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Geometry of twisting cochains. (English) Zbl 0641.32021
The results here extend the work of D. Toledo and Y. L. Tong on the Hirzebruch-Riemann-Roch formula for complex manifolds [Topology 15, 273-301 (1976; Zbl 0355.58014)]. Their proof of the formula involves the construction, for any holomorphic vector bundle E over a smooth complex manifold X, of a certain Čech cocycle which for compact X integrates to give the Euler characteristic \[ \chi (X,E)=\Sigma (-1)^ i\dim_{{\mathbb{C}}}H^ i(X,E). \] The argument is based on Serre- Grothendieck duality for the diagonal embedding \(\Delta\) : \(X\to X\times X\) and comparison of local Koszul resolutions for the sheaf of the diagonal by a ‘twisting cochain’ o) in Hilbert space. We succeed in constructing an operator-valued symbol for finite sections of such operators, and we set up necessary and sufficient conditions for the finite section method to be applicable. In the present paper we extend these results to the Banach space case. Our results essentially generalize well-known statements operators, for singular integral operators and for singular integral operators with Carleman shift.
Reviewer: Z.Binderman
MSC:
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
57R20 Characteristic classes and numbers in differential topology
14C40 Riemann-Roch theorems
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References:
[1] M.F. Atiyah : Complex analytic connections in fibre bundles . Trans. Amer. Math. Soc. 85 (1957) 181-207. · Zbl 0078.16002 · doi:10.2307/1992969
[2] P.F. Baum and R. Bott : On the zeros of meromorphic vector fields . Essays on topology and related topics , memoires dédiés à Georges de Rham. Springer, Berlin, Heidelberg, New York (1970) 29-47. · Zbl 0193.52201
[3] N. Berline and M. Vergne : A computation of the equivariant index of the Dirac operator . Bull. Soc. Math. France 113 (1985) 305-345. · Zbl 0592.58044 · doi:10.24033/bsmf.2036 · numdam:BSMF_1985__113__305_0 · eudml:87490
[4] J.-M. Bismuth : The Atiyah-Singer Theorems: a probabilistic approach. I. The index theorem . J. Functional Analysis 57 (1984) 56-99. · Zbl 0538.58033 · doi:10.1016/0022-1236(84)90101-0
[5] E. Getzler : Pseudodifferential operators on supermanifolds and the Atiyah-Singer Index theorem . Commun. Math. Phys. 92 (1983) 163-178. · Zbl 0543.58026 · doi:10.1007/BF01210843
[6] V. Guillemin and S. Sternberg : Geometric Asymptotics . A.M.S. Mathematical Surveys 14 (1977). · Zbl 0364.53011 · www.ams.org
[7] N.R. O’Brian , D. Toledo and Y.L.L. Tong : The trace map and characteristic classes for coherent sheaves . Am. J. Math. 103 (1981) 225-252. · Zbl 0473.14008 · doi:10.2307/2374215
[8] N.R. O’Brian , D. Toledo and Y.L.L. Tong : Hirzebruch-Riemann-Roch for coherent sheaves . Am. J. Math. 103 (1981) 253-271. · Zbl 0474.14009 · doi:10.2307/2374216
[9] N.R. O’Brian , D. Toledo and Y.L.L. Tong : A Grothendieck-Riemann-Roch formula for maps of complex manifolds . Math. Ann. 271 (1985) 493-526. · Zbl 0539.14005 · doi:10.1007/BF01456132 · eudml:163984
[10] V.K. Patodi : An analytic proof of Riemann-Roch-Hirzebruch theorem for Kaehler manifolds . J. Diff. Geometry 5 (1971) 251-283. · Zbl 0219.53054 · doi:10.4310/jdg/1214429991
[11] D. Toledo and Y.L.L. Tong : A parametrix for \partial and Riemann-Roch in Čech theory. Topology 15 (1976) 273-301. · Zbl 0355.58014 · doi:10.1016/0040-9383(76)90022-7
[12] D. Toledo and Y.L.L. Tong : Duality and intersection theory in complex manifolds I . Math. Ann. 237 (1978) 41-77. · Zbl 0391.32008 · doi:10.1007/BF01351557 · eudml:182776
[13] D. Toledo and Y.L.L. Tong : Duality and intersection theory in complex manifolds II, the holomorphic Lefschetz formula . Ann. Math. 108 (1978) 518-538. · Zbl 0413.32006 · doi:10.2307/1971186 · eudml:182776
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