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## Pseudoholomorphic curves on nearly Kahler manifolds

Communications in Mathematical Physics. 2013. Vol. 324. No. 1. P. 173-177.

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.

Keywords: симплектическая геометриядифференциальная геометрияDifferential GeometrySymplectic Geometry

Publication based on the results of:

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Let M be a closed symplectic manifold of volume V. We say that M admits a full symplectic packing by balls if any collection of symplectic balls of total volume less than V admits a symplectic embedding to M. In 1994 McDuff and Polterovich proved that symplectic packings of Kahler manifolds can be characterized in ...

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Andrey Soldatenkov, Misha Verbitsky, k-symplectic structures and absolutely trianalytic subvarieties in hyperkahler manifolds / Cornell University. Series math "arxiv.org". 2014.

Let $(M,I,J,K)$ be a hyperkahler manifold, and $Z\subset (M,I)$ a complex subvariety in $(M,I)$. We say that $Z$ is trianalytic if it is complex analytic with respect to $J$ and $K$, and absolutely trianalytic if it is trianalytic with respect to any hyperk\"ahler triple of complex structures $(M,I,J',K')$ containing $I$. For a generic complex structure ...

Added: September 5, 2014

Ekaterina Amerik, Misha Verbitsky, Morrison-Kawamata cone conjecture for hyperkahler manifolds / Cornell University. Series math "arxiv.org". 2014.

Let $M$ be a simple holomorphically symplectic manifold, that is, a simply connected holomorphically symplectic manifold of Kahler type with $h^{2,0}=1$. We prove that the group of holomorphic automorphisms of $M$ acts on the set of faces of its Kahler cone with finitely many orbits. This is a version of the Morrison-Kawamata cone conjecture for ...

Added: September 5, 2014

Verbitsky M., Ergodic complex structures on hyperkahler manifolds / Cornell University. Series math "arxiv.org". 2013.

Let M be a compact complex manifold. The corresponding Teichmuller space $\Teich$ is a space of all complex structures on M up to the action of the group of isotopies. The group Γ of connected components of the diffeomorphism group (known as the mapping class group) acts on $\Teich$ in a natural way. An ergodic ...

Added: December 27, 2013

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We propose Gamma Conjectures for Fano manifolds which can be thought of as a square root of the index theorem. Studying the exponential asymptotics of solutions to the quantum differential equation, we associate a principal asymptotic class A_F to a Fano manifold F. We say that F satisfies Gamma Conjecture I if A_F equals the ...

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We propose multidimensional versions of the Painleve VI equation and its degenerations. These field theories are related to the isomonodromy problems on flat holomorphic infinite rank bundles over elliptic curves and take the form of non-autonomous Hamiltonian equations. The modular parameter of curves plays the role of "time". Reduction of the field equations to the ...

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We prove that the wrapped Fukaya category of a punctured sphere ($ S^{2}$ with an arbitrary number of points removed) is equivalent to the triangulated category of singularities of a mirror Landau-Ginzburg model, proving one side of the homological mirror symmetry conjecture in this case. By investigating fractional gradings on these categories, we conclude that ...

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The known upper bounds for the multiplicities of the Laplace-Beltrami operator eigenvalues on the real projective plane are improved for the eigenvalues with even indexes. Upper bounds for Dirichlet, Neumann and Steklov eigenvalues on the real projective plane with holes are also provided. ...

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A hypercomplex manifold M is a manifold equipped with three complex structures satisfying quaternionic relations. Such a manifold admits a canonical torsion-free connection preserving the quaternion action, called Obata connection. A quaternionic Hermitian metric is a Riemannian metric on which is invariant with respect to unitary quaternions. Such a metric is called HKT if it ...

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For every smooth del Pezzo surface $S$, smooth curve $C\in|-K_{S}|$ and $\beta\in(0,1]$, we compute the $\alpha$-invariant of Tian $\alpha(S,(1-\beta)C)$ and prove the existence of K\"ahler--Einstein metrics on $S$ with edge singularities along $C$ of angle $2\pi\beta$ for $\beta$ in certain interval. In particular we give lower bounds for the invariant $R(S,C)$, introduced by Donaldson as ...

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Déev R. N., Compact fibrations with hyperk ̈ahler fibers / Cornell University. Series arXiv "math". 2016.

Essential dimension of a family of complex manifolds is the dimension of the image of its base in the Kuranishi space of the fiber. We prove that any family of hyperk\"ahler manifolds over a compact simply connected base has essential dimension not greater than 1. A similar result about families of complex tori is also ...

Added: September 23, 2016

Galkin S., Golyshev V., Iritani H., Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures / Cornell University. Series math "arxiv.org". 2014. No. 1404.6407.

We propose Gamma Conjectures for Fano manifolds which can be thought of as a square root of the index theorem. Studying the exponential asymptotics of solutions to the quantum differential equation, we associate a principal asymptotic class A_F to a Fano manifold F. We say that F satisfies Gamma Conjecture I if A_F equals the ...

Added: May 4, 2014

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Verbitsky M., Degenerate twistor spaces for hyperkahler manifolds / Cornell University. Series math "arxiv.org". 2013.

Let M be a hyperkaehler manifold, and η a closed, positive (1,1)-form which is degenerate everywhere on M. We associate to η a family of complex structures on M, called a degenerate twistor family, and parametrized by a complex line. When η is a pullback of a Kaehler form under a Lagrangian fibration L, all ...

Added: December 27, 2013

Springer, 2013

Vladimir Arnold was one of the great mathematical scientists of our time. He is famous for both the breadth and the depth of his work. At the same time he is one of the most prolific and outstanding mathematical authors. This second volume of his Collected Works focuses on hydrodynamics, bifurcation theory, and algebraic geometry. ...

Added: February 20, 2013

Ivan Cheltsov, Rubinstein Y., Asymptotically log Fano varieties / Cornell University. Series math "arxiv.org". 2013.

Motivated by the study of Fano type varieties we define a new class of log pairs that we call asymptotically log Fano varieties and strongly asymptotically log Fano varieties. We study their properties in dimension two under an additional assumption of log smoothness, and give a complete classification of two dimensional strongly asymptotically log smooth ...

Added: December 27, 2013