arXiv subjects
Publications and source records attributed to Fedor Bogomolov.
We prove that function fields of varieties of dimension at least two over an algebraic closure of a finite field are determined, modulo purely inseparable extensions, by the quotient by the second term in the lower central series of their pro-$\ell$ Galois groups.
We develop a mixed-characteristic version of the Mori-Mukai technique for producing rational curves on K3 surfaces. We reduce modulo p, produce rational curves on the resulting K3 surface over a finite field, and lift to characteristic zero. As an application, we prove that all complex K3 surfaces with Picard group generated by a class of degree two have an infinite number of rational curves.
We prove that the function field of an algebraic variety of dimension greater than 1 over an algebraically closed field of characteristic zero is determined by its first and second Milnor K-groups.
We give examples of failure of the existence of co-fibered products in the category of algebraic curves.
We prove vanishing results for unramified stable cohomology of finite groups of Lie type.
We study hyperbolic curves and their Jacobians over finite fields in the context of anabelian geometry.
This paper following a geometric approach proves new, and reproves old, vanishing and nonvanishing results on the space of twisted symmetric differentials, $H^0(X,S^mΩ^1_X\otimes \Cal O_X(k))$ with $k\le m$, on subvarieties $X\subset \Bbb P^N$. The case of $k=m$ is special and the nonvanishing results are related to the space of quadrics containing $X$ and lead to interesting geometrical objects associated to $X$, as for example the variety of all tangent trisecant lines of $X$. The same techniques give results on the symmetric differentials of subvarieties of abelian varieties. The paper ends with new results and examples about the jump along smooth families of projective varieties $X_t$ of the symmetric plurigenera, $Q_m(X)= \dim H^0(X,S^mΩ^1_X)$, or of the $α$-twisted symmetric plurigenera, $Q_{α,m}(X)= \dim H^0(X,S^m(Ω^1_X\otimes αK_X))$.
We study the distribution of algebraic points on curves in abelian varieties over finite fields.
We study the structure of abelian subgroups of Galois groups of function fields of surfaces.
We study the distribution of algebraic points on K3 surfaces.
We construct examples of elliptic fibrations of orbifold general type (in the sense of Campana) which have no etale covers dominating a variety of general type.
We study the subgroup B_0(G) of H^2(G,Q/Z) consisting of all elements which have trivial restrictions to every Abelian subgroup of G. The group B_0(G) serves as the simplest nontrivial obstruction to stable rationality of algebraic varieties V/G where V is a faithful complex linear representation of the group G. We prove that B_0(G) is trivial for finite simple groups of Lie type A_{\ell}.
We prove rationality results for moduli spaces of elliptic K3 surfaces and elliptic rational surfaces with fixed monodromy groups.
We construct examples of simply connected nonalgebraic symplectic fourfolds with a prescribed number of nonintersecting symplectic curves with positive self-intersections.
We study correspondences between algebraic curves defined over the separable closure of $\Q$ or $\F_p$.
We study abelian subgroups of Galois groups of function fields.
We construct examples of algebraic surfaces with interesting fundamental groups.
We study monodromy groups of elliptic fibrations over the projective line.