Searcharxiv⌕ Search

arXiv subjects

Misha Verbitsky

Publications and source records attributed to Misha Verbitsky.

At least 19 recordsLinked to original sources
arXiv2026

Apollonian carpets and the boundary of the Kahler cone of a hyperkahler manifold

The ample cone of a compact Kahler $n$-manifold $M$ is the intersection of its Kahler cone and the real subspace generated by integer (1,1)-classes. Its isotropic boundary is the set of all points $η$ on its boundary such that $\int_M η^n=0$. We are interested in the relation between the shape of the isotropic boundary of the ample cone of a hyperkahler manifold and the dynamics of its holomorphic automorphism group $G$. In this case, the projectivization of the ample cone is realized as an open, locally polyhedral subset in a hyperbolic space $H$. The isotropic boundary $S$ is realized as a subset of the hyperbolic boundary (the absolute) $A$ of $H$, which is naturally identified with a Euclidean sphere. It is clear that the isotropic boundary $S$ contains the limit set of $G$ acting on its ample cone. We prove that, conversely, all irrational points on $S$ belong to the limit set. Using a result of N. Shah about limiting distributions of curves under geodesic flow on hyperbolic manifolds, we prove that every real analytic curve in $S$ is contained in a geodesic sphere in $S$,and in presence of such curves the limit set is the closure of the union of these geodesic spheres. We study the geometry of such fractal sets, called Apollonian carpets, and establish the link between the Apollonian carpet and the structure of the automorphism group.

math.AG↗
arXiv2026

Non-commutative Barge-Ghys quasimorphisms: an erratum

This is an erratum to the paper ``Non-commutative Barge-Ghys quasimorphisms''. The effective version of Ambrose-Singer theorem used in this paper was mis-stated, making Theorem 3.11 false as stated. However, this statement remains true for a semidirect product $G=A\rtimes K$ of a non-compact Lie group with a compact Lie group $K$, as long as the curvature of the connection belongs to the Lie algebra of $K$. Theorems 3.12 and 3.22 remain true under this assumptions, the statements of Theorems 4.28 and 4.31 are modified accordingly. Theorem 5.2 is valid only for compact Lie groups.

math.DG↗
arXiv2026

Locally Conformally Kähler Manifolds of Algebraic Codimension One

A locally conformally Kähler (LCK) manifold is a manifold $M$ which admits a Kähler structure on its universal cover $\tilde M$, in such a way that the monodromy acts conformally on $\tilde M$. Let $M$ be an $n$-dimensional compact LCK manifold of algebraic dimension $n-1$. We prove that $M$ is bimeromorphic to the total space of an isotrivial elliptic fibration. Morever, there exists an alteration of $M$ which dominates bimeromorphically a manifold admitting a free action of an elliptic curve.

math.DG↗
arXiv2026

Sections of Lagrangian fibrations on holomorphic symplectic manifolds

Let $M$ be a holomorphically symplectic manifold, equipped with a Lagrangian fibration $π:\; M \to X$. A degenerate twistor deformation (sometimes also called ``a Tate-Shafarevich twist'') is a family of holomorphically symplectic structures on $M$ parametrized by $H^{1,1}(X)$. All members of this family are equipped with a holomorphic Lagrangian projection to $X$, and their fibers are isomorphic to the fibers of $π$. Assume that $M$ is a compact hyperkahler manifold of maximal holonomy, and the general fiber of the Lagrangian projection $π$ is primitive (that is, not divisible) in integer homology. We also assume that $π$ has reduced fibers in codimension 1. Then $M$ has a degenerate twistor deformation $M'$ such that the Lagrangian projection $π:\; M' \to X$ admits a meromorphic section.

math.AG↗
arXiv2026

Projective subvarieties of Bogomolov-Guan manifolds and quasi-diagonals in products of elliptic curves

We study complex subvarieties in certain non-Kahler holomorphically symplectic manifolds $X$, called the Bogomolov-Guan manifolds. Let $E$ be an elliptic curve, $L$ an ample line bundle on $E$, $S\subset E^2$ a complex curve, and $p_1, p_2$ the corresponding projections of $S$ to $E$. The curve $S$ is called a quasi-diagonal if $p_1^*L\otimes p_2^* L^{-1}$ is a torsion line bundle. We show that there are at most countably many quasi-diagonals for any $(E,L)$. Using the quasi-diagonals, we classify the projective subvarieties in the Bogomolov-Guan manifold. The Bogomolov-Guan manifold is equipped with a Lagrangian fibration $π:\; X \to {\Bbb C} P^n$. We show that an irreducible complex subvariety $Z\subset X$ is Moishezon if and only if $π(Z)$ is a point or a certain complex curve which is described in terms of quasi-diagonals. This is used to prove that for a general Bogomolov-Guan manifold, any projective subvariety belongs to a fiber of $π$.

