A Morse complex for the homology of vanishing cycles

Abstract

We construct Morse homology groups associated with any regular function on a smooth complex algebraic variety, allowing singular and non-compact critical loci. These groups are generated by critical points of a certain large pertubation of the function, built from a normal crossing compactification of the variety. They are canonically isomorphic to the homology of vanishing cycles and -- in the absence of bifurcations at infinity -- recover the hypercohomology of the perverse sheaf of vanishing cycles, studied extensively in singularity theory and enumerative geometry. Our construction arises as a special case of a more general construction of Morse homology of non-compact manifolds that admit a compactification by a manifold with corners.