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Misha Verbitsky

Publications and source records attributed to Misha Verbitsky.

At least 109 records · Page 6Linked to original sources
arXiv2013

Trihyperkahler reduction and instanton bundles on CP^3

A trisymplectic structure on a complex 2n-manifold is a triple of holomorphic symplectic forms such that any linear combination of these forms has constant rank 2n, n or 0, and degenerate forms in $Ω$ belong to a non-degenerate quadric hypersurface. We show that a trisymplectic manifold is equipped with a holomorphic 3-web and the Chern connection of this 3-web is holomorphic, torsion-free, and preserves the three symplectic forms. We construct a trisymplectic structure on the moduli of regular rational curves in the twistor space of a hyperkaehler manifold, and define a trisymplectic reduction of a trisymplectic manifold, which is a complexified form of a hyperkaehler reduction. We prove that the trisymplectic reduction in the space of regular rational curves on the twistor space of a hyperkaehler manifold M is compatible with the hyperkaehler reduction on M. As an application of these geometric ideas, we consider the ADHM construction of instantons and show that the moduli space of rank r, charge c framed instanton bundles on CP^3 is a smooth, connected, trisymplectic manifold of complex dimension 4rc. In particular, it follows that the moduli space of rank 2, charge c instanton bundles on CP^3 is a smooth complex manifold dimension 8c-3, thus settling a 30-year old conjecture.

math.AG↗
arXiv2013

A global Torelli theorem for hyperkahler manifolds

A mapping class group of an oriented manifold is a quotient of its diffeomorphism group by the isotopies. We compute a mapping class group of a hypekahler manifold $M$, showing that it is commensurable to an arithmetic subgroup in SO(3, b_2-3). A Teichmuller space of $M$ is a space of complex structures on $M$ up to isotopies. We define a birational Teichmuller space by identifying certain points corresponding to bimeromorphically equivalent manifolds, and show that the period map gives an isomorphism of the birational Teichmuller space and the corresponding period space $SO(b_2-3, 3)/SO(2)\times SO(b_2 -3, 1)$. We use this result to obtain a Torelli theorem identifying any connected component of birational moduli space with a quotient of a period space by an arithmetic subgroup. When $M$ is a Hilbert scheme of $n$ points on a K3 surface, with $n-1$ a prime power, our Torelli theorem implies the usual Hodge-theoretic birational Torelli theorem (for other examples of hyperkahler manifolds the Hodge-theoretic Torelli theorem is known to be false).

math.AG↗
arXiv2013

Holomorphic Lagrangian fibrations on hypercomplex manifolds

A hypercomplex manifold is a manifold equipped with a triple of complex structures satisfying the quaternionic relations. A holomorphic Lagrangian variety on a hypercomplex manifold with trivial canonical bundle is a holomorphic subvariety which is calibrated by a form associated with the holomorphic volume form; this notion is a generalization of the usual holomorphic Lagrangian subvarieties known in hyperkaehler geometry. An HKT (hyperkaehler with torsion) metric on a hypercomplex manifold is a metric determined by a local potential, in a similar way to the Kaehler metric. We prove that a base of a holomorphic Lagrangian fibration is always Kaehler, if its total space is HKT. This is used to construct new examples of hypercomplex manifolds which do not admit an HKT structure.

math.DG↗
arXiv2012

Hyperholomorphic sheaves and new examples of hyperkaehler manifolds

Given a compact hyperkaehler manifold $M$ and a holomorphic bundle B over $M$, we consider a Hermitian connection $\nabla$ on B which is compatible with all complex structures on $M$ induced by the hyperkaehler structure. Such a connection is unique, because it is Yang-Mills. We call the bundles admitting such connections hyperholomorphic bundles. A stable bundle is hyperholomorphic if and only if its Chern classes $c_1$, $c_2$ are SU(2)-invariant, with respect to the natural SU(2)-action on the cohomology. For several years, it was known that the moduli space of stable hyperholomorphic bundles is singular hyperkaehler. More recently, it was proven that singular hyperkaehler varieties admit a canonical hyperkaehler desingularization. In the present paper, we show that a moduli space of stable hyperholomorphic bundles is compact, given some assumptions on Chern classes of B and hyperkaehler geometry of $M$ (we also require $dim_C M>2$). Conjecturally, this leads to new examples of hyperkaehler manifolds. We develop the theory of hyperholomorphic sheaves, which are (intuitively speaking) coherent sheaves compatible with hyperkaehler structure. We show that hyperholomorphic sheaves with isolated singularities can be canonically desingularized by a blow-up. This theory is used to study degenerations of hyperholomorphic bundles.

