Spectral Construction of Non-Holomorphic Eisenstein-Type Series and their Kronecker Limit Formula
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 KX 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 KX , 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.