arXiv research

Control data constructed for smooth weak deformation retraction of stratified spaces.

Study deformations of compact Calabi-Yau conifolds with singularities.

The paper addresses deformations of Kähler spaces with vanishing first Chern class.

Let $X$ be a compact complex manifold with trivial canonical bundle and satisfying the $\partial\bar{\partial}$-Lemma. We show that the Kuranishi space of $X$ is a smooth universal deformation and that small deformations enjoy the same properties as $X$. If, in addition, $X$ admits a complex symplectic form, then the l…

This paper concerns with deformations of noncompact complex hyperbolic manifolds (with locally Bergman metric), varieties of discrete representations of their fundamental groups into $PU(n,1)$ and the problem of (quasiconformal) stability of deformations of such groups and manifolds in the sense of L.Bers and D.Sulliva…

First non-trivial examples of deformed G_2-instantons, distinguishing nearly parallel G_2-structures.

The paper examines bi-Lipschitz triviality of function germs on singular varieties.

Study on deformations of LC Spin(7) instantons simplifies the problem.

Let X be a compact Kahler manifold with a non-trivial holomorphic Poisson structure. Then there exist deformations of non-trivial generalized Kahler structures with one pure spinor on X. We prove that every Poisson submanifold of X is a generalized Kahler submanifold with respect to the deformed generalized Kahler stru…

Study Einstein-Yang-Mills fields on specific manifolds, proving field deformations.

Study on characteristic classes for foliation deformations.

Given a compact complex $n$-fold $X$ satisfying the $\partial\bar\partial$-lemma and supposed to have a trivial canonical bundle $K_X$ and to admit a balanced (=semi-Kähler) Hermitian metric $ω$, we introduce the concept of deformations of $X$ that are {\bf co-polarised} by the balanced class $[ω^{n-1}]\in H^{n-1,\,n-1…

Inspired by Berndtsson's work on the subharmonicity property of the Bergman kernel, we give a local variation formula of the full Bergman kernels associated to deformations of complex manifolds. In compact case, it follows from the reproducing property of the Bergman kernel and the curvature formula of the 0-th direct …

We provide a complete list of two- and three-component Poisson structures of hydrodynamic type with degenerate metric, and study their homogeneous deformations. In the non-degenerate case any such deformation is trivial, that is, can be obtained via Miura transformation. We demonstrate that in the degenerate case this …

We investigate the deformation theory of the simplest bihamiltonian structure of hydrodynamic type, that of the dispersionless KdV hierarchy. We prove that all of its deformations are quasi-trivial in the sense of B. Dubrovin and Y. Zhang, that is, trivial after allowing transformations where the first partial derivati…

First non-trivial examples of deformed Spin(7)-instantons constructed.

Extending the work of G. Székelyhidi and T. Brönnle to Sasakian manifolds we prove that a small deformation of the complex structure of the cone of a constant scalar curvature Sasakian manifold admits a constant scalar curvature structure if it is K-polystable. This also implies that a small deformation of the complex …

This paper deforms complex tori and their mirrors using gerbes.

If a rectangular diagram represents the trivial knot, then it can be deformed into the rectangular diagram with only two vertical edges by a finite sequence of merge operations and exchange operations, without increasing the number of vertical edges, which was shown by I. A. Dynnikov. We show in this paper that we need…

Study of exceptional algebroids in type IIA string theory.

The paper proves a logarithmic partial derivative lemma and applies it to several geometric problems.

The deformation theory of hyperbolic and Euclidean cone-manifolds with all cone angles less then 2π plays an important role in many problems in low dimensional topology and in the geometrization of 3-manifolds. Furthermore, various old conjectures dating back to Stoker about the moduli of convex hyperbolic and Euclidea…

Extends noncommutative deformations of holomorphic line bundles on complex tori and their mirror partners.

We present the extended Kuranishi space for Kodaira surface as a non-trivial example to Kontsevich and Barannikov's extended deformation theory. We provide a non-trivial example of Hertling-Manin's weak Frobenius manifold. In addition, we find that Kodaira surface is its own mirror image. Our computation is done in the…

Global fixed points in low-dimensional surface group space correspond to trivial representations.

We show that the class of CAT(0) spaces is closed under suitable conformal changes. In particular, any CAT(0) space admits a large variety of non-trivial deformations.

