Spectral construction of non-holomorphic Eisenstein-type series and their Kronecker limit formulas

TL;DR

This paper develops a spectral method to construct non-holomorphic Eisenstein-type series on Kähler varieties, generalizing classical elliptic Eisenstein series and deriving their Kronecker limit formulas.

Contribution

It introduces a spectral construction of Eisenstein-type series on Kähler varieties, extending the classical theory to higher dimensions and more general geometric settings.

Findings

Constructed a meromorphic function with a special value related to the log-norm of a holomorphic form.

Generalized elliptic Eisenstein series to higher-dimensional Kähler quotients.

Derived Kronecker limit formulas for these new series.

Abstract

Let $X$ be a smooth, compact, projective K\"ahler variety and $D$ be a divisor of a holomorphic form $F$, and assume that $D$ is smooth up to codimension two. Let $\omega$ be a K\"ahler form on $X$ and $K_{X}$ the corresponding heat kernel which is associated to the Laplacian that acts on the space of smooth functions on $X$. Using various integral transforms of $K_{X}$, we will construct a meromorphic function in a complex variable $s$ whose special value at $s=0$ is the log-norm of $F$ with respect to $\mu$. In the case when $X$ is the quotient of a symmetric space, then the function we construct is a generalization of the so-called elliptic Eisenstein series which has been defined and studied for finite volume Riemann surfaces.