\(L^2\) -cohomology on the coverings of a compact complex manifold. (French) Zbl 1066.32012
The aim of this paper is to define a natural \(L^2\)-cohomology on any unramified covering of a complex analytic space \(X\), with values in the lifting of any coherent analytic sheaf on \(X\). This \(L^2\) cohomology has been constructed independently by P. Eyssidieux [Math. Ann. 317, 527–566 (2000; Zbl 0964.32008)].
It is seen that the usual properties of sheaf cohomology such as cohomology exact sequences or spectral sequences hold in this \(L^2\)-cohomology on \(X\). If \(X\) is projective and non-singular there are \(L^2\) vanishing theorems analogous to those of Kodaira-Serre and Kawamata-Viehweg.
When \(X\) is compact it is possible to define the \(\Gamma\)-dimension for Galois coverings. This \(\Gamma\)-dimension turns out to be finite in this case. An extension of Atiyah’s index theorem is given in this context.
It is seen that the usual properties of sheaf cohomology such as cohomology exact sequences or spectral sequences hold in this \(L^2\)-cohomology on \(X\). If \(X\) is projective and non-singular there are \(L^2\) vanishing theorems analogous to those of Kodaira-Serre and Kawamata-Viehweg.
When \(X\) is compact it is possible to define the \(\Gamma\)-dimension for Galois coverings. This \(\Gamma\)-dimension turns out to be finite in this case. An extension of Atiyah’s index theorem is given in this context.
Reviewer: A. Diaz-Cano (Madrid)
MSC:
| 32C35 | Analytic sheaves and cohomology groups |
| 32C99 | Analytic spaces |
| 32T99 | Pseudoconvex domains |
| 58J20 | Index theory and related fixed-point theorems on manifolds |
Citations:
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