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The Witten complex for algebraic curves with cone-like singularities. (English) Zbl 1166.32015

The main purpose of this short paper is to generalize the Witten deformation to singular complex algebraic curves with cone-like singularities. The author also gives an analytic proof to the Morse inequalities. The contribution of the singular points to the Morse inequalities is closely related to the lack of essential self-adjointness of the Laplace operator (acting on smooth forms with compact support) in the presence of singularities. Moreover, it argued that the singular points contribute to the Morse inequalities in degree 1 and have to be counted with “multiplicities”.

MSC:

32S30 Deformations of complex singularities; vanishing cycles
58H15 Deformations of general structures on manifolds
14B07 Deformations of singularities
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
57R57 Applications of global analysis to structures on manifolds
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References:

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