A lower bound for the second sectional geometric genus of polarized manifolds

Abstract

Let (X, L) be a polarized manifold defined over ℂ with dim X = n ≥ 3 and let (M, A ) be a reduction of (X, L). In this paper, if X is not uniruled and K_M + (n - 2)A is nef and (n - 2)-big, then we give a lower bound for c ₂(M )A n - 2, which improves a result in a previous paper. By using this bound, we prove that g ₂ (X, L) ≥ h ¹(## add figure 'advg.5.3.431_01.gif' ## _x ) if n ≥ 3, X is not uniruled, and K_M + (n - 2)A is nef and (n - 2)-big, where g ₂ (X, L) is the second sectional geometric genus of (X, L). In particular, if n ≥ 3 and κ (X ) ≥ 0, then g ₂ (X, L) ≥ h ¹(## add figure 'advg.5.3.431_01.gif' ## _x ). Furthermore we prove that g ₂ (X, L) ≥ 0 for any polarized 3-folds. For n ≥ 3 we also study (X, L) with g ₂ (X, L) ≥ h ¹(## add figure 'advg.5.3.431_01.gif' ##x ). Finally we consider a lower bound of h ⁰(K_X + L) for polarized 3-folds (X, L ).