A lower bound for sectional genus of quasi-polarized manifolds
Let \(X\) be a smooth projective manifold defined over the complex number field and let \(L\) be a nef-big divisor on \(X\). Then we call the pair \((X,L)\) a quasi-polarized manifold. The sectional genus \(g(L)\) of \((X,L)\) is defined by the following formula: \(2g(L)-2= (K_X+ (n-1)L)L^{n-1}\), where \(n:= \dim X\) and \(K_X\) is the canonical divisor of \(X\). In this paper, we consider the following conjecture: (Conjecture 1) \(g(L)\geq q(X)\), where \(q(X)\) is the irregularity of \(X\). But it is difficult to prove this inequality in general. So we consider the following case: \((X,L)\) has a fiber space \((f,X,Y,L)\), that is, there exists a surjective morphism \(f:X\to Y\) with connected fibers, where \(Y\) is a smooth projective variety with \(\dim X>\dim Y\geq 1\). In this case we consider the following conjecture: (Conjecture 2) \(g(L)\geq q(Y)\) for any quasi-polarized fiber space \((f,X,Y,L)\). By using results with respect to conjecture 2, we study conjecture 1. -- In this paper, we consider the following cases: (1) \(L\) is ample and \(Y\) is a smooth curve, (2) \(\kappa (Y)=0\) or 1 for \(\dim Y\geq 2\). In the case (1), we obtain \(g(L)\geq q(Y)\) and we also characterize \((f,X,Y,L)\) with \(g(L)= q(Y)\). Furthermore we obtain that \(g(L)\geq q(X)\) for any quasi-polarized manifolds \((X,L)\) with \(\kappa (X)=1\) and \(L^n\geq 2\) by using the method of the case (1). In the case (2), we obtain \(g(L)\geq q(Y)\) if \(L^n\geq 2\). In the appendix, we study the weak positivity of \(f_* ((\omega_{X/Y} \otimes L)^{\otimes k})\) with any positive integer \(k\) for a quasi-polarized fiber space \((f,X,Y,L)\), which is used in this paper.
scientific article; zbMATH DE number 1172197