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A vector bundle characterization of pn

A vector bundle characterization of pn Abh. Math. Sem. Univ. Hamburg 58, 89-96 (1988) A Vector Bundle Characterization of P" by A. LANTER! and A. J. SOMMESE The following is a natural conjecture. Conjecture. Let E be an ample rank n vector bundle on a normal, irreducible, n- dimensional projective variety X. If E is spanned at all points by global sections and cn(E) = 1, then X is biholomorphic to projective space P", and E is isomorphic to the direct sum of n copies of OF, (1). The above conjecture is obvious if n = 1, even if X is merely reduced and irreducible. In this paper we prove the above conjecture ifn = 2, or ifn = 3 and X is Gorenstein with isolated singularities. In w 1, we prove our main result (Theorem (1.1)) and use it to deduce our contribution to the conjecture from recent results of the second author [So]. Recently in his Notre Dame thesis (see [W-I), J. WlSNmWSKI has shown how to deduce the full conjecture when X is smooth from (1.1) and Mori theory. When n = 2, the above conjecture has an interpretation that gives nontrivial information about hyperplane sections of threefolds. This corollary, which has http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

A vector bundle characterization of pn

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Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general; Algebra; Differential Geometry; Number Theory; Topology; Geometry
ISSN
0025-5858
eISSN
1865-8784
DOI
10.1007/BF02941370
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Abstract

Abh. Math. Sem. Univ. Hamburg 58, 89-96 (1988) A Vector Bundle Characterization of P" by A. LANTER! and A. J. SOMMESE The following is a natural conjecture. Conjecture. Let E be an ample rank n vector bundle on a normal, irreducible, n- dimensional projective variety X. If E is spanned at all points by global sections and cn(E) = 1, then X is biholomorphic to projective space P", and E is isomorphic to the direct sum of n copies of OF, (1). The above conjecture is obvious if n = 1, even if X is merely reduced and irreducible. In this paper we prove the above conjecture ifn = 2, or ifn = 3 and X is Gorenstein with isolated singularities. In w 1, we prove our main result (Theorem (1.1)) and use it to deduce our contribution to the conjecture from recent results of the second author [So]. Recently in his Notre Dame thesis (see [W-I), J. WlSNmWSKI has shown how to deduce the full conjecture when X is smooth from (1.1) and Mori theory. When n = 2, the above conjecture has an interpretation that gives nontrivial information about hyperplane sections of threefolds. This corollary, which has

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Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg – Springer Journals

Published: Aug 28, 2008

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