Related papers: Algebraic cycles and EPW cubes

We formulate the "real integral Hodge conjecture", a version of the integral Hodge conjecture for real varieties, and raise the question of its validity for cycles of dimension 1 on uniruled and Calabi-Yau threefolds and on rationally…

We develop a theory of abstract arithmetic Chow rings where the role of the fibers at infinity is played by a complex of abelian groups that computes a suitable cohomology theory. This theory allows the construction of many variants of the…

We study consequences and applications of the folklore statement that every double complex over a field decomposes into so-called squares and zigzags. This result makes questions about the associated cohomology groups and spectral sequences…

Consider three normalised cuspidal eigenforms of weight $2$ and prime level $p$. Under the assumption that the global root number of the associated triple product $L$-function is $+1$, we prove that the complex Abel-Jacobi image of the…

Quotients $Y=X/conj$ of complex surfaces by anti-holomorphic involutions $conj\: X\to X$ tend to be completely decomposable when they are simply connected, i.e., split into connected sums, $n CP^2\#m\barCP2$, if $w_2(Y)\ne0$, or into…

For the double of a quiver, the works of Ginzburg, Bocklandt-Le Bruyn and Schedler show that its closed paths, called the necklaces, have a natural Lie bialgebra structure. Schedler also constructed,in [Int. Math. Res. Notices, 2005 (12),…

We show under what conditions the complex computing general Ext-groups carries the structure of a cyclic operad such that Ext becomes a Batalin-Vilkovisky algebra. This is achieved by transferring cyclic cohomology theories for the dual of…

We consider a billiard problem for compact domains bounded by confocal conics on a hyperboloid of one sheet in the Minkowski space. We show that there are two types of confocal families in such setting. Using an algebro-geometric…

We present a general conjecture on congruences between Hecke eigenvalues of parabolically induced and cuspidal automorphic representations of split reductive groups, modulo divisors of critical values of certain $L$-functions. We examine…

We describe some dynamical properties of one parameter families of billiards on convex curves (ovals) which are deformed by the curvature (curve-shortening) flow. We obtain the bifurcations of the period two orbits and some special…

We describe a closed immersion from each representation space of a type A quiver with bipartite (i.e., alternating) orientation to a certain opposite Schubert cell of a partial flag variety. This "bipartite Zelevinsky map" restricts to an…

Thanks to the theory of Coxeter groups, we produce the first family of Calabi-Yau manifolds $X$ of arbitrary dimension $n$, for which $\Bir(X)$ is infinite and the Kawamata-Morrison movable cone conjecture is satisfied. For this family, the…

An index theorem for the anti-self-dual deformation complex on anti-self-dual orbifolds with cyclic quotient singularities is proved. We present two applications of this theorem. The first is to compute the dimension of the deformation…

We consider proper, algebraic semismall maps f from a complex algebraic manifold X. We show that the topological Decomposition Theorem implies a "motivic" decomposition theorem for the rational algebraic cycles of X and, in the case X is…

Let F be a polarized irreducible holomorphic symplectic fourfold, deformation equivalent to the Hilbert scheme parametrizing length-two zero-dimensional subschemes of a K3 surface. The homology group H^2(F,Z) is equipped with an integral…

For a smooth complex projective variety X defined over a number field, we have filtrations on the Chow groups depending of the choice of realizations. If the realization consists of mixed Hodge structure without any additional structure, we…

Let $(W,S)$ be a finite Coxeter system with root system $R$ and with set of positive roots $R^+$. For $\alpha\in R$, $v,w\in W$, we denote by $\partial_\alpha$, $\partial_w$ and $\partial_{w/v}$ the divided difference operators and skew…

We give a formalism of arithmetic mixed sheaves including the case of arithmetic mixed Hodge structures, and show the nonvanishing of certain higher extension groups, and also the nontriviality of the second Abel-Jacobi map for zero cycles…

Let X be a minuscule Schubert variety and $\alpha$ a class of 1-cycle on X. In this article we describe the irreducible components of the scheme of morphisms of class $\alpha$ from a rational curve to X. The irreducible components are…

This is a survey of recent examples of varieties that are not stably rational. We review the specialization method based on properties of the Chow group of zero-cycles used in these examples and explain the point of view of unramified…