TAILIEUCHUNG - Đề tài "Subelliptic SpinC Dirac operators, I "

Let X be a compact K¨hler manifold with strictly pseudoconvex bounda ary, Y. In this setting, the SpinC Dirac operator is canonically identified with ¯ ¯ ∂ + ∂ ∗ : C ∞ (X; Λ0,e ) → C ∞ (X; Λ0,o ). We consider modifications of the classi¯ cal ∂-Neumann conditions that define Fredholm problems for the SpinC Dirac operator. In Part 2, [7], we use boundary layer methods to obtain subelliptic estimates for these boundary value problems. Using these results, we obtain an expression for the finite part of the holomorphic Euler characteristic of a strictly pseudoconvex manifold. | Annals of Mathematics Subelliptic SpinC Dirac operators I By Charles L. Epstein Annals of Mathematics 166 2007 183 214 Subelliptic Spine Dirac operators I By Charles L. Epstein Dedicated to my parents Jean and Herbert Epstein on the occasion of their eightieth birthdays Abstract Let X be a compact Kahler manifold with strictly pseudoconvex boundary Y. In this setting the Spine Dirac operator is canonically identified with d d C X A0 e C X A0 o . We consider modifications of the classical Ỡ-Neumann conditions that define Fredholm problems for the Spine Dirac operator. In Part 2 7 we use boundary layer methods to obtain subelliptic estimates for these boundary value problems. Using these results we obtain an expression for the finite part of the holomorphic Euler characteristic of a strictly pseudoconvex manifold as the index of a Spine Dirac operator with a subellip-tic boundary condition. We also prove an analogue of the Agranovich-Dynin formula expressing the change in the index in terms of a relative index on the boundary. If X is a complex manifold partitioned by a strictly pseudoconvex hypersurface then we obtain formulrn for the holomorphic Euler characteristic of X as sums of indices of Spine Dirac operators on the components. This is a subelliptic analogue of Bojarski s formula in the elliptic case. Introduction Let X be an even dimensional manifold with a Spine-structure see 6 12 . A compatible choice of metric g defines a Spine Dirac operator 3 which acts on sections of the bundle of complex spinors . The metric on X induces a metric on the bundle of spinors. If ơ ơ g denotes a pointwise inner product then we define an inner product of the space of sections of by setting ơ ơ x Ị ơ ơ g dVg . Research partially supported by NSF grants DMS99-70487 and DMS02-03795 and the Francis J. Carey term chair. 184 CHARLES L. EPSTEIN If X has an almost complex structure then this structure defines a Spine-structure. If the complex structure is integrable then the bundle

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