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## Characteristic points, fundamental cubic form and Euler characteristic of projective surfaces

Moscow Mathematical Journal. 2020. Vol. 20. No. 3. P. 511-530.

We define local indices for projective umbilics and godrons (also called cusps of Gauss) on generic smooth surfaces in projective 3-space. By means of these indices, we provide formulas that relate the algebraic numbers of those characteristic points on a surface (and on domains of the surface) with the Euler characteristic of that surface (resp. of those domains). These relations determine the possible coexistences of projective umbilics and godrons on the surface. Our study is based on a "fundamental cubic form" for which we provide a closed simple expression.

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 ...

Added: February 16, 2013

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

Функциональный анализ и его приложения 2018 Т. 52 № 2 С. 78-81

Let G be a finite Abelian group acting (linearly) on space ℝn and, therefore, on its complexification ℂn, and let W be the real part of the quotient ℂn/G (in the general case, W ≠ ℝn/G). The index of an analytic 1-form on the space W is expressed in terms of the signature of the ...

Added: October 27, 2020

Proceedings of the International Geometry Center 2018 Vol. 11 No. 2 P. 1-15

In this paper it is proved that every orientable surface admits an orientation-preserving Morse-Smale diffeomorphism with one saddle orbit. It is shown that these diffeomorphisms have exactly three node orbits. In addition, all possible types of periodic data for such diffeomorphisms are established. ...

Added: October 14, 2017

Moscow Mathematical Journal 2020 Vol. 20 No. 3 P. 475-493

By a famous result of K. Saito, the parameter space of the miniversal deformation of the $A_{r-1}$-singularity carries a Frobenius manifold structure. The Landau-Ginzburg mirror symmetry says that, in the flat coordinates, the potential of this Frobenius manifold is equal to the generating series of certain integrals over the moduli space of $r$-spin curves. In ...

Added: May 22, 2020

Full symplectic packing for tori and hyperkahler manifolds / Cornell University. Series math "arxiv.org". 2014.

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 ...

Added: February 5, 2015

American Mathematical Society, 1996

The book is an introduction to the qualitative theory of dynamical systems on manifolds of low dimension (on the circle and on surfaces). Along with classical results, it reflects the most significant achevements in this area obtained in recent times. The reader of this book need to be familiar only with basic courses in differential ...

Added: October 2, 2014

Dynamic alpha-invariants of del Pezzo surfaces / Cornell University. Series math "arxiv.org". 2014.

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 ...

Added: February 5, 2015

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

Absolutely trianalytic tori in the generalized Kummer variety / Cornell University. Series math "arxiv.org". 2015.

We prove that a generic complex deformation of a generalized Kummer variety contains no complex analytic tori. ...

Added: October 16, 2015

IEEE Transactions on Pattern Analysis and Machine Intelligence 2014 Vol. PP No. 99 P. 1

A new data structure for efficient similarity search in very large datasets of high-dimensional vectors is introduced. This structure called the inverted multi-index generalizes the inverted index idea by replacing the standard quantization within inverted indices with product quantization. For very similar retrieval complexity and pre-processing time, inverted multi-indices achieve a much denser subdivision of ...

Added: December 19, 2014

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

Journal of Geometry and Physics 2012 Vol. 62 P. 2294-2319

We define the notions of trace, determinant and, more generally, Berezinian of matrices over a (Z_2)^n graded commutative associative algebra. The applications include a new approach to the classical theory of matrices with coefficients in a Clifford algebra, in particular of quaternionic matrices. In a special case, we recover the classical Dieudonn\'e determinant of quaternionic ...

Added: September 28, 2015

Mathematische Nachrichten 2018 Vol. 291 No. 17-18 P. 2543-2556

Let a finite abelian group G act (linearly) on the space R^n and thus on its complexification C^n. Let W be the real part of the quotient C^n/G (in general W \neq R^n/G). We give an algebraic formula for the radial index of a 1-form \omega on the real quotient W. It is shown that ...

Added: October 27, 2020

Алгебра и анализ 2021 Т. 33 № 3 С. 73-84

Indices of singular points of a vector field or of a 1-form on a smooth manifold are closely related with the Euler characteristic through the classical Poincar\'e--Hopf theorem. Generalized Euler characteristics (additive topological invariants of spaces with some additional structures) are sometimes related with corresponding analogues of indices of singular points. Earlier, there was defined ...

Added: May 2, 2021

On Hamiltonian and contact isotopy liftings / Cornell University. Series arXiv "math". 2016. No. arXiv:1602.07948.

We construct counterexamples to lifting properties of Hamiltonian and contact isotopies ...

Added: December 7, 2016

Discrete Mathematics 2019 Vol. 342 No. 2 P. 600-614

The second part of the paper is devoted to enumeration of r-regular maps on the torus up to all its homeomorphisms (unsensed maps). We describe in detail the periodic orientation reversing homeomorphisms of the torus which turn out to be representable as glide reflections. We show that considering quotients of the torus with respect to ...

Added: September 21, 2018

Existence of HKT metrics on hypercomplex manifolds of real dimension 8 / Cornell University. Series math "arxiv.org". 2014.

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 ...

Added: September 19, 2014

Chekanov-type theorem for spherized cotangent bundles / Cornell University. Series arXiv "math". 2016. No. arXiv:1602.08743.

We prove a Chekanov-type theorem for the spherization of the cotangent bundle ST∗B of a closed manifold B. It claims that for Legendrian submanifolds in ST∗B the property "to be given by a generating family quadratic at infinity" persists under Legendrian isotopies. ...

Added: December 7, 2016

Kobayashi pseudometric on hyperkahler manifolds / Cornell University. Series math "arxiv.org". 2013.

The Kobayashi pseudometric on a complex manifold $M$ is the maximal pseudometric such that any holomorphic map from the Poincare disk to $M$ is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi-Yau manifolds. Using ergodicity of complex structures, we prove this result for any hyperkaehler manifold if it admits a deformation with a ...

Added: August 28, 2013

The second Betti number of hyperkähler manifolds / Cornell University. Series math "arxiv.org". 2014.

Let M be a compact irreducible hyperkahler manifold, from Bogomolov inequality [V1] we obtain forbidden values of the second Betti number b_2 in arbitrary dimension. ...

Added: February 21, 2014

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

Painleve Field Theory / Cornell University. Series math "arxiv.org". 2013.

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 ...

Added: December 27, 2013

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