On sectional genus of quasi-polarized 3-folds

Let X be a smooth projective variety over C and L a nef-big (resp. ample) d ivisor on X. Then (X, L) is called a quasi-polarized (resp. polarized) mani fold. Then we conjecture that g(L) greater than or equal to q(X), where g(L ) is the sectional genus of L and q(X) = dim H-1(O-X) is the irregularity o f X. In general it is unknown whether this conjecture is true or not, even in the case of dim X = 2. For example, this conjecture is true if dimX = 2 and dimH(0)(L) > 0. But it is unknown if dimX greater than or equal to 3 an d dimH(0)(L) > 0. In this paper, we prove g(L) greater than or equal to q(X ) if dimX = 3 and dimH(0)(L) greater than or equal to 2. Furthermore we cla ssify polarized manifolds (X, L) with dim X = 3, dim H-0(L) greater than or equal to 3, and g(L) = q(X).