math.AG↗
arXiv2026

Coherent sheaves on subvarieties in Hopf manifolds

We prove a version of GAGA theorem for a normal complex analytic variety $X$ equipped with an invertible holomorphic contraction $γ$ with center in $x$. We show that $X$ admits a natural structure of an affine variety, and any $γ$-equivariant complex analytic reflexive coherent sheaf on $X$ admits a natural algebraic structure. We prove a structure theorem for $X_0:=X\backslash x$, showing that it admits a proper action of ${\Bbb C}^*$, and is isomorphic to the space of non-zero vectors in the total space of an ample line bundle over the projective variety $Z:= X_0/{\mathbb C}^*$ equipped with an orbifold structure. We show that the quotient $M:=X_0/γ$ admits a holomorphic embedding to a Hopf manifold, and, conversely, any normal subvariety $M$ in a Hopf manifold is obtained this way. We prove a form of structure theorem, showing that any reflexive coherent sheaf on $M$, $\dim M > 2$, admits a filtration such that its associated graded subquotients, tensored with an appropriate line bundle, are obtained as pullbacks of coherent sheaves on the projective variety $Z=X_0/{\mathbb C}^*$. This is used to show that any reflexive coherent sheaf on $M$ is filtrable, that is, admits a filtration with associated graded quotients of rank $\leq 1$.

math.AG↗
arXiv2026

Reflective lattices and hyperkahler manifolds

Using the results of Nikulin and Vinberg on the groups of isometries generated by reflections, we construct a subvariety called the Nikulin-Vinberg locus in the moduli space of polarized hyperkahler manifolds. It is obtained as a finite union of components of higher Noether-Lefschetz loci which parameterize manifolds with certain special Neron-Severi lattices. The Nikulin-Vinberg locus is the closure of the set of hyperkahler manifolds with Picard number $\geq 3$ which have finite groups of birational automorphisms. Using this construction and a refinement of an argument by Oguiso, we show that any non-trivial family of projective deformations of a hyperkahler manifold with $b_2(M)\geq 6$ has a dense set of fibers which have an infinite group of birational automorphisms.

math.AG↗
arXiv2026

Dolbeault formality for complex nilmanifolds

A quasi-isomorphism of differential graded algebras (DGA) is a multiplicative map inducing an isomorphism on cohomology. A DGA is called formal if it can be connected by a chain of quasi-isomorphisms to its cohomology algebra. We prove that the Dolbeault DGA of a complex nilmanifold is formal only if it is a torus, and the Dolbeault algebra of (0,p)-forms is formal if and only if the complex structure is abelian.

math.DG↗
arXiv2026

On the structure of compact strong HKT manifolds

We study the geometry of compact strong HKT and, more generally, compact BHE manifolds. We prove that any compact BHE manifold with full holonomy must be Kähler and we establish a similar result for strong HKT manifolds. Additionally, we demonstrate a rigidity theorem for strong HKT structures on solvmanifolds and we completely classify those with parallel Bismut torsion. Finally, we introduce the Ricci foliation for hypercomplex manifolds and analyze its properties for compact, simply connected, 8-dimensional strong HKT manifolds, proving that they are always Hopf fibrations over a compact $4$-dimensional orbifold.

math.DG↗
arXiv2025

Minimal multiplicity of fiber components in abelian fibrations

An abelian fibration is a proper projective surjective map of complex varieties with general fiber an abelian variety. Consider a multiple fiber of an abelian fibration, and let $m_1, ..., m_k$ be the multiplicities of its irreducible components. We prove that the minimum of $m_i$ is equal to their greatest common divisor $gcd(m_1, ..., m_k)$

math.AG↗
arXiv2025

The abundance and SYZ conjectures in families of hyperkahler manifolds

Let $L$ be a holomorphic line bundle on a hyperkahler manifold $M$, with $c_1(L)$ nef and not big. SYZ conjecture predicts that $L$ is semiample. We prove that this is true, assuming that $(M,L)$ has a deformation $(M',L')$ with $L'$ semiample. We introduce a version of the Teichmuller space that parametrizes pairs $(M,L)$ up to isotopy. We prove a version of the global Torelli theorem for such Teichmuller spaces and use it to deduce the deformation invariance of semiampleness.

