Premium
On Manifolds Whose Tangent Bundle is Big and 1‐Ample
Author(s) -
Solá Conde Luis Eduardo,
Wiśniewski Jarosław A.
Publication year - 2004
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611504014856
Subject(s) - mathematics , tangent bundle , line bundle , vector bundle , unit tangent bundle , pure mathematics , projective space , normal bundle , tangent space , ample line bundle , morphism , manifold (fluid mechanics) , bundle , frame bundle , symplectic geometry , complex projective space , projective test , mechanical engineering , materials science , engineering , composite material
A line bundle over a complex projective variety is called big and 1‐ample if a large multiple of it is generated by global sections and a morphism induced by the evaluation of the spanning sections is generically finite and has at most 1‐dimensional fibers. A vector bundle is called big and 1‐ample if the relative hyperplane line bundle over its projectivisation is big and 1‐ample. The main theorem of the present paper asserts that any complex projective manifold of dimension 4 or more, whose tangent bundle is big and 1‐ample, is equal either to a projective space or to a smooth quadric. Since big and 1‐ample bundles are ‘almost’ ample, the present result is yet another extension of the celebrated Mori paper ‘Projective manifolds with ample tangent bundles’ ( Ann. of Math. 110 (1979) 593–606). The proof of the theorem applies results about contractions of complex symplectic manifolds and of manifolds whose tangent bundles are numerically effective. In the appendix we re‐prove these results. 2000 Mathematics Subject Classification 14E30, 14J40, 14J45, 14J50.