arXiv research

Two Calabi-Yau theorems for Kähler manifold degenerations.

Paper studies degenerated circle packings in hyperbolic geometry and finds conditions for their existence.

We show that in any spacetime dimension $D\ge 4$, degenerate components of the event horizon do not exist in static vacuum configurations with positive cosmological constant. We also show that without a cosmological constant asymptotically flat solutions cannot possess a degenerate horizon component. Several independen…

The paper classifies equivariant test configurations for spherical varieties.

In this short note, we give a new proof of a theorem of Arezzo-Tian on the existence of smooth geodesic rays tamed by a special degeneration.

Study continuity of Bergman kernels on degenerating varieties.

The Dirichlet problem for a Monge-Ampere equation corresponding to a nonnegative, possible degenerate cohomology class on a Kaehler manifold with boundary is studied. C^{1,α} estimates away from a divisor are obtained, by combining techniques of Blocki, Tsuji, Yau, and pluripotential theory. In particular, C^{1,α} geod…

The paper calculates critical configurations and Morse indices for polygons on circles or ellipses.

We review recent work on the compactification of the moduli space of Hitchin's self-duality equation. We study the degeneration behavior near the ends of this moduli space in a set of generic directions by showing how limiting configurations can be desingularized. Following ideas of Hitchin, we can relate the top bound…

Unique K-polystable degenerations for Fano varieties confirmed.

We investigate the local contribution of the braid monodromy factorization in the context of the links obtained by the closure of these braids. We consider plane curves which are arrangements of lines and conics as well as some algebraic surfaces, where some of the former occur as local configurations in degenerated an…

We study partial collapsing degeneration of Hamiltonian-perturbed Floer trajectories for an adiabatic $\varepsilon$-family and its reversal adiabatic gluing, as the prototype of the partial collapsing degeneration of $2$-dimensional (perturbed) $J$-holomorphic maps to $1$-dimensional gradient segments. We consider the …

We prove a $C^{1,1}$ estimate for solutions of complex Monge-Ampère equations on compact Kähler manifolds with possibly nonempty boundary, in a degenerate cohomology class. This strengthens previous estimates of Phong-Sturm. As applications we deduce the local $C^{1,1}$ regularity of geodesic rays in the space of Kähle…

The seminal work of Eskin-Masur-Zorich described the principal boundary of moduli spaces of abelian differentials that parameterizes flat surfaces with a prescribed generic configuration of short parallel saddle connections. In this paper we describe the principal boundary for each configuration in terms of twisted dif…

Rigidity of critical eigensections on spheres proven.

Study on eigenfunctions on sphere configurations, proving non-existence and construction of critical eigensections.

This article contains a detailed study, in the toric case, of the test configuration geodesic rays defined by Phong-Sturm. We show that the `Bergman approximations' of Phong-Sturm converge in C^1 to the geodesic ray and that the geodesic ray itself is C^{1,1} and no better. The \kahler metrics associated to the geodesi…

We describe typical degenerations of quadratic differentials thus describing ``generic cusps'' of the moduli space of meromorphic quadratic differentials with at most simple poles. The part of the boundary of the moduli space which does not arise from ``generic'' degenerations is often negligible in problems involving …

We show that for a K-unstable Fano variety, any divisorial valuation computing its stability threshold induces a non-trivial special test configuration preserving the stability threshold. When such a divisorial valuation exists, we show that the Fano variety degenerates to a uniquely determined twisted K-polystable Fan…

Proves properness of K-moduli spaces for Fano varieties.

Unified study of homological representations of mapping class groups.

Paper proves linearity of solutions to degenerate elliptic equations in 3D.

We describe the connected components of the complement of a natural "diagonal" of real codimension 1 in a stratum of quadratic differentials on CP1. We establish a natural bijection between the set of these connected components and the set of generic configurations that appear on such "flat spheres". We also prove that…

The abstract presents a new theorem using Ross-Witt Nyström correspondence and Berndtsson's theorem.

