On Sectional Genus of Quasi—Polarized Manifolds with Non-Negative Kodaira Dimension
✍ Scribed by Yoshiaki Fukuma
- Publisher
- John Wiley and Sons
- Year
- 2009
- Tongue
- English
- Weight
- 369 KB
- Volume
- 180
- Category
- Article
- ISSN
- 0025-584X
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✦ Synopsis
Let X be a smooth projective variety over C and L be a nef-big divisor on X. Then ( X , L ) is called a quasi-polarized manifold. Then we conjecture that g ( L ) 2 q ( X ) , where g ( L ) is t.he sectional genus of L and q ( X ) = dim H 1 ( O x ) is the irregularity of X . In general it is unknown I.hat this conjecture is true or not even in the case of dim X = 2. For example, this conjecture is true if dim X = 2 and dim H o ( L ) > 0. But it is unknown if dim X 2 3 and dim H o ( L ) > 0. In this paper, we consider a lower bound for g ( L ) if dim X = 2, dim H o ( L ) 2 2, and n(X) 2 0. We obtain ii stronger result than the above conjecture if dim BslLl 5 0 by a new method which can be applied 1.0 higher dimensional cases. Next we apply this method to the case in which dim X = n 2 3 and we obtain a lower bound for g ( L ) if dim X = 3, dim H o ( L ) 2 2, and n(X) 2 0. This conjecture is true if (X, L ) is the following cases: 1991 Mathematics Subject Classification. Primary 14 C 20.