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The Grothendieck Trace and the de Rham Integral

Published online by Cambridge University Press:  20 November 2018

Pramathanath Sastry  and
Yue Lin L. Tong
Pramathanath Sastry
Affiliation:
University of Toronto at Mississauga Mississauga, Ontario, email: pramath@math.toronto.edu
Yue Lin L. Tong
Affiliation:
Purdue University West Lafayette, IN USA, email: tong@math.purdue.edu
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Abstract

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On a smooth $n$-dimensional complete variety $X$ over $\mathbb{C}$ we show that the trace map ${{\bar{\theta }}_{X}}\,:\,{{H}^{n}}(X,\,\Omega _{X}^{n})\,\to \,\mathbb{C}$ arising from Lipman's version of Grothendieck duality in $\left[ \text{L} \right]$ agrees with

$${{(2\pi i)}^{-n}}{{(-1)}^{n(n-1)/2}}\,\int_{X}{:\,H_{DR}^{2n}\,(X,\,\mathbb{C})\,\to \,\mathbb{C}}$$

under the Dolbeault isomorphism.

Keywords

14F1032A2514A1514F0518E30
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 46 , Issue 3 , 01 September 2003 , pp. 429 - 440
Copyright
Copyright © Canadian Mathematical Society 2003

References

[C] Conrad, B., Grothendieck duality and base change. Lecture Notes in Math. 1750, Springer-Verlag, New York, 2000.Google Scholar
[C1] Conrad, B., Clarifications and corrections to .Grothendieck duality and base change.. Preprint, www-math.mit.edu/.dejong.Google Scholar
[C2] Conrad, B., More Examples. Preprint, www-math.mit.edu/.dejong.Google Scholar
[D] Deligne, P., Intégration sur un cycle évanescent. Invent.Math. 76 (1983), 129–143.Google Scholar
[dR] de Rham, G., Variétés différentiables. Hermann, Paris, 1955.Google Scholar
[GH] Griffiths, P. and Harris, J., Principles of Algebraic Geometry. John Wiley & Sons, New York, 1978.Google Scholar
[G1] Grothendieck, A., Théorèmes de dualité pour les faisceaux algébriques cohérents. Séminaire Bourbaki 149, May 1957.Google Scholar
[G2] Grothendieck, A., Rev.etements étales et groupe fondamental. Lecture Notes in Math. 224, Springer-Verlag, Heidelberg, 1971.Google Scholar
[Ha] Hartshorne, R., Algebraic Geometry. Graduate Texts in Math. 52, Springer-Verlag, 1977.Google Scholar
[H] Harvey, F. R., Integral formulae connected by Dolbeault's isomorphism. Rice Univ. Studies 56 (1970), 77–97.Google Scholar
[I] Iversen, B., Cohomology of sheaves. Universitext, Springer-Verlag, 1986.Google Scholar
[L] Lipman, J., Dualizing sheaves, differentials and residues on algebraic varieties. Astérisque 117, 1984.Google Scholar
[T] Tong, Y. L. L., Integral representation formulae and the Grothendieck residue symbol. Amer. J. Math. 95 (1973), 904–917.Google Scholar
[TT] Toledo, D. and Tong, Y. L. L., A parametrix for ∂ and the Riemann-Roch in Čech theory. Topology 15 (1976), 273–301.Google Scholar