Related papers: Algebraic cycles and Hitchin systems

Let X be an abelian scheme over a base variety S with endomorphism algebra D. We prove that the relative Chow motive R(X/S) has a canonical decomposition as a direct sum of motives R^(\xi)$ where \xi runs over an explicitly determined…

For a reductive group $G$, we prove that complex irreducible rigid $G$-local systems with quasi-unipotent monodromies and finite order abelianization on a smooth curve are motivic, generalizing a theorem of Katz for $GL_n$. We do so by…

The aim of this paper is to connect two important and apparently unrelated theories: motivic homotopy theory and ramification theory. We construct motivic homotopy categories over a qcqs base scheme $S$, in which cohomology theories with…

In this paper, we study rational sections of the relative Picard scheme of a linear system on a smooth projective variety. We prove that if the linear system is basepoint-free and the locus of non-integral divisors has codimension at least…

Let $X$ be a complete intersection inside a variety $M$ with finite dimensional motive and for which the Lefschetz-type conjecture $B(M)$ holds. We show how conditions on the niveau filtration on the homology of $X$ influence directly the…

These notes are an account of a series of lectures I gave at the LMS-CMI Research School `Homotopy Theory and Arithmetic Geometry: Motivic and Diophantine Aspects', in July 2018, at the Imperial College London. The goal of these notes is to…

We prove a deletion-contraction formula for motivic Feynman rules given by the classes of the affine graph hypersurface complement in the Grothendieck ring of varieties. We derive explicit recursions and generating series for these motivic…

Let $M(d,\chi)$ be the moduli space of semistable 1-dimensional sheaves supported at curves of degree $d$ on $\mathbb{P}^2$, with Euler characteristic $\chi$. We have the Hilbert-Chow morphism $\pi: M(d,\chi)\rightarrow |dH|$ sending each…

Let A be an abelian variety and let us fix a Weil cohomology with coefficients in F. Let $H^1(A,F)$ be the first cohomology group of A and $Lef(A) \subset GL(H^1(A,F))$ be its Lefschetz group, i.e. the sub-group of $GL(H^1(A,F))$ of linear…

The rational points of a smooth curve $X$ over a number field $k$ map to the set of augmentations of the associated motivic algebra. An expectation, related to Kim's conjecture, is that for $X$ hyperbolic, the set of augmentations which…

In this paper, we confirm a physical conjecture regarding the parabolic $\mathrm{SO}_{2n}$-Hitchin system, showing that Hitchin map factors through a finite cover of the Hitchin base that is isomorphic to an affine space. We first show that…

Let $X$ be a complex smooth projective variety of dimension $d$. Under some assumption on the cohomology of $X$, we construct mutually orthogonal idempotents in $CH_d(X \times X) \otimes \Q$ whose action on algebraically trivial cycles…

We study a formal deformation problem for rational algebraic cycle classes motivated by Grothendieck's variational Hodge conjecture. We argue that there is a close connection between the existence of a Chow-K\"unneth decomposition and the…

We compute the supports of the perverse cohomology sheaves of the Hitchin fibration for $GL_n$ over the locus of reduced spectral curves. In contrast to the case of meromorphic Higgs fields we find additional supports at the loci of…

Gross and Zagier defined certain `higher Green's functions' on products of modular curves and conjectured that the value of these functions at complex multiplication points should be logarithms of algebraic numbers. This is now a theorem of…

By analogy with work of Hitchin on integrable systems, we construct natural relaxations of several kinds of moduli spaces of difference equations, with special attention to a particular class of difference equations on an elliptic curve…

We prove de Cataldo-Hausel-Migliorini's P=W conjecture in arbitrary rank for parabolic Higgs bundles labeled by the affine Dynkin diagrams $\tilde{A}_0$, $\tilde{D}_4$, $\tilde{E}_6$, $\tilde{E}_7$, and $\tilde{E}_8$. Our proof relies on…

Motivated by the Beauville decomposition of an abelian scheme and the "Perverse = Chern" phenomenon for a compactified Jacobian fibration, we study in this paper splittings of the perverse filtration for compactified Jacobian fibrations. On…

This paper investigates the structure of generic motives and their implications for the motivic cohomology of fields. Originating in Voevodsky's theory of motives and related to Beilinson's vision of a motivic $t$-structure, generic motives…

We compute explicitly the Chow motive of any generalized Kummer variety associated to any abelian surface. In fact, it lies in the rigid tensor subcategory of the category of Chow motives generated by the Chow motive of the underlying…