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

设X是复数域上的n维光滑射影簇，L是X上的一个充要线丛。则对（X，L）称为极化流形。本文利用第i个截面几何亏格gi（X，L）等截面不变量，研究了投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。(The以下是这项研究的主要成果）。1.研究了截Betti数和截Hodge数的性质，得到了如下结果：（1）在L无基点的假设下，得到了（X，L）的一个分类，其中b2（X，L）= dim H^2（X，Z）或0 <$H^<1，1> 2，（X，L）<$1. (2)在L是很满的假设下，给出了（X，L）的一个分类，其中h^<1，1>_2（X，L）= 2.作为截面不变量的一个应用，我们研究了伴随丛Kx + tL的整体截面的维数.我们得到了如下结果，这些结果成为解决伴随丛整体截面维数问题的第一步。（1）当dimX = 3时：若0 <$κ（K_x + L）<$2或κ（X）<$0，则dimH^0（K_x + L）> 0成立。若κ（K_x + L）= 3，则对所有在λ 2中的整数dim H^0（m（K_x + L））> 0成立.此外，若L_1和L_2是X上的样本线丛，且K_x + L_1+ L_2是nef，则有dim H^0（K_x + L_1 + L_2）> 0. (The最后一个结果被认为是Beltrametti和Sommese猜想的推广。）(2)当dim X = 4时：若0 <$κ（K_x + L）<$2且K_x + L为nef，则dim H^0（K_x + L）> 0成立。若κ（K_x + L）<$3且K_x + L是nef，则对任意整数m，dim H^0（m（K_x + L））> 0成立，其中m <$4.

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.