We partially confirm an old conjecture of Donaldson that if there exists a cscK metrics in a given Kähler class, then there is no degenerated geodesic ray which is tamed by a bounded ambient geometry unless it parallels to a holomorphic line consists of cscK metrics only. We also prove that for simple test configuratio…

The paper studies parallel surfaces of cuspidal cross caps and their degeneracy.

Similar simplices can be inscribed in most smoothly embedded spheres.

Graph neural networks fail to distinguish certain 3D atom configurations.

Study finds minimal length networks connecting three points in Heisenberg group.

Let $Δ\subset \mathbb{R}^n$ be an $n$-dimensional integral Delzant polytope. It is well-known that there exist the $n$-dimensional compact toric manifold $X_Δ$ and the very ample $(\mathbb{C}^\times)^n$-equivariant line bundle $L_Δ$ on $X_Δ$ associated with $Δ$. In the present paper, we give a necessary and sufficient …

Abstract framework for two meromorphic forms on punctured surfaces.

Unified geometric perspectives on PDEs, torsion invariants, and moduli theory.

Study chord diagrams and knot theory, proving inevitable complexity in cohomology sequences.

We propose a deterministic numerical method for pricing vanilla options under the SABR stochastic volatility model, based on a finite element discretization of the Kolmogorov pricing equations via non-symmetric Dirichlet forms. Our pricing method is valid under mild assumptions on parameter configurations of the proces…

If $Γ<\mathrm{PSL}(2,\mathbb{C})$ is a lattice, we define an invariant of a representation $Γ\rightarrow \mathrm{PSL}(n,\mathbb{C})$ using the Borel class $β(n)\in \mathrm{H}^3_\mathrm{c}(\mathrm{PSL}(n,\mathbb{C}),\mathbb{R})$. We show that the invariant is bounded and its maximal value is attained by conjugation of t…

In this paper we investigate the asymptotics of forward-start options and the forward implied volatility smile in the Heston model as the maturity approaches zero. We prove that the forward smile for out-of-the-money options explodes and compute a closed-form high-order expansion detailing the rate of the explosion. Fu…

Paper connects geometric and analytic aspects of Higgs bundles and pleated surfaces.

A central question in the study of line arrangements in the complex projective plane $\mathbb{CP}^2$ is: when does the combinatorial data of the arrangement determine its topological properties? In the present work, we introduce a topological invariant of complexified real line arrangements, the chamber weight. This in…

We prove two new results on the K-polystability of Q-Fano varieties based on purely algebro-geometric arguments. The first one says that any K-semistable log Fano cone has a special degeneration to a uniquely determined K-polystable log Fano cone. As a corollary, we combine it with the differential-geometric results to…

Proves stable degeneration preserves symplectic forms and confirms Kaledin's conjecture.

We prove that the degenerate part of the distributive homology of a multispindle is determined by the normalized homology. In particular, when the multispindle is a quandle $Q$, the degenerate homology of $Q$ is completely determined by the quandle homology of $Q$. For this case (and generally for two term homology of …

We study a constrained optimal control problem with possibly degenerate coefficients arising in models of optimal portfolio liquidation under market impact. The coefficients can be random in which case the value function is described by a degenerate backward stochastic partial differential equation (BSPDE) with singula…

Computes fundamental groups of restricted configuration spaces.

An n-dimensional submanifold X of a projective space P^N (C) is called tangentially degenerate if the rank of its Gauss mapping γ: X ---> G (n, N) satisfies 0 < rank γ< n. The authors systematically study the geometry of tangentially degenerate submanifolds of a projective space $P^N (\mathbf{C})$. By means of the foca…

Dirac structures on tangent bundles provide a unified framework for Lagrange--Dirac dynamical systems.

Study degenerate Bianchi transformations for pseudo-spherical submanifolds in 5D space.

New stabilization found in planar elasticae with degenerate diffusion.

Uniform estimates for Calabi-Yau degenerations proved.