math.AG↗
arXiv2025

Birational geometry of hyperkahler manifolds and the Hu-Yau conjecture

Wierzba and Wisniewski proved that in dimension 4, every bimeromorphic map of hyperkahler manifolds is represented as a composition of Mukai flops. Hu and Yau conjectured that this result can be generalized to arbitrary dimension. They defined ``Mukai's elementary transformation'' as the blow-up of a subvariety ruled by complex projective spaces, composed with the contraction of the ruling. Hu and Yau conjectured that any bimeromorphic map of hyperkahler manifolds can be decomposed into a sequence of Mukai's elementary transformations, after possibly removing subvarieties of codimension greater than $2$. We prove this conjecture for compact hyperkahler manifolds of maximal holonomy by decomposing any bimeromorphic map into a composition of wall-crossing flops associated with MBM contractions.

math.AG↗
arXiv2025

Torsor Neron models of hyperkahler manifolds

Let $M$ be a compact hyperkahler manifold equipped with a Lagrangian fibration $π:\; M \to X$, and $M'$ the smooth locus of $π$. We prove that over a complement to a codimension $\geq 2$ subset in $X$, the projection $π:\; M' \to X$ has a natural structure of a torsor over an abelian group bundle, which can be understood as a complex analytic variant of the Néron model construction. This gives an independent proof of a result by Y.-J. Kim.

math.AG↗
arXiv2025

Exotic hypercomplex structures on a torus do not exist

A hypercomplex manifold is a manifold with three complex structures satisfying quaternionic relations. Such a manifold admits a unique torsion-free connection preserving the quaternionic action, called the Obata connection. A compact Kahler manifold admitting a hypercomplex structure always admits a hyperkahler structure as well; however, it is not obvious whether the original hypercomplex structure is hyperkahler. A non-hyperkahler hypercomplex structure on a Kahler manifold is called exotic. We show that the Obata connection for an exotic hypercomplex structure on a torus is flat and classify complete flat affine structures on real tori. We use this classification to prove that exotic hypercomplex structures do not exist.

math.DG↗
arXiv2025

Hermitian-symplectic and Kahler structures on degenerate twistor deformations

Let $(M, Ω)$ be a holomorphically symplectic manifold equipped with a holomorphic Lagrangian fibration $π: M \to B$, and $η$ a closed $(1,1)$-form on $B$. Then $Ω+ π^* η$ is a holomorphically symplectic form on a complex manifold which is called the degenerate twistor deformation of $M$. We prove that degenerate twistor deformations of compact holomorphically symplectic Kähler manifolds are also Kähler. First, we prove that degenerate twistor deformations are Hermitian symplectic, that is, tamed by a symplectic form; this is shown using positive currents and an argument based on the Hahn--Banach theorem, originally due to Sullivan. Then we apply a version of Huybrechts's theorem showing that two non-separated points in the Teichmüller space of holomorphically symplectic manifolds correspond to bimeromorphic manifolds if they are Hermitian symplectic.

math.AG↗
arXiv2025

Special Hermitian structures on suspensions

Motivated by the construction based on topological suspension of a family of compact non-Kähler complex manifolds with trivial canonical bundle given by L. Qin and B. Wang in [QW], we study toric suspensions of balanced manifolds by holomorphic automorphisms. In particular, we show that toric suspensions of Calabi-Yau manifolds are balanced. We also prove that suspensions associated with hyperbolic automorphisms of hyperkähler manifolds do not admit any pluriclosed, astheno-Kähler or p-pluriclosed Hermitian metric. Moreover, we consider natural extensions for hypercomplex manifolds, providing some explicit examples of compact holomorphic symplectic and hypercomplex non-Kähler manifolds. We also show that a modified suspension construction provides examples with pluriclosed metrics.

math.DG↗
arXiv2024

Principles of Locally Conformally Kahler Geometry

An LCK (locally conformally Kahler) manifold is a complex manifold admitting a Kahler covering with monodromy acting by homotheties. Hopf manifolds and their submanifolds are the prime examples. This book presents an introduction to the principles of LCK geometry (the first two parts) and its current situation (the last part). It is supposed to be accessible to master and graduate students of complex geometry. The book contains many exercises of different levels of difficulty. We finish it by a list of open questions.

math.DG↗