Related papers: Algebraic cycles and intersections of 2 quadrics

We introduce and begin to analyse a class of algebras, associated to congruence subgroups, that extend both the algebra of modular forms of all levels and the ring of classical Hecke operators. At the intuitive level, these are algebras of…

We study the relation between a complex projective set C in CP^n and the set R in RP^(2n+1) defined by viewing each equation of C as a pair of real equations. Once C is presented by quadratic equations, we can apply a spectral sequence to…

We construct some analog of cubical Bloch's higher Chow groups. Instead of considering cycles in $X\times\mathbb A^n$ we consider varieties $Y$ over $X$ together with a distinguished element in the $n$-th exterior power of the…

We show, for all $n\ge 2$ even and $d\ge 2+\frac{4}{n}$, that the moduli of smooth degree $d$ hypersurfaces of $\mathbb{P}^{n+1}$ contains infinitely many different Hodge loci whose Zariski tangent space has the same codimension as the…

Let $f : X \rightarrow B$ be a proper flat dominant morphism between two smooth quasi-projective complex varieties $X$ and $B$. Assume that there exists an integer $l$ such that all closed fibres $X_b$ of $f$ satisfy $CH_j(X_b) = \Q$ for…

Let $X$ be a smooth complex projective variety with trivial Chow groups. (By trivial, we mean that the cycle class is injective.) We show (assuming the Lefschetz standard conjecture) that if the vanishing cohomology of a general complete…

We consider the connections among algebraic cycles, abelian varieties, and stable rationality of smooth projective varieties in positive characteristic. Recently Voisin constructed two new obstructions to stable rationality for rationally…

The main purpose of this paper is to show that the mixed Hodge polynomial of the ``space of equations'' for smooth complete intersections of given multidegree in $\mathbb{C} P^n$ is divisible by the mixed Hodge polynomial of the group…

A. Weil identified a 2-dimensional space of rational classes of Hodge type (n,n) in the middle cohomology of every 2n-dimensional abelian variety with a suitable complex multiplication by an imaginary quadratic number field. These abelian…

Let C be a curve over a non-singular base variety S. We study algebraic cycles on the symmetric powers C^[n] and on the Jacobian J. The Chow homology of C^[*], the sum of all C^[n], is a ring using the Pontryagin product. We prove that this…

We review the Hodge theory of some classic examples from mirror symmetry, with an emphasis on what is intrinsic to the A-model, and on interesting open questions and problems. In particular, we illustrate the construction of a quantum…

We study an algebraic cycle of the form $Z_0= r {\mathbb P}^{\frac{n}{2}}+\check r \check{\mathbb P}^{\frac{n}{2}}$, $r \in{\mathbb N},\check r \in{\mathbb Z},\ \ 1\leq r , |\check r |\leq 10,\ \ \gcd ( r ,\check r )=1$, inside the cubic…

Let G be a complex reductive group and X a projective spherical G-variety. Moreover, assume that the subalgebra A of the cohomology ring H^*(X, R) generated by the Chern classes of line bundles has Poincare duality. We give a description of…

We introduce a path-theoretic framework for understanding the representation theory of (quantum) symmetric and general linear groups and their higher level generalisations over fields of arbitrary characteristic. Our first main result is a…

We prove that the intersection cohomology of the Baily-Borel compactification of a complex Shimura variety is identified with the top weight quotient of the mixed Hodge structure on the reductive Borel-Serre compactification. This yields…

This note is about a $16$-dimensional family of surfaces of general type with $p_g=2$ and $q=0$ and $K^2=1$, called "special Horikawa surfaces". These surfaces, studied by Pearlstein-Zhang and by Garbagnati, are related to K3 surfaces. We…

Let $C$ be a smooth projective curve of genus $2$. Following a method by O' Grady, we construct a semismall desingularization $\tilde{\mathcal{M}}_{Dol}^G$ of the moduli space $\mathcal{M}_{Dol}^G$ of semistable $G$-Higgs bundles of degree…

The formal algebraic structures that govern higher-spin theories within the unfolded approach turn out to be related to an extension of the Kontsevich Formality, namely, the Shoikhet-Tsygan Formality. Effectively, this allows one to…

We compute Hochschild cohomology of projective hypersurfaces starting from the Gerstenhaber-Schack complex of the (restricted) structure sheaf. We are particularly interested in the second cohomology group and its relation with…

The moduli spaces of flat $\mathrm{SL}_2$- and $\mathrm{PGL}_2$-connections are known to be singular SYZ-mirror partners. We establish the equality of Hodge numbers of their intersection (stringy) cohomology. In rank two, this answers a…