify holomorphic Jacobi manifolds with homogeneous holomorphic Poisson manifolds and holomorphic contact groupoids with homogeneous complex symplectic groupoids.
Let $A$ be a commutative Banach algebra. Let $M$ be a complex manifold on $A$ (an $A$-manifold). Then, we define an $A$-holomorphic vector bundle $(wedge^kT^*)(M)$ on $M$. For an open set $U$ of $M$, $omega$ is said to be an $A$-holomorphic differential $k$-form on $U$, if $omega$ is an $A$-holomorphic section of $(wedge^kT^*)(M)$ on $U$. So, if the set of all $A$-holomorphic differential $k$-forms on $U$ is denoted by $Omega_{M}^k(U)$, then ${Omega_{M}^k(U)}_{U}$ is a sheaf of modules on the structure sheaf $O_M$ of the $A$-manifold $M$ and the cohomology group $H^l(M,Omega_{M}^k)$ with the coefficient sheaf ${Omega_{M}^k(U)}_{U}$ is an $O_M(M)$-module and therefore, in particular, an $A$-module. There is no new thing in our definition of a holomorphic differential form. However, this is necessary to get the cohomology group $H^l(M,Omega_{M}^k)$ as an $A$-module. Furthermore, we try to define the structure sheaf of a manifold that is locally a continuous family of $mathbb C$-manifolds (and also the one of an analytic family). Directing attention to a finite family of $mathbb C$-manifolds, we mentioned the possibility that Dolbeault theorem holds for a continuous sum of $mathbb C$-manifolds. Also, we state a few related problems. One of them is the following. Let $nin mathbb N$. Then, does there exist a $mathbb C^n$-manifold $N$ such that for any $mathbb C$-manifolds $M_1, M_2, cdots, M_{n-1}$ and $M_n$, $N$ can not be embedded in the direct product $M_1times M_2 times cdots times M_{n-1} times M_n$ as a $mathbb C^n$-manifold ? So, we propose something that is likely to be a candidate for such a $mathbb C^2$-manifold $N$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
