Related papers: Cubic fourfolds, Kuznetsov components and Chow mot…

We prove that the symmetric monoidal category of mixed motives generated by an abelian variety (more generally, an abelian scheme) can be described as a certain module category. More precisely, we describe it as the category of…

We explore the connection between $K3$ categories and 0-cycles on holomorphic symplectic varieties. In this paper, we focus on Kuznetsov's noncommutative $K3$ category associated to a nonsingular cubic 4-fold. By introducing a filtration on…

We develop a motivic cohomology theory, representable in the Voevodsky's triangulated category of motives, for smooth separated Deligne-Mumford stacks and show that the resulting higher Chow groups are canonically isomorphic to the higher…

A well known conjecture asserts that a cubic fourfold X is rational if it has a cohomologically associated K3 surface. G.Ouchi proved that if X admits a finite group G of symplectic automorphisms, whose order is different from 2, then X has…

We consider the polynomial algebra $\mathbb{C}[\mathbf{z}]:=\mathbb{C}[z_1,\,z_2,\,z_3]$ and the polynomial $f:=z_1^3+z_2^3+z_3^3+3qz_1z_2z_3$, where $q\in \mathbb{C}$. Our aim is to compute the Hochschild homology and cohomology of the…

Given a complex manifold endowed with a $\mathbb{C}^\times$-action and a DQ-algebra equipped with a compatible holomorphic Frobenius action (F-action), we prove that if the $\mathbb{C}^\times$-action is free and proper, then the category of…

For a smooth projective variety equipped with a Chow-K\"unneth (abbr. CK) decomposition, the notions of motivic multiple twist-multiplicativity and multiplicativity defect are introduced to interpret the obstruction to the compatibility of…

We compute a complete set of isomorphism classes of cubic fourfolds over $\mathbb{F}_2$. Using this, we are able to compile statistics about various invariants of cubic fourfolds, including their counts of points, lines, and planes; all…

The Beilinson-Bloch type conjectures predict that the low degree rational Chow groups of intersections of quadrics are one dimensional. This conjecture was proved by Otwinowska. Making use of homological projective duality and the recent…

We study isogenies between K3 surfaces in positive characteristic. Our main result is a characterization of K3 surfaces isogenous to a given K3 surface $X$ in terms of certain integral sublattices of the second rational $\ell$-adic and…

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…

Consider the wrapped Fukaya category W of a collection of exact Lagrangians in a Liouville manifold. Under a non-degeneracy condition implying the existence of enough Lagrangians, we show that natural geometric maps from the Hochschild…

Let X be a smooth cubic hypersurface. We prove that a general cubic surface is isomorphic to a hyperplane section of X .

We consider Calabi-Yau threefolds of Borcea-Voisin type over Q. They are constructed from products of K3 surfaces and elliptic curves. We use concrete K3 surfaces and discuss the automorphy of the Galois representations associated to the…

We associate to any element in the Milnor K-theory of a field $k$ modulo 2 an invertible Morava K-theory motive over $k$. Specifically, for $\alpha$ in $\mathrm{K}^{\mathrm{M}}_{n+1}(k)/2$ we construct an invertible $\mathrm{K}(n)$-motive…

We study Grothedieck groups of triangulated categories using weight structures, weight complexes, and the corresponding pure (co)homological functors. We prove some general statements on $K_0$ of weighted categories and apply it to…

Using valuative techniques, we show that a smooth affine surface with a non-elementary automorphism group and completable by a cycle of rational curves is either the algebraic torus or a smooth cubic affine surface of Markov type.…

This thesis studies Frobenius manifolds arising from extended deformations of complex structures on compact Calabi-Yau manifolds, following the construction by Sergey Barannikov and Maxim Kontsevich. The work is based on the investigation…

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 develop a graded version of the theory of cyclotomic q-Schur algebras, in the spirit of the work of Brundan-Kleshchev on Hecke algebras and of Ariki on q-Schur algebras. As an application, we identify the coefficients of the canonical…