On Sectional Genus of Quasi—Polarized Manifolds with Non-Negative Kodaira Dimension

Let X be a smooth projective variety over ? and L be a nef-big divisor on X. Then (X, L) is called a quasi - polarized manifold. Then we conjecture that g(L) ≥ q(X), where g(L) is the sectional genus of L and q(X) = dim H¹(O_x) is the irregularity of X. In general it is unknown that 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≥(L) > 0. But it is unknown if dim X ≥ 3 and dim H⁰(L) > 0. In this paper, we consider a lower bound for g(L) if dim X = 2, dim H⁰(L) ≥ 2, and k(X) ≥ 0. We obtain a stronger result than the above conjecture if dim Bs|L| ≤ 0 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 ≥ 3 and we obtain a lower bound for g(L) if dim X = 3, dim H⁰(L) ≥ 2, and k(X) ≥ 0.