Study on properties of polarirad varieties by their sectional invariants and its applications

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Let X be an n-dimensional smooth projective variety defined over the field of complex numbers, and let L be an ample line bundle on X. Then the pair (X, L) is called a polarized manifold. The purpose of this research is to investigate a polarized manifold's version of the theory of projective surfaces by using several sectional invariants such as the ith sectional geometric genus gi(X, L), and to give its application. We have obtained the following results for three years. (The following are main results of this research.)1. We investigated properties of the sectional Betti numbers and the sectional Hodge numbers, and the following were obtained :(1) A classification of (X, L) with b_2(X, L)= dim H^2(X, Z)or 0 〓H^<1,1>_2,(X, L)〓 1 under the assumption that L is base point free.(2) A classification of (X, L) with h^<1,1>_2(X, L)= 2 under the assumption that L is very ample.2. As an application of sectional invariants, we investigated the dimension of global sections of adjoint bundles K_x + tL. We obtained the following results which become the first step to solve some conjectures concerning the dimension of global sections of adjoint bundles.(1)The case where dimX = 3: If 0 〓 κ(K_x + L) 〓 2 or κ(X)〓 0, then dimH^0(K_x + L)> 0 holds. If κ(K_x + L)= 3, then dim H^0(m(K_x + L))> 0 holds for every integer in with in 〓 2. Moreover if L_1 and L_2 are ample line bunldes on X and K_x + L_1+ L_2 is nef, then we have dim H^0 (K_x + L_1 + L_2)> 0. (The last result is thought to be a generalization of a conjecture of Beltrametti and Sommese.)(2) The case where dim X = 4: If 0 〓 κ(K_x + L)〓 2 and K_x + L is nef, then dim H^0(K_x + L)> 0 holds. If κ(K_x + L)〓 3 and K_x + L is nef, then dim H^0(m(K_x + L))> 0 holds for every integer m with m 〓 4.