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arXiv · 2509.20685

A Morse complex for the homology of vanishing cycles

Aleksander Doan·Juan Muñoz-Echániz
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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.

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Aleksander Doan, Juan Muñoz-Echániz. 2025-09-25. A Morse complex for the homology of vanishing cycles. https://arxiv.org/abs/2509.20685

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