Remarks on the second sectional geometric genus of quasi-polarized manifolds and their applications

2010/03/30 by Yoshiaki Fukuma, Fukuma, Yoshiaki
Mathematics · #14C17 (Secondary) #14C20 (Primary) #14E30 #14J30 #14J35 #14J40 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14C17 #msc:14C20 #msc:14E30 #msc:14J30 #msc:14J35 #msc:14J40

paper · pdf · doi:10.48550/arxiv.1003.5736

20 pages. We improved some results and corrected some errata in the second version of this paper

arxiv created 2010/08/02 · arxiv updated 2010/08/03

Abstract

In our previous papers, we investigated a lower bound for the second sectional geometric genus g₂(X,L) of n-dimensional polarized manifolds (X,L) and by using these, we studied the dimension of global sections of K_X+tL with t≥ 2. In this paper, we consider the case where (X,L) is a quasi-polarized manifold. First we will prove g₂(X,L)≥ h¹(O_X) for the following cases: (a) n=3, κ(X)=-∞ and κ(K_X+L)≥ 0. (b) n≥ 3 and κ(X)≥ 0. Moreover, by using this inequality, we will study h⁰(K_X+tL) for the case where (X,L) is a quasi-polarized 3-fold.

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