alg-geom↗
arXiv2012

Holography principle for twistor spaces

Let $S$ be a smooth rational curve on a complex manifold $M$. It is called ample if its normal bundle is positive. We assume that $M$ is covered by smooth holomorphic deformations of $S$. The basic example of such a manifold is a twistor space of a hyperkahler or a 4-dimensional anti-selfdual Riemannian manifold $X$ (not necessarily compact). We prove "a holography principle" for such a manifold: any meromorphic function defined in a neighbourhood $U$ of $S$ can be extended to $M$, and any section of a holomorphic line bundle can be extended from $U$ to $M$. This is used to define the notion of a Moishezon twistor space: this is a twistor space $\Tw(X)$ admitting a holomorphic embedding to a Moishezon variety $M'$. We show that this property is local on $X$, and the variety $M'$ is unique up to birational transform. We prove that the twistor spaces of hyperkahler manifolds obtained by hyperkahler reduction of flat quaternionic-Hermitian spaces by the action of reductive Lie groups (such as Nakajima's quiver varieties) are always Moishezon.

math.AG↗
arXiv2012

Rational curves and special metrics on twistor spaces

A Hermitian metric $ω$ on a complex manifold is called SKT or pluriclosed if $dd^cω=0$. Let M be a twistor space of a compact, anti-selfdual Riemannian manifold, admitting a pluriclosed Hermitian metric. We prove that in this case M is Kähler, hence isomorphic to $\C P^3$ or a flag space. This result is obtained from rational connectedness of the twistor space, due to F. Campana. As an aside, we prove that the moduli space of rational curves on the twistor space of a K3 surface is Stein.

math.DG↗
arXiv2012

Families of Lagrangian fibrations on hyperkaehler manifolds

A holomorphic Lagrangian fibration on a holomorphically symplectic manifold is a holomorphic map with Lagrangian fibers. It is known that a given compact manifold admits only finitely many holomorphic symplectic structures, up to deformation. We prove that a given compact manifold with $b_2 \geq 7$ admits only finitely many deformation types of holomorphic Lagrangian fibrations. We also prove that all known hyperkahler manifolds are never Kobayashi hyperbolic.

math.AG↗
arXiv2012

Pseudoholomorphic curves on nearly Kahler manifolds

Let M be an almost complex manifold equipped with a Hermitian form such that its de Rham differential has Hodge type (3,0)+(0,3), for example a nearly Kahler manifold. We prove that any connected component of the moduli space of pseudoholomorphic curves on M is compact. This can be used to study pseudoholomorphic curves on a 6-dimensional sphere with the standard (G_2-invariant) almost complex structure.

math.DG↗
arXiv2012

Calibrations in hyperkahler geometry

We describe a family of calibrations arising naturally on a hyperkähler manifold $M$. These calibrations calibrate the holomorphic Lagrangian, holomorphic isotropic and holomorphic coisotropic subvarieties. When $M$ is an HKT (hyperkaehler with torsion) manifold with holonomy $SL(n, {\Bbb H})$, we construct another family of calibrations $Φ_i$, which calibrates holomorphic Lagrangian and holomorphic coisotropic subvarieties. The calibrations $Φ_i$ are (generally speaking) not parallel with respect to any torsion-free connection on $M$.

math.DG↗
arXiv2012

Subvarieties of hypercomplex manifolds with holonomy in SL(n,H)

