| Softcover ISBN: | 978-2-85629-908-1 |
| Product Code: | SMFMEM/162 |
| List Price: | $52.00 |
| AMS Member Price: | $41.60 |
| Softcover ISBN: | 978-2-85629-908-1 |
| Product Code: | SMFMEM/162 |
| List Price: | $52.00 |
| AMS Member Price: | $41.60 |
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Book DetailsMémoires de la Société Mathématique de FranceVolume: 162; 2019; 140 ppMSC: Primary 32; 58
Among the transversally elliptic operators initiated by Atiyah and Singer, Kohn's \(\square_{b}\) operator on CR manifolds with \(S^1\) action is a natural one of geometric significance for complex analysts.
The authors' first main result establishes an asymptotic expansion for the heat kernel of such an operator with values in its Fourier components, which involves a contribution in terms of a distance function from lower dimensional strata of the \(S^1\)-action. The second main result computes a local index density, in terms of tangential characteristic forms, on such manifolds as Sasakian manifolds (of interest in string theory), by showing that certain non-trivial contributions from strata in the heat kernel expansion will eventually cancel out by applying Getzler's rescaling technique to off-diagonal estimates. This leads to a local result which can be thought of as a type of local index theorem on these CR manifolds.
The authors give examples of these CR manifolds, some of which arise from Brieskorn manifolds. Moreover, in some cases, the authors reinterpret Kawasaki's Hirzebruch-Riemann-Roch formula for a complex orbifold equipped with an orbifold holomorphic line bundle as an index theorem obtained by a single integral over a smooth CR manifold. They achieve this without the use of equivariant cohomology methods.
A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.
ReadershipGraduate students and research mathematicians.
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Among the transversally elliptic operators initiated by Atiyah and Singer, Kohn's \(\square_{b}\) operator on CR manifolds with \(S^1\) action is a natural one of geometric significance for complex analysts.
The authors' first main result establishes an asymptotic expansion for the heat kernel of such an operator with values in its Fourier components, which involves a contribution in terms of a distance function from lower dimensional strata of the \(S^1\)-action. The second main result computes a local index density, in terms of tangential characteristic forms, on such manifolds as Sasakian manifolds (of interest in string theory), by showing that certain non-trivial contributions from strata in the heat kernel expansion will eventually cancel out by applying Getzler's rescaling technique to off-diagonal estimates. This leads to a local result which can be thought of as a type of local index theorem on these CR manifolds.
The authors give examples of these CR manifolds, some of which arise from Brieskorn manifolds. Moreover, in some cases, the authors reinterpret Kawasaki's Hirzebruch-Riemann-Roch formula for a complex orbifold equipped with an orbifold holomorphic line bundle as an index theorem obtained by a single integral over a smooth CR manifold. They achieve this without the use of equivariant cohomology methods.
A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.
Graduate students and research mathematicians.