We describe on any finitely generated group G the space of maps G->C which satisfy the parallelogram identity, f(xy)+f(xy^{-1})=2f(x)+2f(y). It is known (but not well-known) that these functions correspond to Zariski-tangent vectors at the trivial character of the character variety of G in SL_2(C). We study the obstruc…

Compactification of M- / string theory on manifolds with $G_2$ structure yields a wide variety of 4D and 3D physical theories. We analyze the local geometry of such compactifications as captured by a gauge theory obtained from a three-manifold of ADE singularities. Generic gauge theory solutions include a non-trivial g…

A long-standing open problem in systolic geometry asks whether a Riemannian metric on the real projective space whose volume equals that of the canonical metric, but is not isometric to it, must necessarily carry a periodic geodesic of length smaller than π. A contact-geometric reformulation of systolic geometry and th…

A C_k-move is a local move that involves (k+1) strands of a link. A C_k-move is called a C_k^d-move if these (k+1) strands belong to mutually distinct components of a link. Since a C_k^d-move preserves all k-component sublinks of a link, we consider the converse implication: are two links with common k-component sublin…

We consider some natural infinitesimal Einstein deformations on Sasakian and 3-Sasakian manifolds. Some of these are infinitesimal deformations of Killing spinors and further some integrate to actual Killing spinor deformations. In particular, on 3-Sasakian 7 manifolds these yield infinitesimal Einstein deformations pr…

The article provides conditions for unobstructedness of ASD manifolds.

In this paper, we demonstrate that the complete hyperbolic structure of various two-bridge knots and links cannot be deformed to an inequivalent strictly convex projective structure. We also prove a complementary result showing that under certain rigidity hypotheses, branched covers of amphicheiral knots admit non-triv…

New moves help untangle complex knots.

The paper analyzes systoles of complex projective spaces under various metrics.

We establish the correspondence between tame harmonic bundles and $μ_L$-stable parabolic Higgs bundles with trivial characteristic numbers. We also show the Bogomolov-Gieseker type inequality for $μ_L$-stable parabolic Higgs bundles. Then we show that any local system on a smooth quasi projective variety can be deforme…

The abstract proves a Moser-like theorem for C-symplectic structures and applies it to complex manifolds.

The paper studies deformations of Filippov algebroids using cohomology and DGLA.

This paper studies global webs on the projective plane with vanishing curvature. The study is based on an interplay of local and global arguments. The main local ingredient is a criterium for the regularity of the curvature at the neighborhood of a generic point of the discriminant. The main global ingredient, the Lege…

Constructs universal local deformations for curves and differential forms.

Quantum neural networks need both data-dependent and trainable unitaries for effective geometric deformation.

We consider formal deformations of the Poisson algebra of functions (with singularities) on $T^*M$ which are Laurent polynomials of fibers. Tn the case: $\dim M=1$ ($M=S^1, {\bf R}$), there exists a non-trivial $\star$-product on this algebra non-equivalent to the standard Moyal product.

We discuss a deformation of Sasakian structure in the presence of totally skew-symmetric torsion by introducing odd dimensional manifolds whose metric cones are Kähler with torsion. It is shown that such a geometry inherits similar properties to those of Sasakian geometry. As an example of them, we present an explicit …

Paper discusses conditions for deforming coupled Kähler-Einstein metrics.

Completes the classification of Moebius deformable hypersurfaces for dimensions 5 and above.

We propose a new approach to the Mirror Symmetry Conjecture in a form suitable to possibly non-Kähler compact complex manifolds whose canonical bundle is trivial. We apply our methods by proving that the Iwasawa manifold $X$, a well-known non-Kähler compact complex manifold of dimension $3$, is its own mirror dual to t…

The theory of surfaces in Euclidean space can be naturally formulated in the more general context of Legendre surfaces into the space of contact elements. We address the question of deformability of Legendre surfaces with respect to the symmetry group of Lie sphere contact transformations from the point of view of the …

In this paper we are interested in non trivial bi-Hamiltonian deformations of the Poisson pencil $ω_λ=ω_2+λω_1=uδ'(x-y)+\f{1}{2}u_xδ(x-y)+λδ'(x-y)$. Deformations are generated by a sequence of vector fields $\{X_2, X_4,...\}$, where each $X_{2k}$ is homogenous of degree $2k$ with respect to a grading induced by rescali…