Mixed Hodge Structures

With a basic knowledge of cohomology theory, the background necessary to understand Hodge theory and polarization, Deligne's Mixed Hodge Structure on cohomology of complex algebraic varieties is described.Comment: pages 8

A REMARK ON VANISHING CYCLES WITH TWO STRATA

International audienceSuppose that the critical locus Σ of a complex analytic function f on affine space is, itself, a space with an isolated singular point at the origin 0, and that the Milnor number of f restricted to normal slices of Σ − {0} is constant. Then, the general theory of perverse sheaves puts severe restrictions on the cohomology of the Milnor fiber of f at 0, and even more surprising restrictions on the cohomology of the Milnor fiber of generic hyperplane slices.

Decomposition, purity and fibrations by normal crossing divisors

We give a simple geometric proof of the decomposition theorem in terms of Thom-Whitney stratifications by reduction to fibrations by normal crossings divisors over the strata and explain the relation with the local purity theorem an unpublished result of Deligne and Gabber.Comment: arXiv admin note: text overlap with arXiv:1302.581

De la Pureté locale à la décomposition

The decomposition theorem is deduced from local purity

SIMPLE SINGULARITIES AND SIMPLE LIE ALGEBRAS

International audienceIn this paper we state Grothendieck's conjectures concerning simple Lie Algebras. We survey the results known about them

Du théoréme de décomposition á la Pureté locale

A new proof of the decomposition theorem is established using a relation with a version of the local purity theorem of Deligne and Gabber adapted to complex algebraic varieties

Picard groups for line bundles with connections

International audienceWe study analogues of the usual Picard group for complex manifolds or non-singular complex algebraic varieties but instead of line bundles we study line bundles with connections. We choose an approach which works for both cases. We identify obstructions for the existence of a connection, or a connection which is even integrable or regular (integrable), and point out where one should be careful when passing from the analytic to the algebraic case