Related papers: Motivic Poincar\'e series of cusp surface singular…
We give here a description of the motivic Poincare series in case of irreducible quasi-ordinary hypersurfaces in all dimension. We give an explicit formula in a particular case. Finally, for such singularities, we give a constructive proof…
The geometric motivic Poincar\'e series of a germ $(S,0)$ of complex algebraic variety takes into account the classes in the Grothendieck ring of the jets of arcs through $(S,0)$. Denef and Loeser proved that this series has a rational…
The arithmetic motivic Poincar\'e series of a variety $V$ defined over a field of characteristic zero, is an invariant of singularities which was introduced by Denef and Loeser by analogy with the Serre-Oesterl\'e series in arithmetic…
Lejeune-Jalabert and Reguera computed the geometric Poincare series P_{geom}(T) for toric surface singularities. They raise the question whether this series equals the arithmetic Poincare series. We prove this equality for a class of toric…
We study the scheme of formal arcs on a singular algebraic variety and its images under truncations. We prove a rationality result for the Poincare series of these images which is an analogue of the rationality of the Poincare series…
Our purpose is to investigate all defined Poincar\'e series associated with multi-index filtrations and value semigroups of curve singularities---not necessarily complex---with regard to the property of forgetting variables, i.e., by making…
We associate with an infinite cyclic cover of a punctured neighborhood of a simple normal crossing divisor on a complex quasi-projective manifold (assuming certain finiteness conditions are satisfied) an element in the Grothendieck ring…
We study endomorphisms of complete Noetherian local rings in the context of motivic integration. Using the notion of an auto-arc space, we introduce the (reduced) auto-Igusa zeta series at a point, which appears to measure the degree to…
We define Poincar\'e series associated to a toric or analytically irreducible quasi-ordinary hypersurface singularity, (S,0), by a finite sequence of monomial valuations, such that at least one of them is centered at the origin 0. This…
The geometric motivic Poincar\'e series of a variety, which was introduced by Denef and Loeser, takes into account the classes in the Grothendieck ring of the sequence of jets of arcs in the variety. Denef and Loeser proved that this series…
We give the explicit algorithm computing the motivic generalization of the Poincare series of the plane curve singularity introduced by A. Campillo, F. Delgado and S. Gusein-Zade. It is done in terms of the embedded resolution of the curve.…
We first recall the construction of the Chow motive modelling intersection cohomology of a proper surface and study its fundamental properties. Using Voevodsky's category of effective geometrical motives, we then study the motive of the…
We provide sharp lower bounds for the multiplicity of a local holomorphic foliation defined in a complex surface in terms of data associated to a germ of invariant curve. Then we apply our methods to invariant curves whose branches are…
We compute the motivic Milnor fiber of a complex plane curve singularity in an inductive and combinatoric way using the extended simplified resolution graph. The method introduced in this article has a consequence that one can study the…
In recent years, there has been a development in approaching rationality problems through motivic methods (cf. [Kontsevich--Tschinkel'19], [Nicaise--Shinder'19], [Nicaise--Ottem'21]). This method requires the explicit construction of…
We prove the motivic classes in the motivic cohomology groups of Picard modular surfaces with non-trivial coefficients constructed in a paper of Loeffler\textendash Skinner\textendash Zerbes are in the motivic cohomology groups of the…
We provide an explicit description of the Poincar\'e dual of each generator of the rational cohomology ring of the $SU(2)$ character variety for a genus $g$ surface with central extension -- equivalently, that of the moduli space of stable…
We study Poincar\'e series associated to a finite collection of divisors on i. a finite graph and ii. a certain family of metric graphs called chain of loops. Our main results are proofs of rationality of the Poincar\'e series and…
In this thesis, we use logarithmic methods to study motivic objects. Let R be a complete discrete valuation ring with perfect residue field k, and denote by K its fraction field. We give in chapter 2 a new construction of the motivic Serre…
We study the Poincar\'e series of the mixed and pure trace rings of generic matrices. These series are known to be rational functions. We obtain an explicit formula in lowest terms in the case of $2\times2$ matrices; a denominator, which we…