\(L^ 2\)-Dolbeault complexes on singular curves and surfaces. (English) Zbl 0684.58040
Summary: Give the smooth part of a singular curve or normal surface the metric induced from the ambient projective space. On this incomplete manifold the minimal \(L^ 2{\bar \partial}\)-complex of (0,q)-forms has finite- dimensional cohomology groups. The Euler characteristic of this cohomology equals the Todd genus of any desingularization of the singular variety.
MSC:
58J20 | Index theory and related fixed-point theorems on manifolds |
58J10 | Differential complexes |
57R20 | Characteristic classes and numbers in differential topology |
32J25 | Transcendental methods of algebraic geometry (complex-analytic aspects) |