Sectional geometric genera for ample vector bundles

Abstract

Two invariants, called the $c_1$-sectional geometric genus and the $\co(1)$-sectional geometric genus, are introduced for generalized polarized manifolds, i.e., pairs of a projective manifold and an ample vector bundle on it. They are generalizations of the $c_1$-sectional genus and the $\co(1)$-sectional genus introduced by T.~Fujita for generalized polarized manifolds. They are also generalizations of the sectional geometric genus introduced by Y.~Fukuma for polarized manifolds. We classify generalized polarized manifolds of the smallest $c_1$ or $\co(1)$-sectional geometric genus under the spannedness condition for ample vector bundles.