arXiv subjects

Misha Verbitsky

Publications and source records attributed to Misha Verbitsky.

At least 37 records · Page 2Linked to original sources

arXiv2024

Rigid currents on compact hyperkahler manifolds

A rigid cohomology class on a complex manifold is a class that is represented by a unique closed positive current. The positive current representing a rigid class is also called rigid. For a compact Kahler manifold $X$ all eigenvectors of hyperbolic automorphisms acting on $H^{1,1}(X)$ that have non-unit eigenvalues are rigid classes. Such classes are always parabolic, namely, they belong to the boundary of the Kahler cone and have vanishing volume. We study parabolic $(1,1)$-classes on compact hyperkahler manifolds with $b_2 \geq 7$. We show that a parabolic class is rigid if it is not orthogonal to a rational vector with respect to the BBF form. This implies that a general parabolic class on a hyperkahler manifold is rigid.

math.AG↗

arXiv2024

Balanced metrics and Gauduchon cone of locally conformally Kahler manifolds

A complex Hermitian $n$-manifold $(M,I, ω)$ is called locally conformally Kahler (LCK) if $dω=θ\wedgeω$, where $θ$ is a closed 1-form, balanced if $ω^{n-1}$ is closed, and SKT if $dIdω=0$. We conjecture that any compact complex manifold admitting two of these three types of Hermitian forms (balanced, SKT, LCK) also admits a Kahler metric, and prove partial results towards this conjecture. We conjecture that the (1,1)-form $-d(Iθ)$ is Bott--Chern homologous to a positive (1,1)-current. This conjecture implies that $(M,I)$ does not admit a balanced Hermitian metric. We verify this conjecture for all known classes of LCK manifolds.

math.DG↗

arXiv2024

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↗

arXiv2023

Do products of compact complex manifolds admit LCK metrics?

An LCK (locally conformally Kahler) manifold is a Hermitian manifold which admits a Kahler cover with deck group acting by holomorphic homotheties with respect to the Kahler metric. The product of two LCK manifolds does not have a natural product LCK structure. It is conjectured that a product of two compact complex manifolds is never LCK. We classify all known examples of compact LCK manifolds onto three not exclusive classes: LCK with potential, a class of manifolds we call of Inoue type, and those containing a rational curve. In the present paper, we prove that a product of an LCK manifold and an LCK manifold belonging to one of these three classes does not admit an LCK structure.

math.DG↗

arXiv2023

Roundness of the ample cone and existence of double Lagrangian fibrations on hyperkahler manifolds

Let $M$ be a hyperkahler manifold of maximal holonomy (that is, an IHS manifold), and let $K$ be its Kahler cone, which is an open, convex subset in the space $H^{1,1}(M, R)$ of real (1,1)-forms. This space is equipped with a canonical bilinear symmetric form of signature $(1,n)$ obtained as a restriction of the Bogomolov-Beauville-Fujiki form. The set of vectors of positive square in the space of signature $(1,n)$ is a disconnected union of two convex cones. The "positive cone" is the component which contains the Kahler cone. We say that the Kahler cone is "round" if it is equal to the positive cone. The manifolds with round Kahler cones have unique bimeromorphic model and correspond to Hausdorff points in the corresponding Teichmuller space. We prove thay any maximal holonomy hyperkahler manifold with $b_2 > 4$ has a deformation with round Kahler cone and the Picard lattice of signature (1,1), admitting two non-collinear integer isotropic classes. This is used to show that all known examples of hyperkahler manifolds admit a deformation with two transversal Lagrangian fibrations, and the Kobayashi metric vanishes unless the Picard rank is maximal.

math.AG↗

arXiv2023

Parabolic automorphisms of hyperkahler manifolds

A parabolic automorphism of a hyperkahler manifold is a holomorphic automorphism acting on $H^2(M)$ by a non-semisimple quasi-unipotent linear map. We prove that a parabolic automorphism which preserves a Lagrangian fibration acts on its fibers ergodically. The invariance of a Lagrangian fibration is automatic for manifolds satisfying the hyperkahler SYZ conjecture; this includes all known examples of hyperkahler manifolds. When there are two parabolic automorphisms preserving two distinct Lagrangian fibration, it follows that the group they generate acts on $M$ ergodically. Our results generalize those obtained by S. Cantat for K3 surfaces.

