Mordell-Weil lattices of type E8 and deformation of singularities

1991 ◽  
pp. 177-202 ◽  
Author(s):  
Tetsuji Shioda
Keyword(s):  
Deformation Of Singularities

scholarly journals DEFORMATION OF SINGULARITIES AND THE HOMOLOGY OF INTERSECTION SPACES

2012 ◽  
Vol 04 (04) ◽  
pp. 413-448 ◽  
Author(s):  
MARKUS BANAGL ◽  
LAURENTIU MAXIM
Keyword(s):  
Hodge Structure ◽  
Ring Homomorphism ◽  
Intersection Cohomology ◽  
Mixed Hodge Structure ◽  
Intersection Homology ◽  
Continuous Map ◽  
Rational Cohomology ◽  
Deformation Of Singularities ◽  
Complex Projective ◽  
Singular Hypersurface

While intersection cohomology is stable under small resolutions, both ordinary and intersection cohomology are unstable under smooth deformation of singularities. For complex projective algebraic hypersurfaces with an isolated singularity, we show that the first author's cohomology of intersection spaces is stable under smooth deformations in all degrees except possibly the middle, and in the middle degree precisely when the monodromy action on the cohomology of the Milnor fiber is trivial. In many situations, the isomorphism is shown to be a ring homomorphism induced by a continuous map. This is used to show that the rational cohomology of intersection spaces can be endowed with a mixed Hodge structure compatible with Deligne's mixed Hodge structure on the ordinary cohomology of the singular hypersurface. Regardless of monodromy, the middle degree homology of intersection spaces is always a subspace of the homology of the deformation, yet itself contains the middle intersection homology group, the ordinary homology of the singular space, and the ordinary homology of the regular part.

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