arXiv research

New class of singular complex manifolds studied with degenerate theory.

Classifies degenerations of complex projective plane with rational singularities.

Paper generalizes a theorem for real analytic singularities.

Rectifies singular set of harmonic maps into complex.

For isolated complex hypersurface singularities with real defining equation we show the existence of a monodromy vector field such that complex conjugation intertwines the local monodromy diffeomorphism with its inverse. In particular, it follows that the geometric monodromy is the composition of the involution induced…

Survey on the topology of singular foliations in complex 2-space.

We study fundamental groups of projective varieties with normal crossing singularities and of germs of complex singularities. We prove that for every finitely-presented group G there is a complex projective surface S with simple normal crossing singularities only, so that the fundamental group of S is isomorphic to G. …

Constructs Kahler-Einstein metrics near isolated log canonical singularities.

The paper studies complex affine structures near irregular singularities.

Gradient and Laplacian estimates for complex Monge-Ampère equations found.

Compact RCD spaces derived from singular Kahler metrics on 3D projective varieties.

Proves immediate transversality for conic singularities.

New Calabi-Yau metrics with conical singularities are created near complex lines.

Study differentiable maps on hypersurface links, finding fold maps with circle singular value sets.

Proves Kähler-Ricci shrinkers are complex analytic varieties.

The paper builds complex hyperbolic 2-manifolds with isolated singularities.

We construct and classify, in the case of two complex dimensions, the possible tangent cones at points of limit spaces of non-collapsed sequences of Kähler-Einstein metrics with cone singularities.

Detects singularities in complex data to improve machine learning models.

Study on singularities of Chern-Ricci flow on complex manifolds.

Algorithm classifies saddle-focus singularities in Hamiltonian systems.

We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irreducible singularities (rational cuspidal curves) in the complex proje…

Maps with many singularities found in complex space.

Study two types of singular Kähler-Einstein metrics on complex varieties.

Study families of Lie algebroids on complex spaces, introducing unfoldings.

Paper proves conditions for rational homology complex projective planes with singularities.

Explains complex analytic invariants of vector fields and foliations.

Study on Hölder continuity of complex Monge-Ampère solutions on Stein spaces.

Let $(X,ω)$ be a compact Kähler manifold of dimension $n$ and fix $1\leq m\leq n$. We prove that the total mass of the complex Hessian measure of $ω$-$m$-subharmonic functions is non-decreasing with respect to the singularity type. We then solve complex Hessian equations with prescribed singularity, and prove a Hodge i…

We discuss the bi-Lipschitz geometry of an isolated singular point of a complex surface which particular emphasis on when it is metrically conical.

The original de Rham cohomology due to Souriau and the singular cohomology in diffeology are not isomorphic to each other in general. This manuscript introduces a singular de Rham complex endowed with an integration map into the singular cochain complex which gives the de Rham theorem for every diffeological space. It …

The paper proves isomorphisms between two complexes related to singular foliations.

It has long been known that every quasi-homogeneous normal complex surface singularity with Q-homology sphere link has universal abelian cover a Brieskorn complete intersection singularity. We describe a broad generalization: First, one has a class of complete intersection normal complex surface singularities called "s…

We give a geometric construction of the BGG resolutions in singular infinitesimal character in the case of 1-graded complex Lie algebras of type A.

Study on rational projective planes with small index singularities.

We show that a weighted homogeneous complex surface singularity is metrically conical (i.e., bi-Lipschitz equivalent to a metric cone) only if its two lowest weights are equal. We also give an example of a pair of weighted homogeneous complex surface singularities that are topologically equivalent but not bi-Lipschitz …

We investigate the face numbers of simplicial complexes with Buchsbaum vertex links, especially pseudomanifolds with isolated singularities. This includes deriving Dehn-Sommerville relations for pseudomanifolds with isolated singularities and establishing lower bound theorems when the singularities are also homological…

Localizes smooth spaces to study their homotopy properties.

Unified framework for complex, split-complex, and dual numbers.

The complex Lie superalgebras $\mathfrak{g}$ of type $D(2,1;a)$ - also denoted by $\mathfrak{osp}(4,2;a)$ - are usually considered for "non-singular" values of the parameter $a$, for which they are simple. In this paper we introduce five suitable integral forms of $\mathfrak{g}$, that are well-defined at singular valu…

We prove an existence theorem for Asymptotically Conical Ricci Flat Kahler metrics in $\mathbb{C}^2$ with cone singularities along a smooth complex curve. These metrics are expected to arise as blow up limits of non collapsed sequences of Kahler Einstein metrics with cone singularities.

In this article we introduce a generalization of locally conformally Kaehler metrics from complex manifolds to complex analytic spaces with singularities and study which properties of locally conformally Kaehler manifolds still hold in this new setting. We prove that if a complex analytic space has only quotient singul…

Continuity of Kähler-Einstein potentials at singularities proven.

The study characterizes 3-pseudomanifolds with up to two singularities.

Solves complex Monge-Ampère equations on Kähler manifolds.

We define a torus action on the (complex) Cayley Grassmannian $X$. Using this action, we prove that $X$ is a singular variety. We also show that the singular locus is smooth and has the same cohomology ring as that of $\mathbb{CP}^5$. Furthermore, we identify the singular locus with a quotient of $G_2^\mathbb{C}$ by a …

In this paper, we prove the asymptotic expansion of the solutions to some singular complex Monge-Ampère equation which arise naturally in the study of the conical Kähler-Einstein metric.

Bayesian models' singular fluctuation is shown to be akin to specific heat, influencing model complexity and generalization.

Extended Vaisman theorem to compact spaces with singularities.