arXiv research

Proves non-positivity of Hirzebruch form on stable weights and connects to flat logarithmic connections.

Study logarithmic flat connections on principal bundles using Lie groupoids.

Study flat connections with logarithmic singularities on complex plane curves.

Abstract: New geometric incarnation of isomonodromy functors.

The paper establishes a Poisson Poincaré-Dulac theorem for Poisson-flat connections.

Normal forms and moduli stacks for flat connections on complex manifolds.

Geodesics spiral around compact subsets in CAT(0) spaces.

We show that Masur's logarithmic law of geodesics in the moduli space of translation surfaces does not imply unique ergodicity of the translation flow, but that a similar law involving the flat systole of a Teichmüller geodesic does imply unique ergodicity. It shows that the flat geometry has a better control on ergodi…

Paper studies statistical manifolds with logarithmic divergences.

Study Bergman kernels on Kähler manifolds, answering Lu-Tian's question.

In this paper, we investigate representations of $\operatorname{At}(N)$, the Atiyah algebroids of a holomorphic line bundles $N$ over a complex manifold $Y$. In particular, we relate $\operatorname{At}(N)$-modules with logarithmic connections through two functors. On the one hand, we use these functors to the define in…

For a closed surface M with metric g, the Robin mass m(p) at the point p is the value of the Green function G(p,q) at p=q after the logarithmic singularity has been removed. The Laplacian-mass is the average value of the Robin mass, minus the value of the Robin mass for the round sphere of the same area. The Laplacian-…

Given an irreducible well-generated complex reflection group, we construct an explicit basis for the module of vector fields with logarithmic poles along its reflection arrangement. This construction yields in particular a Hodge filtration of that module. Our approach is based on a detailed analysis of a flat connectio…

Given a smooth manifold $M$ equipped with a properly and discontinuous smooth action of a discrete group $G$, the nerve $M_{\bullet}G$ is a simplicial manifold and its vector space of differential forms $\operatorname{Tot}_{N}\left(A_{DR}(M_{\bullet}G)\right)$ carry a $C_{\infty}$-algebra structure $m_{\bullet}$. We sh…

In this paper, we extend Deligne's functorial Riemann-Roch isomorphism for hermitian holomorphic line bundles on Riemann surfaces to the case of flat, not necessarily unitary connections. The Quillen metric and star-product of Gillet-Soule are replaced with complex valued logarithms. On the determinant of cohomology si…

We study the logarithmic $L^{(α)}$-divergence which extrapolates the Bregman divergence and corresponds to solutions to novel optimal transport problems. We show that this logarithmic divergence is equivalent to a conformal transformation of the Bregman divergence, and, via an explicit affine immersion, is equivalent t…

Log-conformal projective pairs restrict to simple geometric structures.

Green functions play an important role in conformal geometry. In this paper, we explain how to compute explicitly the logarithmic singularities of the Green functions of the conformal powers of the Laplacian. These operators include the Yamabe and Paneitz operators, as well as the conformal fractional powers of the Lap…

The conformal Codazzi structure is an intrinsic geometric structure on strictly convex hypersufaces in a locally flat projective manifold. We construct the GJMS operators and the Q-curvature for conformal Codazzi structures by using the ambient metric. We relate the total Q-curvature to the logarithmic coefficient in t…

We show that bi-flat $F$-manifolds can be interpreted as natural geometrical structures encoding the almost duality for Frobenius manifolds without metric. Using this framework, we extend Dubrovin's duality between orbit spaces of Coxeter groups and Veselov's $\vee$-systems, to the orbit spaces of exceptional well-gene…

We show that the residue density of the logarithm of a generalised Laplacian on a closed manifold defines an invariant polynomial valued differential form. We express it in terms of a finite sum of residues of classical pseudodifferential symbols. In the case of the square of a Dirac operator, these formulae provide a …

Given a weighted line arrangement in the projective plane, with weights satisfying natural constraint conditions, we show the existence of a Ricci-flat Kähler metric with cone singularities along the lines asymptotic to a polyhedral Kähler cone at each multiple point. Moreover, we discuss a Chern-Weil formula that expr…

We consider the volume expansion of the Blaschke metric, which is a projectively invariant metric on a strictly convex domain in a locally flat projective manifold. When the boundary is even dimensional, we express the logarithmic coefficient L as the integral of affine invariants over the boundary. We also formulate a…

Study local models for special Kähler metrics near discriminant locus components.

We introduce a new version of a curvature-dimension inequality for non-negative curvature. We use this inequality to prove a logarithmic Li-Yau inequality on finite graphs. To formulate this inequality, we introduce a non-linear variant of the calculus of Bakry and Émery. In the case of manifolds, the new calculus and …

Introduces new info-geometric structure for dynamics on graphs and hypergraphs.

CR 3-sphere rigidity proven through curvature invariant.

In this article we discuss the distribution of asset price movements by the market potential function. From the principle of free energy minimization we analyze two different kinds of market potentials. We obtain a U-shaped potential when market reversion (i.e. contrarian investors) is dominant. On the other hand, if t…

New metrics defined for full-rank correlation matrices, ensuring unique operations.

Reconstructs fundamental groups from liquid local systems.

Two types of differentials are shown equivalent for compactifying moduli spaces.

In this paper, we introduce the notions of logarithmic Poisson structure and logarithmic principal Poisson structure; we prove that the latter induces a representation by logarithmic derivation of the module of logarithmic Kahler differentials; therefore, it induces a differential complex from which we derive the notio…

The paper studies hybrid connections on Hessian manifolds and their properties.

Estimates for geodesics on hyperbolic tori improve previous bounds.

Study rigidity by logarithmic capacity and related functions.

Study real logarithms of semi-simple matrices, focusing on differential structure.

Study excess logarithmic residues for foliations to bound invariant hypersurfaces and test log canonicity.

Logarithmic connections on principal bundles over normal varieties are studied.

We present a new method to solve certain $\bar{\partial}$-equations for logarithmic differential forms by using harmonic integral theory for currents on Kahler manifolds. The result can be considered as a $\bar{\partial}$-lemma for logarithmic forms. As applications, we generalize the result of Deligne about closedness…

Local logarithmic Brunn-Minkowski holds for zonoids.

New framework for logarithmically divergent integrals on manifolds with corners.

Directly proves logarithmic systolic growth for all hyperbolic surfaces.

Motivated by the Strong Cosmic Censorship Conjecture for asymptotically AdS spacetimes, we initiate the study of massive scalar waves satisfying $\Box_g ψ- μψ=0$ on the interior of Anti-de Sitter (AdS) black holes. We prescribe initial data on a spacelike hypersurface of a Reissner--Nordström--AdS black hole and impose…

Logarithmic separation profile in hyperbolic groups shows hierarchical structure.

Paper uses ABP method to prove logarithmic Sobolev inequalities on curved spaces.

Investment and consumption strategy optimized under uncertain conditions.

The paper constructs a Saito basis for a specific class of divisors and applies it to logarithmic Poisson geometry.

Extended logarithm for solvable elements in mapping class groups.