Related papers: Chai's Conjecture and Fubini properties of dimensi…

We obtain new partial results supporting the spectral set conjecture in dimension 1.

Let X be a complex surface with no nontrivial 2-forms. Then we show that Bloch's conjecture is true (i.e. the Albanese map in this case is injective) if and only if any homologically trivial idempotent in the ring of correspondences…

Let A be a complete discrete valuation ring with possibly imperfect residue field. The purpose of this paper is to give a notion of conductor for Galois representations over A that generalizes the classical Artin conductor. The definition…

We confirm the quasi-projective case of Saito's conjecture, namely that the cohomological characteristic classes defined by Abbes and Saito can be computed in terms of the characteristic cycles. We construct a cohomological characteristic…

Classifying Frobenius algebras is a key question that has been addressed in various contexts. The structure of finite-dimensional Frobenius algebras depends on the base field and the dimension of the algebra, leading to different…

We discuss the role of additive polynomials and $p$-polynomials in the theory of valued fields of positive characteristic and in their model theory. We outline the basic properties of rings of additive polynomials and discuss properties of…

The aim of this paper is to build a theory of commutative and noncommutative {\it injective} valuations of various algebras (including algebras with zero divisors). The targets of our valuations are (well-)ordered commutative and…

Let G be a semiabelian variety defined over a finite subfield of an algebraically closed field K of prime characteristic. We describe the intersection of a subvariety X of G with a finitely generated subgroup of G(K).

In despair, as Deligne (2000) put it, of proving the Hodge and Tate conjectures, we can try to find substitutes. For abelian varieties in characteristic zero, Deligne (1982) constructed a theory of Hodge classes having many of the…

We prove several results about integral versions of Fourier duality for abelian schemes, making use of Pappas's work on integral Grothendieck-Riemann-Roch. If $S$ is smooth quasi-projective of dimension $d$ over a field and $\pi \colon X\to…

We define a motivic conductor for any presheaf with transfers $F$ using the categorical framework developed for the theory of motives with modulus by Kahn-Miyazaki-Saito-Yamazaki. If $F$ is a reciprocity sheaf this conductor yields an…

The necessary number of commuting vector fields for the Chaplygin ball in the absolute space is constructed. We propose to get these vector fields in framework of the Poisson geometry similar to the Hamiltonian mechanics.

We prove the geometric Bogomolov conjecture over a function field of characteristic zero.

In this article, we use a class of harmonic functions (maybe multi-valued) to study the equality part in a weighted version of Suita conjecture for higher derivatives and finite points case, and we obtain some sufficient and necessary…

Let $X$ be a projective curve of genus 2 over an algebraically closed field of characteristic 2. The Frobenius map on X induces a rational map on the moduli space of rank-2 bundles. We show that up to isomorphism, there is only one (up to…

Recently M. Mustata and V. Srinivas related a natural conjecture about the Frobenius action on the cohomology of the structure sheaf after reduction to characteristic $p > 0$ with another conjecture connecting multiplier ideals and test…

The theory of fields that are equipped with a countably infinite family of commuting derivations is not companionable; but if the axiom is added whereby the characteristic of the fields is zero, then the resulting theory is companionable.…

We study the coinduction functor on the category of FI-modules and its variants. Using the coinduction functor, we give new and simpler proofs of (generalizations of) various results on homological properties of FI-modules. We also prove…

We prove the semisimplicity conjecture for A-motives over finitely generated fields K. This conjecture states that the rational Tate modules V_p(M) of a semisimple A-motive M are semisimple as representations of the absolute Galois group of…

We classify the unipotent character sheaves on a fixed connected component of a reductive algebraic group under a mild hypothesis on the characteristic of the ground field.