Abstract:Finding the Noether inequality for varieties of general type is a natural problem in the study of algebraic geometry, which dates back to the work of M. Noether for surfaces. Later, Debarre proved that irregular surfaces of general type satisfy a stronger Noether inequality. Recently, J. Chen, M. Chen and C. Jiang established the optimal Noether inequality in dimension three. In this ...
Abstract:The systole of a closed Riemannian manifold is defined to be the length of a shortest noncontractible loop. Gromov's systolic inequality relates systole to volume, which is a curvature free inequality. Gromov proved that systolic inequality holds on closed essential manifolds. Gromov's further work indicates that systolic inequality is related to topological complicatedness of manifol...