Abstract: Let (C^n,0)->(C,0) the germ of an holomorphic function with an isolated singularity. The Milnor formula computes the difference of the Euler characteristics of the generic and special fiber in terms of the Milnor number, i.e. the (complex) dimension of the Jacobian algebra.
Let S an henselian DVR with residue characteristics p and let X a flat, separated S-scheme of finite type, that is moreover regular. Let us suppose that the structural morphism has only an isolated singularity.
The Deligne-Milnor formula, conjectured by P. Deligne, generalizes the Milnor formula to the positive and mixed characteristics case. It computes the difference of the Euler characteristics (l-adiques, for l a prime number different from p) of the geometric fibers of X/S in terms of a number of algebraic nature (which is also called the Milnor number) and of a number of arithmetic nature (the Swan conductor of the generic fiber).
The case of equicharacteristis has been proved by P. Deligne (together with the case of relative dimension 0 and of ordinary quadratic singularities).
In mixed characteristics, the conjecture is still open outside the case of relative dimension 1, which follows from theorems due to S. Bloch and F. Orgogozo.
In this talk, we will discuss how to prove the conjecture under the additional assumption of unipotent action of the inertia group on the cohomology of the geometric generic fiber using derived and non commutative geometry, following ideas of B. Toen and G. Vezzosi. This provides us with new cases in the mixed characteristics situation.
This talk is based on work in collaboration with D. Beraldo: arXiv:2211.11717.
ID: 664 5939 7723