arXiv research

New class of singular complex manifolds studied with degenerate theory.

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

Detects singularities in complex data to improve machine learning models.

Solves complex Monge-Ampère equations on Kähler manifolds.

CR singularities in 3-manifolds can be cancelled by an isotopy supported in an arbitrarily small neighborhood of a Seifert surface.

In this paper, we study the convergence of Calabi-Yau manifolds under Kähler degeneration to orbifold singularities and complex degeneration to canonical singularities (including the conifold singularities), and the collapsing of a family of Calabi-Yau manifolds.

Rectifies singular set of harmonic maps into complex.

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…

The paper studies invariant complex manifolds in holomorphic slow-fast systems.

New structures allow for self-crossing singularities, leading to new families of stable generalized complex manifolds.

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

Extended Vaisman theorem to compact spaces with singularities.

In this paper, we develop results in the direction of an analogue of Sjamaar and Lerman's singular reduction of Hamiltonian symplectic manifolds in the context of reduction of Hamiltonian generalized complex manifolds (in the sense of Lin and Tolman). Specifically, we prove that if a compact Lie group acts on a general…

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…

In this article we introduce the notion of Polyhedral Kahler manifolds, even dimensional polyhedral manifolds with unitary holonomy. We concentrate on the 4-dimensional case, prove that such manifolds are smooth complex surfaces, and classify the singularities of the metric. The singularities form a divisor and the res…

Newlander-Nirenberg theorem extended to complex b-manifolds.

We study variuos homological structures associated with Poisson algebra, the canonical differential complex for singular Poisson structure and the analogue of the star operator for such manifolds. Give the interpretation of the classical Koszul differential of exterior forms, as the supercommutator with some second ord…

Study on hyperelliptic Lefschetz fibrations, finding singular fiber counts.

Conic singular sub-manifolds are Lipschitz Normally Embedded in compact non-Euclidean manifolds.

Explains complex analytic invariants of vector fields and foliations.

Develops Bialynicki-Birula and Morse-Bott theory for complex analytic spaces.

The paper connects fibrations to generalized complex structures in semi-toric geometry.

Compactness results for Hermitian manifolds help understand Type IIB flow.

Using an approach based on the heat kernel we prove an Atiyah-Bott-Lefschetz theorem for the $L^2-$Lefschetz numbers associated to an elliptic complex of cone differential operators over a compact manifold with conical singularities. We then apply our results to the case of the de Rham complex.

Formal Normal Form created for special CR singularities.

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…

In this paper, we first classify singular fibers of proper $C^\infty$ stable maps of 3-dimensional manifolds with boundary into surfaces. Then, we compute the cohomology groups of the associated universal complex of singular fibers, and obtain certain cobordism invariants for Morse functions on compact surfaces with bo…

Quantizes symplectic manifolds with toric singularities using Toeplitz operators.

Diffeology extends differential geometry to complex spaces.

We study the singularities of Legendrian subvarieties of contact manifolds in the complex-analytic category and prove two rigidity results. The first one is that Legendrian singularities with reduced tangent cones are contactomorphically biholomorphic to their tangent cones. This result is partly motivated by a problem…

The paper proves infinitely many strong symplectic fillings for cusp singularity links.

Extends geometric quantization to singular spaces.

Study controls bifurcations in Eulerian flows with multiple Hopf singularities.

Simplicial sets deformation retract onto transverse simplices.

Given a real analytic function $f$ from $\mathbb{R}^4$ to $\mathbb{R}^2$ with isolated critical point at the origin, the link $L_f$ of the singularity is a real fibred knot in $\mathbb{S}^{3}$. From this singularities, we construct a family of real isolated suspension singularities from $\mathbb{R}^6$ to $\mathbb{R}^2$…

Study classifies bubbles of Type I singularities in Kähler-Ricci flow on compact surfaces.

Constructs Morse homology for complex algebraic varieties.

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

Estimates for complex Monge-Ampère equations lead to insights on moduli spaces and singular metrics.

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

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 simplest case of a manifold with singularities is a manifold M with boundary, together with an identification of the boundary with a product M1 x P, where P is a fixed manifold. The associated singular space is obtained by collapsing P to a point. When P = Z/k or S^1, we show how to attach to such a space a noncomm…

We prove the existence and uniqueness of the solutions of some very general type of degenerate complex Monge-Ampère equations. This type of equations is precisely what is needed in order to construct Kähler-Einstein metrics over irreducible singular Kähler spaces with ample or trivial canonical sheaf and singular Kähle…

We establish the monotonicity property for the mass of non-pluripolar products on compact Kahler manifolds, and we initiate the study of complex Monge-Ampere type equations with prescribed singularity type. Using the variational method of Berman-Boucksom-Guedj-Zeriahi we prove existence and uniqueness of solutions with…

New Calabi-Yau metrics found on complex symmetric spaces.

Holomorphic vector fields and anti-canonical divisors on complex manifolds are studied.

In this article we study the deformation theory of conically singular Cayley submanifolds. In particular, we prove a result on the expected dimension of a moduli space of Cayley deformations of a conically singular Cayley submanifold. Moreover, when the Cayley submanifold is a two-dimensional complex submanifold of a C…

In this paper, we prove that a sequence of weak almost Kähler-Ricci solitons under further suitable conditions converge to a Kähler-Ricci soliton with complex codimension of singularities at least 2 in the Gromov-Hausdorff topology. As a corollary, we show that on a Fano manifold with the modified K-energy bounded belo…