Related papers: Motivic Integration and Logarithmic Geometry

We prove a recognition principle for motivic infinite P1-loop spaces over a perfect field. This is achieved by developing a theory of framed motivic spaces, which is a motivic analogue of the theory of E-infinity-spaces. A framed motivic…

We put forward in this paper a uniform narrative that weaves together several variants of Hrushovski-Kazhdan style integral, and describe how it can facilitate the understanding of the Denef-Loeser motivic Milnor fiber and closely related…

In this paper, we construct four different theories of integration, two that are for Voevodsky motives, one for mixed $\ell$-adic sheaves, and a fourth theory of integration for rational mixed Hodge structures. We then show that they…

We target multivariable series associated with resolutions of complex analytic normal surface singularities. In general, the equivariant multivariable analytical and topological Poincar\'e series are well-defined and have good properties…

We generalize the motivic incarnation morphism from the theory of arithmetic integration to the relative case, where we work over a base variety S over a field k of characteristic zero. We develop a theory of constructible effective Chow…

Motivic Serre invariants defined by Loeser and Sebag are elements of the Grothendieck ring of varities modulo $\mathbb{L}-1$. In this paper, we show that we can lift these invariants to modulo the square of $\mathbb{L}-1$ after tensoring…

In this thesis we compute motivic classes of hypertoric varieties, Nakajima quiver varieties and open de Rham spaces in a certain localization of the Grothendieck ring of varieties. Furthermore we study the $p$-adic pushforward of the Haar…

We prove in this paper the original version of Kontsevich and Soibelman's motivic integral identity conjecture for formal functions by developing a novel framework for equivariant motivic integration on special rigid varieties. This theory…

Given a smooth complex threefold X, we define the virtual motive of the Hilbert scheme of n points on X. In the case when X is Calabi-Yau, this gives a motivic refinement of the n-point degree zero Donaldson-Thomas invariant of X. The key…

We propose a suitable substitute for the classical Grothendieck ring of an algebraically closed field, in which any quasi-projective scheme is represented, while maintaining its non-reduced structure. This yields a more subtle invariant,…

This paper introduces a new cohomology theory for schemes of finite type over an arithmetic ring. The main motivation for this Arakelov-theoretic version of motivic cohomology is the conjecture on special values of $L$-functions and zeta…

A well-known conjecture of Gross and Zagier states that the values of the higher automorphic Green's function at pairs of points with complex multiplication in the upper half-plane are proportional to the logarithm of an algebraic number.…

We use a formula of Bultot to compute the motivic zeta function for the toric degeneration of the quartic K3 and its Gross-Siebert mirror dual degeneration. We check for this explicit example that the identification of the logarithm of the…

We formulate and prove an analog of Poonen's finite-field Bertini theorem with Taylor conditions that holds in the Grothendieck ring of varieties. This gives a broad generalization of the work of Vakil-Wood, who treated the case of smooth…

In this article we give an expression of the motivic Milnor fiber at infinity and the motivic nearby cycles at infinity of a polynomial $f$ in two variables with coefficients in an algebraic closed field of characteristic zero. This…

We introduce a new notion of $\boxast$-product of two integrable series with coefficients in distinct Grothendieck rings of algebraic varieties, preserving the integrability and commuting with the limit of rational series. In the same…

In this paper we use motivic integration and non-archimedean analytic geometry to study the singularities at infinity of the fibers of a polynomial map $f\colon \mathbb A^d_\mathbb C \to \mathbb A^1_\mathbb C$. We show that the motive…

For any scheme $M$ with a perfect obstruction theory, Jiang and Thomas associate a scheme $N$ with symmetric perfect obstruction theory. The scheme $N$ is a cone over $M$ given by the dual of the obstruction sheaf of $M$, and contains $M$…

In this paper we prove motivic versions of the Langlands-Shelstad Fundamental Lemma and Ng\^o's Geometric Stabilization. To achieve this, we follow the strategy from the recent proof by Groechenig, Wyss and Ziegler which avoided the use of…

In this paper, we construct some maps related to the motivic Galois action on depth-graded motivic multiple zeta values. And from these maps we give some short exact sequences about depth-graded motivic multiple zeta values in depth two and…