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

Let $X$ be a smooth, compact, projective Kähler variety and $D$ be a divisor of a holomorphic form $F$, and assume that $D$ is smooth up to codimension two. Let $ω$ be a Kähler 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 $μ$. 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.