相关论文: Stable rationality of hypersurfaces in sch\"{o}n a…

We introduce and study the question how can stable birational types vary in a smooth proper family. Our starting point is the specialization for stable birational types of Nicaise and the author and our emphasis is on stable birational…

One of the emerging problems in algebraic geometry is to characterize the affine $n$-space $\mathbb{A}^n$ among smooth affine schemes up to $\mathbb{A}^1$-contractibility. Recent efforts show that this characterization holds in dimensions…

We study the stable rationality problem for quadric and cubic surface bundles over surfaces from the point of view of the degeneration method for the Chow group of 0-cycles. Our main result is that a very general hypersurface X of bidegree…

Making use of the recent theory of noncommutative motives, we construct a new motivic measure, which we call the Tits' motivic measure. As a first application, we prove that two Severi-Brauer varieties (or more generally twisted…

We find a relation between a cubic hypersurface $Y$ and its Fano variety of lines $F(Y)$ in the Grothendieck ring of varieties. We prove that if the class of an affine line is not a zero-divisor in the Grothendieck ring of varieties, then…

We prove that stable rationality specializes in regular families whose fibers are integral and have at most ordinary double points as singularities. Our proof is based on motivic specialization techniques and the criterion of Larsen and…

Let G be a semisimple affine algebraic group over a field F. Assuming that G becomes of inner type over some finite field extension of F of degree a power of a prime p, we investigate the structure of the Chow motives with coefficients in a…

We study pairs $(f, \Gamma)$ consisting of a non-Archimedean rational function $f$ and a finite set of vertices $\Gamma$ in the Berkovich projective line, under a certain stability hypothesis. We prove that stability can always be attained…

This is the first of a series of papers, where we investigate hierarchies of generalized {L}\"{u}roth problems on the hierarchy of rationality, starting with the obvious hierarchy between the rationality and the ruledness. Our primary goal…

We show that representations of convolution algebras such as Lustzig's graded affine Hecke algebra or the quiver Hecke algebra and quiver Schur algebra in (affine) type A can be realised in terms of certain equivariant motivic sheaves…

We introduce a mock toric variety, a generalization of a toric variety. For a non-toric example, Del-Pezzo surfaces are mock toric varieties. These new varieties inherit some properties of mock toric varieties. In application, we give…

We prove a rigidity result for certain $p$-complete \'etale $\mathbf{A}^{1}$-invariant sheaves of anima over a qcqs finite-dimensional base scheme $S$ of bounded \'etale cohomological dimension with $p$ invertible on $S$. This generalizes…

Let $S$ be a smooth projective surface over a field. We introduce the notion of integral decomposability and, respectively, the opposite notion of integral indecomposability, of the transcendental motive $M^2_{\rm tr}(S)$. If the…

We construct a theory of motivic integration for smooth rigid varieties. As an application new invariants of degenerations are obtained.

We prove that the Chow motives of twisted derived equivalent K3 surfaces are isomorphic, not only as Chow motives (due to Huybrechts), but also as Frobenius algebra objects. Combined with a recent result of Huybrechts, we conclude that two…

Given a smooth quasi-projective complex algebraic variety $\mathcal{S}$, we prove that there are only finitely many Hodge-generic non-isotrivial families of smooth projective hypersurfaces over $\mathcal{S}$ of degree $d$ in…

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…

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…

The Shafarevich conjecture for a class of varieties over a number field posits the finitude of those with good reduction outside a finite set of primes. In the case of hypersurfaces in the torus $\mathbb{G}_m^n$, a natural class to consider…

We prove a formula expressing the motivic integral (\cite{ls}) of a K3 surface over $\bC((t))$ with semi-stable reduction in terms of the associated limit Hodge structure. Secondly, for every smooth variety over a non-archimedean field we…