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 J. Sib. Fed. Univ. Math. Phys., 2008, Volume 1, Issue 2, Pages 105–124 (Mi jsfu12)

Multi-Logarithmic Differential Forms on Complete Intersections

Alexandr G. Aleksandrova, Avgust K. Tsikhb

a Institute of Control Sciences, Russian Academy of Sciences
b Institute of Mathematics, Siberian Federal University

Abstract: We construct a complex $\Omega_S^\bullet(\log C)$ of sheaves of multi-logarithmic differential forms on a complex analytic manifold $S$ with respect to a reduced complete intersection $C\subset S$, and define the residue map as a natural morphism from this complex onto the Barlet complex $\omega_C^\bullet$ of regular meromorphic differential forms on $C$. It follows then that sections of the Barlet complex can be regarded as a generalization of the residue differential forms defined by Leray. Moreover, we show that the residue map can be described explicitly in terms of certain integration current.

Keywords: complete intersection, multi-logarithmic differential forms, regular meromorphic differential forms, Poincaré residue, logarithmic residue, Grothendieck duality, residue current.

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UDC: 517.55
Accepted: 12.04.2008
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Citation: Alexandr G. Aleksandrov, Avgust K. Tsikh, “Multi-Logarithmic Differential Forms on Complete Intersections”, J. Sib. Fed. Univ. Math. Phys., 1:2 (2008), 105–124

Citation in format AMSBIB
\Bibitem{AleTsi08} \by Alexandr~G.~Aleksandrov, Avgust~K.~Tsikh \paper Multi-Logarithmic Differential Forms on Complete Intersections \jour J. Sib. Fed. Univ. Math. Phys. \yr 2008 \vol 1 \issue 2 \pages 105--124 \mathnet{http://mi.mathnet.ru/jsfu12} \elib{https://elibrary.ru/item.asp?id=11482590} 

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This publication is cited in the following articles:
1. N. A. Bushueva, “On isotopies and homologies of subvarieties of toric varieties”, Siberian Math. J., 51:5 (2010), 776–788
2. A. G. Aleksandrov, “The Multiple Residue and the Weight Filtration on the Logarithmic de Rham Complex”, Funct. Anal. Appl., 47:4 (2013), 247–260
3. Schulze M., Tozzo L., “A Residual Duality Over Gorenstein Rings With Application to Logarithmic Differential Forms”, J. Singul., 18 (2018), 272–299
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