arXiv research

Applying a local Gauss-Bonnet formula for closed subanalytic sets to the complex analytic case, we obtain characterizations of the Euler obstruction of a complex analytic germ in terms of the Lipschitz-Killing curvatures and the Chern forms of its regular part. We also prove analogous results for the global Euler obstr…

We study the problem of desingularizing coassociative conical singularities via gluing, allowing for topological and analytic obstructions, and discuss applications. This extends the author's earlier work on the unobstructed case. We interpret the analytic obstructions geometrically via the obstruction theory for defor…

The paper explores properties of CR hypersurfaces and their flatness.

We give series of explicit examples of Levi-nondegenerate real-analytic hypersurfaces in complex spaces that are not transversally holomorphically embeddable into hyperquadrics of any dimension. For this, we construct invariants attached to a given hypersurface that serve as obstructions to embeddability. We further st…

Research characterizes traces of Sobolev mappings into manifolds, identifying analytical obstructions and proving surjectivity under certain conditions.

Introduce generalized Ueda obstruction classes for line bundles and apply them to non-semi-positivity.

Analyzes complex structure deformations using cohomology contraction methods.

We carry out the programme of R. Liouville \cite{Liouville} to construct an explicit local obstruction to the existence of a Levi--Civita connection within a given projective structure $[Γ]$ on a surface. The obstruction is of order 5 in the components of a connection in a projective class. It can be expressed as a poi…

Study shows a universal local obstruction to the Samuelson condition for tangent Lagrangian 2-webs.

We calculate relations on characteristic classes which are obstructions preventing closed Kähler manifolds from carrying holomorphic Cartan geometries. We apply these relations to give global constraints on the phase spaces of complex analytic determined and underdetermined systems of differential equations.

Paper proves index theorem for self-adjoint elliptic boundary problems.

Lagrangian submanifolds in strict nearly Kähler 6-manifolds are related to special Lagrangian submanifolds in Calabi-Yau 6-manifolds and coassociative cones in $G_2$-manifolds. We prove that the mean curvature of a Lagrangian submanifold $L$ in a nearly Kähler manifold $(M^{2n}, J, g)$ is symplectically dual to the Mas…

We provide a new topological obstruction for complete stable minimal hypersurfaces in R^n. For $n\geq 4$, we prove that any complete orientable stable hypersurfaces in R^n has only one end. This follows from a more general analytic theorem we prove in the paper.

Paper proves all Lagrangians unobstructed if one is, using non-archimedean analytic structure.

Paper excludes the lowest energy level as an accumulation point for harmonic maps into analytic manifolds.

The entropy-degree theorem applies to Alexandrov spaces with curvature constraints.

Methods of parabolic geometries have been recently used to construct a class of elliptic complexes on quaternionic manifolds, the Salamon's complex being the simplest case. The purpose of this paper is to describe an algorithm how to compute their analytical indices in terms of characteristic classes. Using this, we ar…

On a Hermitian manifold we construct a symmetric $(1,1)$- tensor $H$ using the torsion and the curvature of the Chern connection. On a compact balanced Hermitian manifold we find necessary and sufficient conditions in terms of the tensor $H$ for a harmonic $1$-form to be analytic and for an analytic $1$-form to be harm…

We study the deformation of spherical conical metrics with at least some of the cone angles larger than $2π$. We show in this note via synthetic geometry that for one family of such metrics, there is local rigidity in the choice of cone positions if angles are fixed. This gives an evidence of the analytic obstruction c…

Study uncovers complex critical points in tensor decomposition.

The paper extends complex structure existence to manifolds of dimension 8.

The paper introduces new invariants to study topological properties of map germs.

Study on conformal harmonic coordinates on manifolds, proving existence and properties.

The projective metrizability problem can be formulated as follows: under what conditions the geodesics of a given spray coincide with the geodesics of some Finsler space, as oriented curves. In Theorem 3.8 we reformulate the projective metrizability problem for a spray in terms of a first-order partial differential ope…

New method normalizes Milnor fibrations for real analytic maps.

The paper studies Vafa-Witten equations on Kaehler manifolds and identifies obstructions to nontrivial solutions.

Proves uniqueness of certain spacetime solutions with extremal horizons.

This paper studies torsion obstructions to complex sections on manifolds.

The paper introduces Lagrangian vanishing cycles to prove obstructions for symplectic foliations.

New examples show positive scalar curvature metrics on manifolds with boundary that cannot be extended.

We study bounded pseudoconvex domains in complex Euclidean spaces. We find analytical necessary conditions and geometric sufficient conditions for a domain being of trivial Diederich--Fornæss index (i.e. the index equals to 1). We also connect a differential equation to the index. This reveals how a topological conditi…

The maximal analytic Schwarzschild spacetime is manifestly inextendible as a Lorentzian manifold with a twice continuously differentiable metric. In this paper, we prove the stronger statement that it is even inextendible as a Lorentzian manifold with a continuous metric. To capture the obstruction to continuous extens…

Fewer obstructions for small graphs in knotless embedding.

3028 obstructions found for embedding without knots.

The paper develops obstructions for embedding 2D complexes into 4D space.

Global obstructions found for conformally Einstein metrics in 6D.

Complete surgery obstructions for manifolds with finite fundamental group, disproving a conjecture.

Given a polarized manifold there are obstructions for asymptotic Chow semistability described as integral invariants. One of them is an obstruction to the existence for the first Chern class of the polarization to admit a constant scalar curvature Kähler (cscK) metric. A natural question is whether or not the other obs…

The article classifies cubiquitous sublattices and applies them to branched covers.

For a 3-manifold with torus boundary admitting an appropriate involution, we show that Khovanov homology provides obstructions to certain exceptional Dehn fillings. For example, given a strongly invertible knot in S^3, we give obstructions to lens space surgeries, as well as obstructions to surgeries with finite fundam…

New obstructions found for smooth desingularization of compact Einstein orbifolds.

We continue our study, initiated in our earlier paper, of Riemann surfaces with constant curvature and isolated conic singularities. Using the machinery developed in that earlier paper of extended configuration families of simple divisors, we study the existence and deformation theory for spherical conic metrics with s…

For $\ell >1$, we develop $L^{(2)}$-signature obstructions for $(4\ell-3)$-dimensional knots with metabelian knot groups to be doubly slice. For each $\ell>1$, we construct an infinite family of knots on which our obstructions are non-zero, but for which double sliceness is not obstructed by any previously known invari…

Study identifies obstructions for solving a 4th-order boundary problem.

Developing a singular dimension descent method for positive scalar curvature obstructions

Study Euler obstruction of 1-forms on determinantal singularities.

Proves Massey's theorems on complex structure obstructions.

Study on a specific obstruction in four-dimensional geometry.