Degeneration of the Leray spectral sequence for certain geometric quotients

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Degeneration of the Leray spectral sequence for certain geometric quotients C.A.M. Peters Department of Mathematics, University of Grenoble I UMR 5582 CNRS-UJF, 38402-Saint-Martin d'Heres France, email: J.H.M. Steenbrink? Department of Mathematics, University of Nijmegen Toernooiveld, NL-6525 ED Nijmegen The Netherlands, email: 25th March 2004 Abstract We prove that the Leray spectral sequence in rational cohomology for the quotient map Un,d ? Un,d/G where Un,d is the affine variety of equations for smooth hypersurfaces of degree d in Pn(C) and G is the general linear group, degenerates at E2. Key Words and Phrases: Geometric quotient, hypersurfaces, Leray spectral sequence Math. Subj. Class.: 14D20, 14L35, 14J70 1 Introduction We consider an affine complex algebraic group G which acts on a smooth algebraic variety X. Assume that a geometric quotient f : X ? Y for the ?The second author thanks the University of Grenoble I for its hospitality and financial support 1

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AbstractWe prove that the Leray spectral sequence in rational cohomologyfor the quotient mapUn,d→Un,d/GwhereUn,dis the aﬃne varietyof equations for smooth hypersurfaces of degreedinPn(C) andGisthe general linear group, degenerates atE2.Key Words and Phrases: Geometric quotient, hypersurfaces, Lerayspectral sequenceMath. Subj. Class.: 14D20, 14L35, 14J70

1 IntroductionWe consider an aﬃne complex algebraic groupGwhich acts on a smoothalgebraic varietyX. Assume that a geometric quotientf:X→Yfor the∗The second author thanks the University of Grenoble I for its hospitality and ﬁnancialsupport

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