Related papers: On the p-adic weight-monodromy conjecture for comp…

Let $k$ be a perfect field of characteristic $p > 0$. For a strictly semi-stable scheme over $k[[t]]$, we construct the weight spectral sequence in $p$-adic cohomology using the theory of arithmetic $\mathcal{D}$-modules, whose $E_1$ terms…

We prove a number of results on the \'etale cohomology of rigid analytic varieties over $p$-adic non-archimedean local fields. Among other things, we establish bounds for Frobenius eigenvalues, show a strong version of Grothendieck's local…

We prove that under semismall smooth flops, smooth projective varieties have (non-canonically) isomorphic Chow motives with coefficients in any noetherian local ring $\Lambda$ by comparing the pushforward of the constant intersection…

In this paper, we show that the infinitesimal Torelli theorem implies the existence of deformations of automorphisms. In the first part, we use Hodge theory and deformation theory to study the deformations of automorphisms of complex…

We generalize Givental's Theorem for complete intersections in smooth toric varieties in the Fano case. In particular, we find Gromov--Witten invariants of Fano varieties of dimension $\geq 3$, which are complete intersections in weighted…

We develop a $p$-adic theory of periods for 1-motives, extending the classical theory of complex periods into the non-archimedean setting. For 1-motives with good reduction over $p$-adic local fields, we construct a $p$-adic integration…

Via the relative fundamental exact sequence of $p$-adic Hodge theory, we determine the geometric $p$-adic pro-\'etale cohomology of the Drinfeld symmetric spaces defined over a $p$-adic field, thus giving an alternative proof of a theorem…

We introduce the notion of cracked polytope, and - making use of joint work with Coates and Kasprzyk - construct the associated toric variety $X$ as a subvariety of a non-singular toric variety $Y$ under certain conditions. Restricting to…

The aim of this paper is to study the behavior of Hodge-theoretic (intersection homology) genera and their associated characteristic classes under proper morphisms of complex algebraic varieties. We obtain formulae that relate (parametrized…

This article explores to which extent the algebro-geometric theory of rational descendant Gromov-Witten invariants can be carried over to the tropical world. Despite the fact that the tropical moduli-spaces we work with are non-compact, the…

We prove that the d\'evissage property holds for periodic cyclic homology for a local complete intersection embedding into a smooth scheme. As a consequence, we show that the complexified topological Chern character maps for the bounded…

We prove analogs of Whitehead's theorem (from algebraic topology) for both the Chow groups and for the Grothendieck group of coherent sheaves: a morphism between smooth projective varieties whose pushforward is an isomorphism on the Chow…

We give the full classification of smooth toric Legendrian subvarieties in projective space. We also prove that under some minor assumptions the group of linear automorphisms preserving given Legendrian subvariety preserves the contact…

In this paper, we verify a part of the Mirror Symmetry Conjecture for Schoen's Calabi-Yau 3-fold, which is a special complete intersection in a toric variety. We calculate a part of the prepotential of the A-model Yukawa couplings of the…

Let $\mathbb{V}$ be a motivic variation of Hodge structure on a $K$-variety $S$, let $\mathcal{H}$ be the associated $K$-algebraic Hodge bundle, and let $\sigma \in \textrm{Aut}(\mathbb{C}/K)$ be an automorphism. The absolute Hodge…

In [R2] and [RO] the Arnold conjecture for closed symplectic manifolds with trivial second homotopy group was proved. This proof used surgery and cobordism theory. Here we give a purely cohomological proof of this result.

Let $K$ be a sub-$p$-adic field. We show that the functor sending a finite type $K$-scheme to its \'etale topos is fully faithful after localizing at the class of universal homeomorphisms. This generalizes a result of Voevodsky, who proved…

Given a smooth projective variety $X$ over a number field $k$ and $P\in X(k)$, the first author conjectured that in a precise sense, any sequence that approximates $P$ sufficiently well must lie on a rational curve. We prove this conjecture…

Any smooth projective variety contains many complete intersection subvarieties with ample cotangent bundles, of each dimension up to half its own dimension.

Type-A toric varieties may be obtained as GIT quotients with respect to a torus action with weights corresponding to roots of the group $SL(k)$ for some $k>1$. These varieties appear in various important applications, in particular, as…