相关论文: Deformation principle and Andr\'e motives of proje…

We show that the Andr\'{e} motive of a hyper-K\"{a}hler variety $X$ over a field $K \subset \mathbb{C}$ with $b_2(X)>6$ is governed by its component in degree $2$. More precisely, we prove that if $X_1$ and $X_2$ are deformation equivalent…

We study the Mumford--Tate conjecture for hyperk\"{a}hler varieties. We show that the full conjecture holds for all varieties deformation equivalent to either an Hilbert scheme of points on a K3 surface or to O'Grady's ten dimensional…

We prove the conjectures of Hodge and Tate for any six-dimensional hyper-K\"ahler variety that is deformation equivalent to a generalized Kummer variety.

We investigate how the motive of hyper-K\"ahler varieties is controlled by weight-2 (or surface-like) motives via tensor operations. In the first part, we study the Voevodsky motive of singular moduli spaces of semistable sheaves on K3 and…

We prove that the projectors arising from the decomposition theorem applied to a projective map of quasi projective varieties are absolute Hodge, Andr\'e motivated, Tate and Ogus classes. As a by-product, we introduce, in characteristic…

One of the main tools for the study of compact hyperk\"ahler manifolds is the natural action of the Looijenga-Lunts-Verbitsky Lie algebra on the cohomology of such manifolds. This also applies to the mildly singular holomorphic symplectic…

Given a smooth projective variety $M$ endowed with a faithful action of a finite group $G$, following Jarvis-Kaufmann-Kimura and Fantechi-G\"ottsche, we define the orbifold motive (or Chen-Ruan motive) of the quotient stack $[M/G]$ as an…

Let $Y$ be an abelian variety over a subfield $k \subset \mathbb{C}$ that is of finite type over $\mathbb{Q}$. We prove that if the Mumford-Tate conjecture for $Y$ is true, then also some refined integral and adelic conjectures due to Serre…

We prove the conjectures of Hodge and Tate for any four-dimensional hyper-K\"ahler variety of generalized Kummer type. For an arbitrary variety $X$ of generalized Kummer type, we show that all Hodge classes in the subalgebra of the rational…

Let $X$ be a smooth projective variety over a finitely generated field $K$ of characteristic~$0$ and fix an embedding $K \subset \mathbb{C}$. The Mumford--Tate conjecture is a precise way of saying that certain extra structure on the…

Let $X$ be a compact K\"ahler manifold with vanishing Riemann curvature. We prove that there exists a manifold $X'$, deformation equivalent to $X$, which is not an analytification of any projective variety, if and only if $H^0(X, \Omega^2)…

We introduce the notion of a Hyper-K\"{a}hler manifold $X$ induced by a Hodge structure of K3-type. We explore this notion for the known deformation types of hyper-K\"{a}hler manifolds studying those that are induced by a K3 or abelian…

Motivated by the theory of Inoue-type varieties, we give a structure theorem for projective manifolds $W_0$ with the property of admitting a 1-parameter deformation where $W_t$ is a hypersurface in a projective smooth manifold $Z_t$. Their…

Let $X$ be a cubic fourfold in $P^5_{C}$. We prove that, assuming the Hodge conjecture for the product $S \times S$, where $S$ is a complex surface, and the finite dimensionality of the Chow motive $h(S)$, there are at most a countable…

We prove by induction on dimension the Hodge conjecture for smooth complex projective varieties. Let $X$ be a smooth complex projective variety. Then $X$ is birational to a possibly singular projective hypersurface, hence to a smooth…

We prove, assuming the generalized Riemann hypothesis, the Andre-Oort conjecture for Hilbert modular surfaces. More precisely, let K be a real quadratic field and let S be the coarse moduli space of complex abelian surfaces with…

Let $X$ be a non-singular projective variety over a number field $K$, $i$ a non-negative integer, and $V_{\A}$, the etale cohomology of $\bar X$ with coefficients in the ring of finite adeles $\A_f$ over $\Q$. Assuming the Mumford-Tate…

We give a new proof of the Hodge conjecture for abelian fourfolds of Weil type with discriminant 1 and all of their powers. The Hodge conjecture for these abelian fourfolds was proven by Markman using hyperholomorphic sheaves on…

We show that the Andre motive of an irreducible, symplectic variety, which is deformation equivalent to the Hilbert scheme of points on a K3 surface, is an object the category generated by its motive truncated in degree two.

We show that the motive of the Hilbert scheme of length-$n$ subschemes on a K3 surface or on an abelian surface admits a decomposition similar to the decomposition of the motive of an abelian variety obtained by Shermenev, Beauville, and…