arXiv subjects

Vasily Rogov

Publications and source records attributed to Vasily Rogov.

8 recordsLinked to original sources

arXiv2026

Topology of higher Albanese maps and aspherical varieties with nilpotent fundamental group

Let $X$ be a normal complex algebraic variety. Let $\mathcal{G}^s_{\mathbb{Z}}(X)$ be the maximal torsion free nilpotent quotient of $\pi_1(X)$ of nilpotency class at most $s$. Let $F^{\bullet}\mathfrak{g}^s$ be the Morgan--Hain Hodge filtration on the Lie algebra of the $s$-th lower central quotient of the complex Malcev completion of $\pi_1(X)$. We show that the natural map $H^k(\mathcal{G}^s_{\mathbb{Z}}(X), \mathbb{Z}) \to H^k(X, \mathbb{Z})$ vanishes for $k > \dim F^1\mathfrak{g}^s$. If $\mathcal{G}^s_{\mathbb{Z}}(X)$ is of nilpotency class greater than two, this includes the top nonvanishing degree of $H^{\bullet}(\mathcal{G}^s_{\mathbb{Z}}(X), \mathbb{Z})$. We deduce that if the fundamental group of an aspherical normal variety is virtually nilpotent, it is virtually two-step nilpotent. This gives a positive answer to a question of Aguilar and Campana in this case of aspherical varieties. The ingredients of the proof are the $q$-convexity of higher Albanese manifolds and the definability of higher Albanese maps.

math.AG↗

arXiv2026

Holomorphic families of knots

Let $(M, [g])$ be a $3$-dimensional conformal manifold. The space of knots $\operatorname{Kn}(M)$ in $M$ is an infinite-dimensional manifold that is known to carry an almost complex structure. This structure is formally integrable by a result of Brylinski. We study finite dimensional holomorphic submanifolds in $\operatorname{Kn}(M)$. We define an holomorphic family of knots in $(M, [g])$ parametrised by a finite-dimensional complex manifold $(X, I_X)$, and construct several series of examples. We show that the base $(X, I_X)$ is K\"ahler, and if $X$ is compact, it is a projective variety of complex dimension at most $2$. In this case the conformal structure $[g]$ uniquely determines the complex structure $I_X$ and vice versa. We prove that if an holomorphic family of knots in $(M, [g])$ over a compact base $(X, I_X)$ defines a foliation on $\mathbf{S}(TM)$, then $(X, I_X) \simeq \mathbb{C}\mathbf{P}^1 \times \mathbb{C}\mathbf{P}^1$, the manifold $(M, [g])$ is conformally equivalent to either $S^3$ or $\mathbb{R}\mathbf{P}^3$ with round metric, and all knots are geodesic in some round metric in the class.

math.DG↗

arXiv2025

O-minimal geometry of higher Albanese manifolds

Let X be a normal quasi-projective variety over $\mathbb{C}$. We study its higher Albanese manifolds, introduced by Hain and Zucker, from the point of view of o-minimal geometry. We show that for each $s$ the higher Albanese manifold $\operatorname{Alb}^s(X)$ can be functorially endowed with a structure of an $\mathbb{R}_{\operatorname{alg}}$-definable complex manifold in such a way that the natural projections $\operatorname{Alb}^s(X) \to \operatorname{Alb}^{s-1}(X)$ are $\mathbb{R}_{\operatorname{alg}}$-definable and the higher Albanese maps $\operatorname{alb}^s \colon X^{\operatorname{an}} \to \operatorname{Alb}^s(X)$ are $\mathbb{R}_{\operatorname{an}, \operatorname{exp}}$-definable. Suppose that for some $s \ge 3$ the definable manifold $\operatorname{Alb}^s(X)$ is definably biholomorphic to a quasi-projective variety. We show that in this case the higher Albanese tower stabilises at the second step, i.e. the maps $\operatorname{Alb}^r (X) \to \operatorname{Alb}^{r-1}(X)$ are isomorphisms for $r\ge 3$. It follows that if $\operatorname{alb}^s \colon X^{\operatorname{an}} \to \operatorname{Alb}^s(X)$ is dominant for some $s \ge 3$, then the higher Albanese tower stabilises at the second step and the pro-unipotent completion of $\pi_1(X)$ is at most 2-step nilpotent. This confirms a special case of a conjecture by Campana on nilpotent fundamental groups of algebraic varieties. As another application, we prove the existence and quasi-projectivity of unipotent Shafarevich reductions.

