Relative Polar Multiplicities and the Real Link

TL;DR

This paper establishes a connection between the topology of the real link of a complex hypersurface and relative polar multiplicities through a chain complex, leading to Morse-type inequalities relating Betti numbers and multiplicities.

Contribution

It introduces a new chain complex framework linking relative polar multiplicities to the cohomology of the real link of a hypersurface, providing novel inequalities.

Findings

Chain complex with ranks given by relative polar multiplicities

Isomorphism between cohomology of the complex and real link

Morse-type inequalities relating Betti numbers and multiplicities

Abstract

For a hypersurface defined by a complex analytic function, we obtain a chain complex of free abelian groups, with ranks given in terms of relative polar multiplicities, which has cohomology isomorphic to the reduced cohomology of the real link. This leads to Morse-type inequalities between the Betti numbers of the real link of the hypersurface and the relative polar multiplicities of the function.