math.AG↗

arXiv2023

Kahler-type embeddings of balls into symplectic manifolds

Consider a symplectic embedding of a disjoint union of domains into a symplectic manifold $M$. Such an embedding is called Kahler-type, or respectively tame, if it is holomorphic with respect to some (not a priori fixed, Kahler-type) complex structure on $M$ compatible with the symplectic form, or respectively tamed by it. Assume that $M$ either of the following: a complex projective space (with the standard symplectic form), an even-dimensional torus or a K3 surface equipped with an irrational Kahler-type symplectic form. Then any two Kahler-type embeddings of a disjoint union of balls into $M$ can be mapped into each other by a symplectomorphism acting trivially on the homology. If the embeddings are holomorphic with respect to complex structures compatible with the symplectic form and lying in the same connected component of the space of Kahler-type complex structures on $M$, then the symplectomorphism can be chosen to be smoothly isotopic to the identity. For certain $M$ and certain disjoint unions of balls we describe precisely the obstructions to the existence of Kahler-type embeddings of the balls into $M$. In particular, symplectic volume is the only obstruction for the existence of Kahler-type embeddings of $l^n$ equal balls (for any $l$) into the $n$-dimensional complex projective space with the standard symplectic form and of any number of possibly different balls into a torus or a K3 surface, equipped with an irrational symplectic form. We also show that symplectic volume is the only obstruction for the existence of tame embeddings of disjoint unions of equal balls, polydisks, or parallelepipeds, into a torus equipped with a generic Kahler-type symplectic form. For balls and parallelepipeds the same is true for K3 surfaces.

math.SG↗

arXiv2023

Algebraic cones of LCK manifolds with potential

A complex manifold $X$ is called "LCK manifolds with potential" if it can be realized as a complex submanifold of a Hopf manifold. Let $Y$ its $\Z$-covering, considered as a complex submanifold in $C^n \backslash 0$. We prove that $Y$ is algebraic. We call the manifolds obtained this way the algebraic cones, and show that the affine algebraic structure on $Y$ is independent from the choice of $X$. We give several intrinsic definitions of an algebraic cone, and prove that these definitions are equivalent.

math.AG↗

arXiv2023

Bimeromorphic geometry of LCK manifolds

A locally conformally Kähler (LCK) manifold is a complex manifold $M$ which has a Kähler structure on its cover, such that the deck transform group acts on it by homotheties. Assume that the Kähler form is exact on the minimal Kähler cover of $M$. We prove that any bimeromorphic map $M'\rightarrow M$ is in fact holomorphic; in other words, $M$ has a unique minimal model. This can be applied to a wide class of LCK manifolds, such as the Hopf manifolds, their complex submanifolds and to OT manifolds.

math.DG↗

arXiv2023

Holomorphic tensors on Vaisman manifolds

An LCK (locally conformally Kahler) manifold is a complex manifold admitting a Hermitian form $ω$ which satisfies $dω=ω\wedge θ$, where $θ$ is a closed 1-form, called the Lee form. An LCK manifold is called Vaisman if the Lee form is parallel with respect to the Levi-Civita connection. The dual vector field, called the Lee field, is holomorphic and Killing. We prove that any holomorphic tensor on a Vaisman manifold is invariant with respect to the Lee field. This is used to compute the Kodaira dimension of Vaisman manifolds. We prove that the Kodaira dimension of a Vaisman manifold obtained as a $Z$-quotient of an algebraic cone over a projective manifold $X$ is equal to the Kodaira dimension of $X$. This can be applied to prove the deformational stability of the Kodaira dimension of Vaisman manifolds.

math.AG↗

arXiv2022

A Calabi-Yau theorem for Vaisman manifolds

A compact complex Hermitian manifold $(M, I, w)$ is called Vaisman if $dw=w\wedge θ$ and the 1-form $θ$, called the Lee form, is parallel with respect to the Levi-Civita connection. The volume form of $M$ is invariant with respect to the action of the vector field $X$ dual to $θ$ (called the Lee field) and the vector field $I(X)$, called { the anti-Lee field}. The cohomology class of $θ$, called the Lee class, plays the same role as the Kahler class in Kahler geometry. We prove that a Vaisman metric is uniquely determined by its volume form and the Lee class, and, conversely, for each Lee class $[θ]$ and each Lee- and anti-Lee-invariant volume form $V$, there exists a Vaisman structure with the volume form $V$ and the Lee class $c[θ]$. This is an analogue of the Calabi-Yau theorem claiming that the Kahler form is uniquely determined by its volume and the cohomology class.