math.AG↗

arXiv2024

The Bieri-Neumann-Strebel sets of quasi-projective groups

Let $X$ be a smooth complex quasi-projective variety and $\Gamma=\pi_1(X)$. Let $\chi \colon \Gamma \to \mathbb{R}$ be an additive character. We prove that the ray $[\chi]$ does not belong to the BNS set $\Sigma(\Gamma)$ if and only if it comes as a pullback along an algebraic fibration $f \colon X \to \mathcal{C}$ over a quasi-projective hyperbolic orbicurve $\mathcal{C}$. We also prove that if $\pi_1(X)$ admits a solvable quotient which is not virtually nilpotent, there exists a finite \'etale cover $X_1 \to X$ and a fibration $f \colon X_1 \to \mathcal{C}$ over a quasi-projective hyperbolic orbicurve $\mathcal{C}$. Both of these results were proved by Delzant in the case when $X$ is a compact K\"ahler manifold. We deduce that $\Gamma$ is virtually solvable if and only if it is virtually nilpotent, generalising the theorems of Delzant and Arapura-Nori. As a byproduct, we prove a version of Simpson's Lefschetz Theorem for the integral leaves of logarithmic $1$-forms that do not extend to any partial compactification. We give two applications of our results. First, we strengthen the recent theorem of Cadorel-Deng-Yamanoi on virtual nilpotency of fundamental groups of quasi-projective $h$-special and weakly special manifolds. Second, we prove the sharpness of Suciu's tropical bound for the fundamental groups of smooth quasi-projective varieties and answer a question of Suciu on the topology of hyperplane arrangements.

math.AG↗

arXiv2024

On the metric Kollár-Pardon problem

Let $(M, g)$ be a compact real analytic Riemannian manifold and $π\colon \widetilde{M} \to M$ its universal cover. Assume that $\widetilde{M}$ can be realised as a manifold definable in an o-minimal structure $Σ$ expanding $\mathbb{R}_{\mathrm{an}}$ in such a way that the pullback metric $\widetilde{g}:=π^*g$ is $Σ$-definable. For instance, this is the case when $\widetilde{M}$ can be realised as a semi-algebraic submanifold in $\mathbb{R}^n$ in such a way that the coefficients of the metric $\widetilde{g}$ are semi-algebraic. We show that there exists a definable smooth map $\widetilde{M} \to \widetilde{K}$ to a compact simply connected $Σ$-definable space $\widetilde{K}$ such that its regular fibres are Riemann locally homogeneous with respect to the metric $\widetilde{g}$. We deduce that under these assumptions $π_1(M)$ is quasi-isometric to a locally homogeneous space. In the case when $M$ is aspherical we show that $(\widetilde{M}, \widetilde{g})$ is a homogeneous Riemannian manifold. A similar result in the setting of complex algebraic geometry was earlier conjectured by Kollár and Pardon (\cite{KP}). Using our results, we prove the conjecture of Kollár-Pardon in the special case of smooth aspherical varieties admitting a bi-definable Kähler metric and discuss the analogues of this conjecture in other branches of geometry.

math.DG↗

arXiv2023

Non-algebraic deformations of flat Kähler manifolds

Let $X$ be a compact Kähler manifold with vanishing Riemann curvature. We prove that there exists a manifold $X'$, deformation equivalent to $X$, which is not an analytification of any projective variety, if and only if $H^0(X, Ω^2) \neq 0$. Using this, we recover a recent theorem of Catanese and Demleitner, which states that a rigid smooth quotient of a complex torus is always projective. We also produce many examples of non-algebraic flat Kähler manifolds with vanishing first Betti number.

math.DG↗

arXiv2021

Shafarevich-Tate groups of holomorphic Lagrangian fibrations

Consider a Lagrangian fibration $\pi\colon X\to \mathbb P^n$ on a hyperk\"ahler manifold $X$. There are two ways to construct a holomorphic family of deformations of $\pi$ over $\mathbb C$. The first one is known under the name Shafarevich-Tate family while the second one is the degenerate twistor family constructed by Verbitsky. We show that both families coincide. We prove that for a very general $X$ all members of the Shafarevich-Tate family are K\"ahler. There is a related notion of the Shafarevich-Tate group associated to a Lagrangian fibration. Its connected component of unity can be shown to be isomorphic to $\mathbb C/\Lambda$ where $\Lambda$ is a finitely generated subgroup of $\mathbb C$ and $\mathbb C$ is thought of as the base of the Shafarevich-Tate family. We show that for a very general $X$, projective deformations in the Shafarevich-Tate family correspond to the torsion points in the connected component of unity of the Shafarevich-Tate group. A sufficient condition for a Lagrangian fibration $X$ to be projective is existence of a holomorphic section. We find sufficient cohomological conditions for existence of a deformation in the Shafarevich-Tate family that admits a section.

math.AG↗

arXiv2018

Kähler submanifolds in Iwasawa manifolds

Iwasawa manifold is a compact complex homogeneous manifold isomorphic to a quotient of the group of complex unipotent $3 \times 3$ matrices by a cocompact lattice. We prove that any compact complex curve in an Iwasawa manifold is contained in a holomorphic torus. We also prove that any Kähler surface in an Iwasawa manifold is either an abelian surface or a non-projective isotrivial elliptic surface of Kodaira dimension one. In the Appendix we show that any complex torus in an Iwasawa manifold carries complex multiplication.

math.DG↗