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

We provide explicit formulas for computing the motivic Chern and Hirzebruch classes of degeneracy loci, especially those coming from the symplectic and odd orthogonal Grassmannians. The Chern-Schwartz-MacPherson classes, K-theory classes,…

Let $G$ be a semi-simple algebraic group over a perfect field $k$. A lot of progress has been made recently in computing the Chow motives of projective $G$-homogenous varieties. When $k$ has positive characteristic, a broader class of…

We prove that there exist rational but not uniformly rational smooth algebraic varieties. The proof is based on computing a certain numerical obstruction developed in the case of compactifications of affine spaces. We show that for some…

For a reductive group $G$, we prove that complex irreducible rigid $G$-local systems with quasi-unipotent monodromies and finite order abelianization on a smooth curve are motivic, generalizing a theorem of Katz for $GL_n$. We do so by…

Faltings proved that there are finitely many abelian varieties of genus $g$ over a number field $K$, with good reduction outside a finite set of primes $S$. Fixing one of these abelian varieties $A$, we prove that there are finitely many…

We construct an upgrade of the motivic volume by keeping track of dimensions in the Grothendieck ring of varieties. This produces a uniform refinement of the motivic volume and its birational version introduced by Kontsevich and Tschinkel…

In this paper, we study the moduli space of rational curves in a general low degree hypersurface in the Orthogonal Grassmanian $OG(k,n+1)$ of $k$-dimensional isotropic subspaces of an $n+1$-dimensional vector space equipped with a…

For any affine variety equipped with coordinates, there is a surjective, continuous map from its Berkovich space to its tropicalisation. Exploiting torus actions, we develop techniques for finding an explicit, continuous section of this…

Using Voisin's method we prove that a very general hypersurface of degree at least 4 in complex projective space of dimension 6, 7, 8 or 9 is not stably rational and so, in particular, not rational. We obtain the same conclusion for the…

We solve a motivic version of the Adams conjecture with the exponential characteristic of the base field inverted. In the way of the proof we obtain a motivic version of mod k Dold theorem and give a motivic version of Brown's trick…

We develop a notion of exponential motives on general prestacks equipped with a $\mathbf{G}_a$-action, and compare them with Whittaker motives via Gaitsgory's Kirillov model. We then establish foundational results for exponential motives on…

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…

Grothendieck-Chow motives of quadric hypersurfaces have provided many insights into the theory of quadratic forms. Subsequently, the landscape of motives of more general projective homogeneous varieties has begun to emerge. In particular,…

We show that the generic fiber of a family of smooth $\mathbb{A}^{1}$-ruled affine surfaces always carries an $\mathbb{A}^{1}$-fibration, possibly after a finite extension of the base. In the particular case where the general fibers of the…

We prove that any affine, resp. polarized projective, spherical variety admits a flat degeneration to an affine, resp. polarized projective, toric variety. Motivated by Mirror Symmetry, we give conditions for the limit toric variety to be a…

The circle method has been successfully used over the last century to study rational points on hypersurfaces. More recently, a version of the method over function fields, combined with spreading out techniques, has led to a range of results…

O'Grady constructed a 6-dimensional irreducible holomorphic symplectic variety by taking a crepant resolution of some moduli space of stable sheaves on an abelian surface. In this paper, we naturally extend O'Grady's construction to fields…

We study the existence of a decomposition of the diagonal for bidegree hypersurfaces in a product of projective spaces. Using a cycle theoretic degeneration technique due to Lange, Pavic and Schreieder, we develop an inductive procedure…

Let A be an abelian surface over a fixed number field. If A is principally polarised, then it is known that the order of the Tate-Shafarevich group of A must, if finite, be a square or twice a square. The situation for A not principally…

This paper studies curves on quartic K3 surfaces, or more generally K3 surfaces which are complete intersection in weighted projective spaces. A folklore conjecture concerning rational curves on K3 surfaces states that all K3 surfaces…