Cohomological invariants of complex manifolds coming from extremal rays

Abstract

In the present paper Mori extremal rays of a smooth projective manifold X are divided into two classes: L-supported and L-negligible (where ``L'' stands for ``Lefschetz'' since the division comes from Hard Lefschetz Theorem). Roughly speaking: L-supported rays are strongly distinguishable in topology while L-negligible rays have very mild geometry. Each L-supported ray R defines hyperplane in H^2(X,R) on which Lefschetz duality degenerates so it is a cohomology ring invariant. The hyperplane carries a multiplicity (cohomology ring invariant) which is related to the geometry of the ray R. The number of L-supported rays is bounded. Although the number of L-negligible rays may be infinite and they are invisible in the cohomology ring, their geometry is easier than that of L-supported rays. They are classifieable in low dimensions. Each L-negligible ray contains lots of ``good'' rational curves whose deformation is of expected dimension. In effect, L-negligible rays are invariant under deformations of complex structure and can be used to compute Gromov-Witten invariants in symplectic geometry.