Abstract
We give a new proof of the vanishing of H 1(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 nontrivial cocycles α ∈ H 1 (X, V) of negative vector bundles. Following the same circle of ideas, we use the analytic characteristics of affine bundles to obtain convexity properties of coverings of projective varieties. We suggest a weakened version of the Shafarevich conjecture: the universal covering X̄ of a projective manifold X is holomorphically convex modulo the pre-image ρ -1(Z) of a subvariety Z ⊂ X. We prove this conjecture for projective varieties X whose pullback map ρ* identifies a nontrivial extension of a negative vector bundle V by script O sign with the trivial extension.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 207-222 |
| Number of pages | 16 |
| Journal | Journal of Algebraic Geometry |
| Volume | 15 |
| Issue number | 2 |
| State | Published - Apr 2006 |
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ASJC Scopus subject areas
- Algebra and Number Theory
- Geometry and Topology
Cite this
Vanishing theorems of negative vector bundles on projective varieties and the convexity of coverings. / Bogomolov, Fedor; De Oliveira, Bruno.
In: Journal of Algebraic Geometry, Vol. 15, No. 2, 04.2006, p. 207-222.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Vanishing theorems of negative vector bundles on projective varieties and the convexity of coverings
AU - Bogomolov, Fedor
AU - De Oliveira, Bruno
PY - 2006/4
Y1 - 2006/4
N2 - We give a new proof of the vanishing of H 1(X, V) for negative vector bundles V on normal projective varieties X satisfying rank V 1 (X, V) of negative vector bundles. Following the same circle of ideas, we use the analytic characteristics of affine bundles to obtain convexity properties of coverings of projective varieties. We suggest a weakened version of the Shafarevich conjecture: the universal covering X̄ of a projective manifold X is holomorphically convex modulo the pre-image ρ -1(Z) of a subvariety Z ⊂ X. We prove this conjecture for projective varieties X whose pullback map ρ* identifies a nontrivial extension of a negative vector bundle V by script O sign with the trivial extension.
AB - We give a new proof of the vanishing of H 1(X, V) for negative vector bundles V on normal projective varieties X satisfying rank V 1 (X, V) of negative vector bundles. Following the same circle of ideas, we use the analytic characteristics of affine bundles to obtain convexity properties of coverings of projective varieties. We suggest a weakened version of the Shafarevich conjecture: the universal covering X̄ of a projective manifold X is holomorphically convex modulo the pre-image ρ -1(Z) of a subvariety Z ⊂ X. We prove this conjecture for projective varieties X whose pullback map ρ* identifies a nontrivial extension of a negative vector bundle V by script O sign with the trivial extension.
UR - http://www.scopus.com/inward/record.url?scp=33645376465&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=33645376465&partnerID=8YFLogxK
M3 - Article
AN - SCOPUS:33645376465
VL - 15
SP - 207
EP - 222
JO - Journal of Algebraic Geometry
JF - Journal of Algebraic Geometry
SN - 1056-3911
IS - 2
ER -