ON SECTIONAL GENUS OF QUASI-POLARIZED MANIFOLDS WITH NONNEGATIVE KODAIRA DIMENSION

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 conje cture that g(L) greater than or equal to q(X), where g(L) is the secti onal genus of L and q(X) = dim H-1(O-X) is the irregularity of X. In g eneral it is unknown that this conjecture is true or not even in the c ase of dim X = 2. For example, this conjecture is true if dim X = 2 an d dim X(0)(L) > 0. But it is unknown if dim X greater than or equal to 3 and dim H-0(L) > 0. In this paper, we consider a lower bound for g( L) if dim X = 2, dim H-0(L) greater than or equal to 2, and kappa(X) g reater than or equal to 0. We obtain a stronger result than the above conjecture if dim Bs\L\ less than or equal to 9 by a new method which can be applied to higher dimensional cases. Next we apply this method to the case in which dim X = n greater than or equal to 3 and we obtai n a lower bound for g(L) if dim X = 3, dim H-0(L) greater than or equa l to 2, and kappa(X) greater than or equal to 0.