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

设X是定义在复数域上的n维光滑射影簇，L是X上的充分线丛，则(X，L)对称为极化流形。利用第i个截断几何亏格gi(X，L)等几个截断不变量，研究了射影曲面理论的极化流形形式，并给出了它的应用.三年来，我们取得了以下成果。1.研究了截断Betti数和截断Hodge数的性质，得到了如下结果：(1)(X，L)的一个分类，其中b_2(X，L)=dim H^2(X，Z)或0〓H^&lt；1，1&gt；_2，(X，L)〓1。_2(X，L)=2，假设L很大。作为截口不变量的一个应用，我们研究了伴随丛Kx+Tl的整体截口的维数。我们得到了以下结果，这些结果是解决关于伴随丛整体截面的一些猜想的第一步：(1)当dimX=3时：如果0〓κ(K_x+L)〓2或κ(X)〓0，则dimH^0(K_x+L)&gt；0成立。如果κ(K_x+L)=3，则对〓2中Win中的每一个整数，都有dim H^0(m(K_x+L))&gt；0。此外，如果L_1和L_2是X上的充分线团，且K_x+L_1+L_2是nef，则有dim H^0(K_x+L_1+L_2)&gt；0。(2)若Dim X=4：若0〓κ(K_x+L)〓2且K_x+L为NEF，则Dim H^0(K_x+L)&gt；0成立。如果κ(K_x+L)〓3和K_x+L是nef，则对m个〓为4的整数m，dim H^0(m(K_x+L))&gt；0成立。

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.