BOUNDING CODIMENSION-ONE SUBVARIETIES AND A GENERAL INEQUALITY BETWEEN CHERN NUMBERS

We extend the Miyaoka-Yau inequality for a surface to an arbitrary non uniruled normal complex projective variety, eliminating the hypothesis that the variety must be minimal. The inequality is sharp in dimensio n three and is also sharp among minimal varieties. For nonminimal vari eties in dimension four or higher, an error term is picked up which ca n be controlled. As a consequence. we bound codimension one subvarieti es in a variety of general type linearly in terms of their Chern numbe rs. In particular, we show that there are only a finite number of smoo th Fano, Abelian and Calabi-Yau subvarieties of codimension one in any variety of general type.