Vanishing Theorems and Universal Coverings of Projective Varieties
نویسندگان
چکیده
This article contains a new argument which proves vanishing of the first cohomology for negative vector bundles over a complex projective variety if the rank of the bundle is smaller than the dimension of the base. Similar argument is applied to the construction of holomorphic functions on the universal covering of the complex projective variety .
منابع مشابه
Vanishing Theorems of Negative Vector Bundles on Projective Varieties and the Convexity of Coverings
We give a new proof of the vanishing of H1(X, V ) for negative vector bundles V on normal projective varieties X satisfying rank V < dim X. Our proof is geometric, it uses a topological characterization of the affine bundles associated with non-trivial cocycles α ∈ H1(X, V ) of negative vector bundles. Following the same circle of ideas, we use the analytic characteristics of affine bundles to ...
متن کاملConvexity of Coverings of Projective Varieties and Vanishing Theorems
Let X be a projective manifold, ρ : X̃ → X its universal covering and ρ∗ : V ect(X) → V ect(X̃) the pullback map for isomorphism classes of vector bundles. This article makes the connection between the properties of the pullback map ρ∗ and the properties of the function theory on X̃. Our approach motivates a weakened version of the Shafarevich conjecture: the universal covering X̃ of a projective m...
متن کاملNilpotent groups and universal coverings of smooth projective varieties
Characterizing the universal coverings of smooth projective varieties is an old and hard question. Central to the subject is a conjecture of Shafarevich according to which the universal cover X̃ of a smooth projective variety is holomorphically convex, meaning that for every infinite sequence of points without limit points in X̃ there exists a holomorphic function unbounded on this sequence. In t...
متن کاملHolomorphic Functions and Vector Bundles on Coverings of Projective Varieties
Let X be a projective manifold, ρ : X̃ → X its universal covering and ρ∗ : V ect(X) → V ect(X̃) the pullback map for the isomorphism classes of vector bundles. This article establishes a connection between the properties of the pullback map ρ∗ and the properties of the function theory on X̃. We prove the following pivotal result: if a universal cover of a projective variety has no nonconstant holo...
متن کاملBranched Coverings and Minimal Free Resolution for Infinite-dimensional Complex Spaces
We consider the vanishing problem for higher cohomology groups on certain infinite-dimensional complex spaces: good branched coverings of suitable projective spaces and subvarieties with a finite free resolution in a projective space P(V ) (e.g. complete intersections or cones over finitedimensional projective spaces). In the former case we obtain the vanishing result for H. In the latter case ...
متن کامل