A hypercomplex manifold M is a manifold with a triple I,J,K of complex structure operators satisfying quaternionic relations. For each quaternion L=aI +bJ+cK, L^2=-1, L is also a complex structure operator on M, called an induced complex structure. We are studying compact complex subvarieties of (M,L), when L is a generic induced complex structure. Under additional assumptions (Obata holonomy contained in SL(n,H), existence of an HKT metric), we prove that (M,L) contains no divisors, and all complex subvarieties of codimension 2 are trianalytic (that is, also hypercomplex).

math.AG↗
arXiv2011

Blow-ups of locally conformally Kahler manifolds

A locally conformally Kahler (LCK) manifold is a manifold which is covered by a Kahler manifold, with the deck transform group acting by homotheties. We show that the blow-up of a compact LCK manifold along a complex submanifold admits an LCK structure if and only if this submanifold is globally conformally Kahler. We also prove that a twistor space (of a compact 4-manifold, a quaternion-Kahler manifold or a Riemannian m anifold) cannot admit an LCK metric, unless it is Kahler.

math.AG↗
arXiv2011

Hodge theory on nearly Kaehler manifolds

Let (M,I, ω, Ω) be a nearly Kaehler 6-manifold, that is, an SU(3)-manifold with the (3,0)-form Ωand the Hermitian form ωwhich satisfies $dω=3λ\ReΩ, d\ImΩ=-2λω^2$, for a non-zero real constant λ. We develop an analogue of Kaehler relations on M, proving several useful identities for various intrinsic Laplacians on M. When M is compact, these identities bring powerful results about cohomology of M. We show that harmonic forms on M admit the Hodge decomposition, and prove that H^{p,q}(M)=0 unless p=q or (p=1, q=2) or (p=2, q=1).

math.DG↗
arXiv2011

A formally Kahler structure on a knot space of a G2-manifold

A knot space in a manifold M is a space of oriented immersions from a circle S^1 to M up to Diff(S^1). Brylinski has shown that a knot space of a Riemannian threefold is formally Kahler. We prove that a space of knots in a holonomy G2 manifold is formally Kahler.

math.DG↗
arXiv2011

Oeljeklaus-Toma manifolds admitting no complex subvarieties

The Oeljeklaus-Toma (OT-) manifolds are complex manifolds constructed by Oeljeklaus and Toma from certain number fields, and generalizing the Inoue surfaces $S_m$. On each OT-manifold we construct a holomorphic line bundle with semipositive curvature form and trivial Chern class. Using this form, we prove that the OT-manifolds admitting a locally conformally Kahler structure have no non-trivial complex subvarieties. The proof is based on the Strong Approximation theorem for number fields, which implies that any leaf of the null-foliation of w is Zariski dense.

math.CV↗
arXiv2011

Hyperholomorpic connections on coherent sheaves and stability

Let $M$ be a hyperkaehler manifold, and $F$ a torsion-free and reflexive coherent sheaf on $M$. Assume that $F$ (outside of its singularities) admits a connection with a curvature which is invariant under the standard SU(2)-action on 2-forms. If the curvature is square-integrable, then $F$ is stable and its singularities are hyperkaehler subvarieties in $M$. Such sheaves (called hyperholomorphic sheaves) are well understood. In the present paper, we study sheaves admitting a connection with SU(2)-invariant curvature which is not necessarily square-integrable. This situation arises often, for instance, when one deals with higher direct images of holomorphic bundles. We show that such sheaves are stable.

math.AG↗
arXiv2011

Moduli spaces of framed instanton bundles on CP^3 and twistor sections of moduli spaces of instantons on C^2

We show that the moduli space $M$ of holomorphic vector bundles on $CP^3$ that are trivial along a line is isomorphic (as a complex manifold) to a subvariety in the moduli of rational curves of the twistor space of the moduli space of framed instantons on $\R^4$, called the space of twistor sections. We then use this characterization to prove that $M$ is equipped with a torsion-free affine connection with holonomy in $Sp(2n,\C)$.

math.AG↗
arXiv2010

A CR twistor space of a G2-manifold

Let M be a G2-manifold. We consider an almost CR-structure on the sphere bundle of unit tangent vectors on M, called the CR twistor space. This CR-structure is integrable if and only if M is a holonomy G2 manifold. We interpret G2-instanton bundles as CR-holomorphic bundles on its twistor space.

math.DG↗