math.DG↗

arXiv2022

Mall bundles and flat connections on Hopf manifolds

A Mall bundle on a Hopf manifold H is a holomorphic vector bundle whose pullback to the universal cover of H is trivial. We define resonant and non-resonant Mall bundles, generalizing the notion of the resonance in ODE, and prove that a non-resonant Mall bundle always admits a flat holomorphic connection. We use this observation to prove a version of Poincare-Dulac linearization theorem, showing that any non-resonant invertible holomorphic contraction of a complex space is linear in appropriate holomorphic coordinates. We define the notion of resonance in Hopf manifolds, and show that all non-resonant Hopf manifolds are linear; previously, this result was obtained by Kodaira using the Poincare-Dulac theorem.

math.DG↗

arXiv2022

Rigidity of Lagrangian embeddings into symplectic tori and K3 surfaces

A Kahler-type form is a symplectic form compatible with an integrable complex structure. Let M be either a torus or a K3-surface equipped with a Kahler-type form. We show that the homology class of any Maslov-zero Lagrangian torus in M has to be non-zero and primitive. This extends previous results of Abouzaid-Smith (for tori) and Sheridan-Smith (for K3-surfaces) who proved it for particular Kahler-type forms on M. In the K3 case our proof uses dynamical properties of the action of the diffeomorphism group of M on the space of the Kahler-type forms. These properties are obtained using Shah's arithmetic version of Ratner's orbit closure theorem.

math.SG↗

arXiv2022

Non-linear Hopf manifolds are locally conformally Kahler

A Hopf manifold is a quotient of $C^n\backslash 0$ by the cyclic group generated by a holomorphic contraction. Hopf manifolds are diffeomorphic to $S^1\times S^{2n-1}$ and hence do not admit Kahler metrics. It is known that Hopf manifolds defined by linear contractions (called linear Hopf manifolds) have locally conformally Kahler (LCK) metrics. In this paper we prove that the Hopf manifolds defined by non-linear holomorphic contractions admit holomorphic embeddings into linear Hopf manifolds, and, moreover they admit LCK metrics.

math.DG↗

arXiv2022

Lee classes on LCK manifolds with potential

An LCK (locally conformally Kahler) manifold is a complex manifold $(M,I)$ equipped with a Hermitian form $ω$ and a closed 1-form $θ$, called the Lee form, such that $dω=θ\wedgeω$. An LCK manifold with potential is an LCK manifold with a positive Kahler potential on its cover, such that the deck group multiplies the Kahler potential by a constant. A Lee class of an LCK manifold is the cohomology class of the Lee form. We determine the set of Lee classes on LCK manifolds admitting an LCK structure with potential, showing that it is an open half-space in $H^1(M,{\mathbb R})$. For Vaisman manifolds, this theorem was proven in 1994 by Tsukada; we give a new self-contained proof of his result.

math.DG↗

arXiv2021

The Moser isotopy for holomorphic symplectic and C-symplectic structures

A C-symplectic structure is a complex-valued 2-form which is holomorphically symplectic for an appropriate complex structure. We prove an analogue of Moser's isotopy theorem for families of C-symplectic structures and list several applications of this result. We prove that the degenerate twistorial deformation associated to a holomorphic Lagrangian fibration is locally trivial over the base of this fibration. This is used to extend several theorems about Lagrangian fibrations, known for projective hyperkähler manifolds, to the non-projective case. We also exhibit new examples of non-compact complex manifolds with infinitely many pairwise non-birational algebraic compactifications.

math.AG↗

arXiv2021

Compact homogeneous locally conformally Kahler manifolds are Vaisman. A new proof

An LCK manifold with potential is a complex manifold with a Kahler potential on its cover, such that any deck transformation multiplies the Kahler potential by a constant multiplier. We prove that any homogeneous LCK manifold admits a metric with LCK potential. This is used to give a new proof that any compact homogeneous LCK manifold is Vaisman.

math.DG↗

arXiv2021

Supersymmetry and Hodge theory on Sasakian and Vaisman manifolds

Sasakian manifolds are odd-dimensional counterpart to Kahler manifolds. They can be defined as contact manifolds equipped with an invariant Kahler structure on their symplectic cone. The quotient of this cone by the homothety action is a complex manifold called Vaisman. We study harmonic forms and Hodge decomposition on Vaisman and Sasakian manifolds. We construct a Lie superalgebra associated to a Sasakian manifold in the same way as the Kahler supersymmetry algebra is associated to a Kahler manifold. We use this construction to produce a self-contained, coordinate-free proof of the results by Tachibana, Kashiwada and Sato on the decomposition of harmonic forms and cohomology of Sasakian and Vaisman manifolds. In the last section, we compute the supersymmetry algebra of Sasakian manifolds explicitly.

math.DG↗