Debasish Sadhukhan | Searcharxiv

arXiv2024

Estimates and asymptotics of Teichmüller modular forms

In this article, we derive estimates of Teichmüller modular forms, and associated invariants. Let $\mathcal{M}_{g}$ denote the moduli space of compact hyperbolic Riemann surfaces of genus $g\geq 2$, and let $\overline{M}_{g}$ be the Deligne-Mumford compactification of $\mathcal{M}_{g}$, and we denote its boundary by $\partial\mathcal{M}_{g}$. Let $π:\mathcal{C}_{g}\longrightarrow\mathcal{M}_{g}$ be the universal surface. For any $n\geq 1$, let $Λ_{n}:=π_{\ast}(T_{v}\mathcal{C}_{g})^{n}$, where $T_{v}\mathcal{C}_{g}$ denotes the vertical holomorphic tangent bundle of the fibration $π$, and the fiber of $Λ_{n}$ over any $X\in\mathcal{M}_{g}$ is equal to $H^{0}(X,Ω_{X}^{\otimes n})$, the space of holomorphic differentials of degree-$n$, defined over the Riemann surface $X$. Let $λ_{n}:=\mathrm{det}(Λ_{n})$ denote the determinant line bundle of the vector bundle $Λ_{n}$, whose sections are known as Teichmüller modular forms. The complex vector space of Teichmüller modular forms is equipped with Quillen metric, which is denoted by $\|\cdot\|_{\mathrm{Qu}}$.

math.CV↗

arXiv2023

Estimates of Bergman Kernels and Bergman metric on compact Picard surfaces

Let $Γ\subset \mathrm{SU}((2,1),\mathbb{C})$ be a torsion-free cocompact subgroup. Let $\mathbb{B}^{2}$ denote the $2$-dimensional complex ball endowed with the hyperbolic metric $μ_{\mathrm{hyp}}$, and let $X_Γ:=Γ\backslash \mathbb{B}^{2}$ denote the quotient space, which is a compact complex manifold of dimension $2$. Let $Λ:= Ω_{X_Γ}^{2}$ denote the line bundle on $X_Γ$, whose sections are holomorphic $(2,0)$-forms. For any $k\geq 1$, the hyperbolic metric induces a point-wise metric on $H^{0}(X_Γ,Λ^{\otimes k })$, which we denote by $|\cdot|_{\mathrm{hyp}}$. For any $k\geq 1$, let $\mathcal{B}_Λ^{ k}$ denote the Bergman kernel of the complex vector space $H^{0}(X_Γ,Λ^{\otimes k })$. For any $k\geq 3$, and $z,w\in X_Γ$, the first main result of the article is an off-diagonal estimate of the Bergman kernel $ \mathcal{B}_Λ^{ k}$. For any $k\geq 1$, let $μ_{\mathrm{ber}}^{k}(z):=-\frac{i}{2π}\partial_{z}\partial_{\overline{z}}\log| \mathcal{B}_Λ^{ k}(z,z)|_{\mathrm{hyp}}$ denote the Bergman metric associated the line bundle $Λ^{\otimes k}$, and let $μ_{\mathrm{ber}}^{k,\mathrm{vol}}(z)$ denote the associated volume form. For $k\gg 1$ sufficiently large, and $ε>0$, the second main result of the article is the following estimate \begin{align*} \sup_{z\in X_Γ}\bigg|\frac{μ_{\mathrm{ber}}^{k,\mathrm{vol}}(z)}{μ_{\mathrm{hyp}}^{\mathrm{vol}}}\bigg|=O_{X_Γ,ε}\big(k^{4+ε}\big), \end{align*} where $μ_{\mathrm{hyp}}^{\mathrm{vol}}$ denotes the volume form associated to the hyperbolic metric $μ_{\mathrm{hyp}}$, and the implied constant depends on the Picard surface $X_Γ$, and on the choice of $ε>0$. \end{abstract}

math.CV↗