Institut de Mathématiques de Luminy

Singularities in Geometry and Topology

1 - Singularities and noncommutative geometry.
2 - Characteristic classes of singular varieties.
3 - Toric varieties.
4 - Topology of complex singularities.
5 - Singularities of spaces of representations.
6 - Steenrod operations and Singularities.
7 - Analytic geometry

5.1. Singularities and noncommutative geometry.

With Andre Legrand (Toulouse), Jean-Paul Brasselet established the relation between noncommutative geometry and singularities. More precisely, for a singular space provided with "a good" stratification, one defines a mixed complex whose periodic cyclic homology corresponds to intersection homology of the singular variety. This result generalizes a well-known result of A. Connes saying that the periodic cyclic homology of the algebra of differentiable functions on a smooth variety corresponds to the de Rham cohomology of this variety.

5.2. Characteristic classes of singular varieties.

With D. Lehmann (Montpellier), J Seade (Mexico City) and T Suwa (Sapporo), Jean-Paul Brasselet showed the relation between Schwartz-MacPherson classes and Fulton-Jonhson classes in the case of complete intersections. This relation is expressed using a generalization of the Milnor numbers, already introduced and studied by A. Parusinski and P. Pragacz.

Related works concern the study of bivariant constructible functions (with S. Yokura, Kagoshima) and an interpretation of the Milnor classes in Algebraic Geometry (with P. Aluffi, Tallahassee).

In another work, with Le Dung Trang (Marseilles) and J Seade, we show that the Euler local obstruction in a point of a singular variety is equal to the index of a radial field on a general hyperplane section passing by this point. Importance of this result lies in the fact that the Euler local obstruction is one of the main ingredients of the construction of MacPherson and is a fundamental local invariant of singular varieties.

5.3. Toric varieties.

The toric varieties are objects for which it is possible to express many topological and geometrical properties in terms of combinatorial geometry. With K Fieseler (Uppsala) and G Barthel and L Kaup (Konstanz), Jean-Paul Brasselet studied the equivariant intersection homology of toric varieties (vanishing theorems) and the corresponding Poincaré polynomial. From this study, one deduces a notion of combinatorial sheaves on (non necessarily rational) fans letting hope to show a "hard Lefschetz Theorem" for these fans.

5.4. Topology of the complex singularities.

Anne Pichon worked on several research programmes concerning the topology of complex singularities of surfaces:

In collaboration with I Luengo (Complutense University of Madrid), she described the topological action of the normalisation on the link of singularities of complex surfaces, then applied this result by giving an explicit description of the topology of the link of singularities of hypersurfaces in the origin of C 3 of equations f 1 (x, y, z) + f 2 (x, y, z)=0, where f 1 and f 2 are two homogeneous polynomials. Anne Pichon and I Luengo also worked on the characterization of the singularities of surfaces whose link is a sphere. They showed the conjecture of Lź D. T. (the entrelacs is a sphere if and only if the singularity is equisingular unibranch) for several important families of singularities.

In collaboration with F Michel (Toulouse), Anne Pichon showed that an analytical germ f: (C 3 ,0) -> (C,0) has isolated singularity if and only if the edge of its Milnor fibre is homeomorphic with the link of f -1 (0) .

Anne Pichon also studied the degenerated families of complex curves of fixed genus. She gave a topological interpretation of works of M. Artin and G. Winters in terms of fibrations on the circle of Waldhausen varieties thus highlighting their relationships to more recent work of Y. Matsumoto and J. Montesinos.

5.5. Singularities of representations spaces.

The work of Laetitia Ladurelli relates to the geometry of representations spaces. She studied in an explicit way the relation between intersection homology and Casson invariant. This one is an invariant of closed varieties of dimension 3. In a more precise way, it counts the intersection points between spaces of SU(2)-representations of the fundamental groups of the components of a Heegaard decomposition of the variety. These representations spaces are seen like cycles in a (singular) representations space of a fundamental group. A fine study of the geometry of singularities of this space made it possible to give a characterization of the local geometry in the neighborhood of the strata of "a good" stratification in terms of symplectic geometry.

5.6. Steenrod operations and Singularities.

Adriana Ciampella (Post-Doc. TMR, Napoli) studies the possibility of defining Steenrod operations in a systematic way in cyclic homology. The goal is to define Steenrod operations in the singular case (generalizing in that works of Loday and Karoubi).

5.7. Analytical geometry.

Pasquale Zito (Post-Doc. TMR, Roma) and Jorg Schurmann (Post-Doc. TMR, Hamburg) currently work within the team "Singularities", respectively on C * - algebras and cyclic Homology and on the definitions of the perverse sheaves, this within the framework of the TMR contract "Analytical Geometry".

Last update : September 22, 2001, EL.