Related papers: Lichtenbaum-van Hamel duality for singular varieti…

These are notes of my lectures at the summer school "Higher-dimensional geometry over finite fields" in Goettingen, June--July 2007. We present a proof of Tate's theorem on homomorphisms of abelian varieties over finite fields (including…

In our previous study of duality for complete discrete valuation fields with perfect residue field, we treated coefficients in finite flat group schemes. In this paper, we treat abelian varieties. This in particular implies Grothendieck's…

We study homological multiplicities of spherical varieties of reductive group $G$ over a $p$-adic field $F$. Based on Bernstein's decomposition of the category of smooth representations of a $p$-adic group, we introduce a sheaf that…

We give a description of the cohomology groups of the structure sheaf on smooth compactifications $\overline{X}(w)$ of Deligne--Lusztig varieties $X(w)$ for ${\rm GL}_n$, for all elements $w$ in the Weyl group. As a consequence, we obtain…

We first prove Bosch-L\"utkebohmert-Raynaud's conjectures on existence of global N\'eron models of not necessarily semi-abelian algebraic groups in the perfect residue fields case. We then give a counterexample to the existence in the…

Lind and Schmidt have shown that the homoclinic group of a cyclic $\Z^k$ algebraic dynamical system is isomorphic to the dual of the phase group. We show that this duality result is part of an exact sequence if $k=1$. The exact sequence is…

For a curve $C$ over a perfect field $k$ of characteristic $p > 0$ we study the tame cohomology of $X = \textit{Spa}(C,k)$ introduced in arXiv:1801.04776. We prove that the tame cohomology groups of $X$ with $p$-torsion coefficients satisfy…

Let $p$ be an algebraic point of a projective variety $X$ defined over a number field. Liouville inequality tells us that the norm at $p$ of a non vanishing integral global section of an hermitian line bundle over $X$ is either zero or it…

This is an English translation of the author's 1989 note in Russian, published in a collection "Arithmetic and Geometry of Varieties" (V.E. Voskresenski, ed.), Kuibyshev State University, Kuibyshev, 1989, pp. 57--67. Let $X$ be be an…

We prove that the theory of a Henselian valued field of characteristic zero, with finite ramification, and whose value group is a $Z$-group, is model-complete in the language of rings if the theory of its residue field is model-complete in…

For a proper, smooth scheme $X$ over a $p$-adic field $K$, we show that any proper, flat, semistable $\mathcal{O}_K$-model $\mathcal{X}$ of $X$ whose logarithmic de Rham cohomology is torsion free determines the same $\mathcal{O}_K$-lattice…

We solve the problem of constructing a genus-zero full conformal field theory (a conformal field theory on genus-zero Riemann surfaces containing both chiral and antichiral parts) from representations of a simple vertex operator algebra…

We revisit the computation, due to Hesselholt and Madsen, of the K-theory of truncated polynomial algebras for perfect fields of positive characteristic. The resulting K-groups are expressed in terms of big Witt vectors of the field. The…

We investigate the algebra of a Hausdorff ample groupoid, introduced by Steinberg, over a commutative semiring S. In particular, we obtain a complete characterization of congruence-simpleness for such Steinberg algebras, extending the…

For a finite totally ramified extension $L$ of a complete discrete valuation field $K$ with the perfect residue field of characteristic $p>0$, it is known that $L/K$ is an abelian extension if the upper ramification breaks are integers and…

We use the theory of Condensed Mathematics to build a condensed cohomology theory for the Weil group of a $p$-adic field. The cohomology groups are proved to be locally compact abelian groups of finite ranks in some special cases. This…

We prove that when Hodge theory survives on non-compact symplectic manifolds, a compact symplectic Lie group action having fixed points is necessarily Hamiltonian, provided the associated almost complex structure preserves the space of…

We prove that smooth, projective, $K$-trivial, weakly ordinary varieties over a perfect field of characteristic $p>0$ are not geometrically uniruled. We also show a singular version of our theorem, which is sharp in multiple aspects. Our…

In the 2002 Durham Symposium, Markus Linckelmann [1] conjectured the existence of a regular central k*-extension of the full subcategory over the selfcentralizing Brauer pairs of the Frobenius P-category F_{(b,G)} associated with a block b…

Auel-Bigazzi-B\"ohning-Graf von Bothmer proved that if a proper smooth variety $X$ over a field $k$ of characteristic $p>0$ has universally trivial Chow group of $0$-cycles, the cohomological Brauer group of $X$ is universally trivial as…