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# The Grothendieck Trace and the de Rham Integral

Published:2003-09-01
Printed: Sep 2003
• Pramathanath Sastry
• Yue Lin L. Tong
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## Abstract

On a smooth $n$-dimensional complete variety $X$ over ${\mathbb C}$ we show that the trace map ${\tilde\theta}_X \colon\break H^n (X,\Omega_X^n) \to {\mathbb C}$ arising from Lipman's version of Grothendieck duality in \cite{ast-117} agrees with $$(2\pi i)^{-n} (-1)^{n(n-1)/2} \int_X \colon H^{2n}_{DR} (X,{\mathbb C}) \to {\mathbb C}$$ under the Dolbeault isomorphism.
 MSC Classifications: 14F10 - Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials [See also 13Nxx, 32C38] 32A25 - Integral representations; canonical kernels (Szegoo, Bergman, etc.) 14A15 - Schemes and morphisms 14F05 - Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20] 18E30 - Derived categories, triangulated categories