Related papers: Cohomological Methods in Intersection Theory

We construct the Weil restriction map for l-adic cohomology and, more generally, for mixed Weil cohomology theories. We study its compatibility with the motivic cycle class map and show that these constructions admit a natural…

Let $K$ and $L$ be algebraic extensions of the rational numbers inside the field of complex numbers. An $L$-de Rham-Betti class on a smooth projective variety $X$ over $K$ is a class in the Betti cohomology with $L$-coefficients of the…

This text is a set of lecture notes for a series of four talks given at I.P.A.M., Los Angeles, on March 18-20, 2003. The first lecture provides a quick overview of symplectic topology and its main tools: symplectic manifolds, almost-complex…

The subring of the Grothendieck ring of varieties generated by the graph hypersurfaces of quantum field theory maps to the monoid ring of stable birational equivalence classes of varieties. We show that the image of this map is the copy of…

The de Rham stack construction of Simpson shows that D-modules are quasicoherent sheaves on a modified geometry. Drinfeld furthermore introduced the ring stack perspective (aka transmutation), which asserts that a coefficient theory is…

In this note we relate the notions of Lefschetz type, decomposability, and isomorphism, on Chow motives with the notions of unit type, decomposability, and isomorphism, on noncommutative motives. Examples, counter-examples, and applications…

We prove a motivic version of the Lefschetz hyperplane theorem for a smooth ample divisor $\Theta$ on an Abelian variety. We use this to construct a motive $P$ that realizes the primitive cohomology of $\Theta$.

To an arbitrary variety over a field of characteristic zero, we associate a complex of Chow motives, which is, up to homotopy, unique and bounded. We deduce that any variety has a natural Euler characteristic in the Grothendieck group 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…

The purpose of this article is to construct a Hecke-equivariant Chow motive whose realizations equal interior (or intersection) cohomology of Picard surfaces with regular algebraic coefficients. As a consequence, we are able to define…

Let $H$ be a Hopf algebra. We consider $H$-equivariant modules over a Hopf module category $\mathcal C$ as modules over the smash extension $\mathcal C\# H$. We construct Grothendieck spectral sequences for the cohomologies as well as the…

These are (not updated) notes from the lectures I gave in St.Petersburg in July of 2001. Their goal is to give an expository account of the proof of Kontsevich's combinatorial formula for intersections on moduli spaces of curves following…

We develop Milne's theory of Lefschetz motives for general adequate equivalence relations and over a not necessarily algebraically closed base field. The corresponding categories turn out to enjoy all properties predicted by standard and…

We study the Gauss and Jacobi sums from a viewpoint of motives. We exhibit isomorphisms between Chow motives arising from the Artin-Schreier curve and the Fermat varieties over a finite field, that can be regarded as (and yield a new proof…

In this dissertation, we discuss mainly the corresponding geometric and representation theoretic aspects of relative $p$-adic Hodge theory and $p$-adic motives. To be more precise, we study the corresponding analytic geometry of the…

The main goal of this article is to study the cohomology rings and their applications of moment-angle complexes associated to Gorenstein* complexes, especially, the applications in combinatorial commutative algebra and combinatorics. First,…

We describe the homology intersection form associated to regular holonomic GKZ systems in terms of the combinatorics of regular triangulations. Combining this result with the twisted period relation, we obtain a formula of cohomology…

In this article a higher order support theory, called the cohomological jump loci, is introduced and studied for dg modules over a Koszul extension of a local dg algebra. The generality of this setting applies to dg modules over local…

Since the subject of noncommutative geometry is now entering maturity, we felt there is need for presentation of the material at an undergraduate course level. Our review is a zero order approximation to this project. Thus, the present…

Recent works of the authors have demonstrated the usefulness of considering moduli spaces of Artinian reductions of a given ring when studying standard graded rings and their Lefschetz properties. This paper illuminates a key aspect of…