Regularization of divergent integrals in complex geometry

Doktorsavhandling, 2026

A unifying theme of this thesis is the study of Archimedean zeta functions defined by complex-geometric data. Classically, Archimedean zeta functions over C are parameter-dependent integrals over a domain in Cⁿ, where the parameter is a complex power of the modulus of a holomorphic function. We consider a global complex-geometric generalization in which the integration is over a complex manifold or reduced analytic space, and where the modulus of a holomorphic function is replaced by the norm of a holomorphic section of a vector bundle.

The first two papers concern finite parts of divergent integrals on reduced complex analytic spaces. Given a singular differential form whose singularities are determined a holomorphic section of a vector bundle, a Hermitian metric induces a natural regularization of the divergent integral, giving rise to an Archimedean zeta function. A finite part of the divergent integral is defined as the constant term in the Laurent expansion of this zeta function about 0. Paper I establishes an explicit formula describing the dependence of the resulting finite part on the choice of Hermitian metric. Paper II develops a current calculus adapted to this setting and derives decomposition formulas that permit explicit computations of certain finite parts. We illustrate these formulas with a family of examples on projective space, where the resulting finite parts turn out to be multiple zeta values.

The second pair of papers studies Archimedean zeta functions arising as partition functions of Gibbs ensembles on compact Kähler manifolds. In Paper III, we consider systems of particles on the two-dimensional sphere, interacting through logarithmic pair potentials. Depending on the numerical values of the coupling constants, the resulting partition functions are either examples of Archimedean zeta functions or slight generalizations thereof. Using techniques from complex algebraic geometry, in particular the Fulton—MacPherson compactification of configuration space, we establish the meromorphic continuation of these partition functions and relate the location of their critical inverse temperatures to a discrete optimization problem governing both integrability and particle clustering.

Paper IV concerns Berman's probabilistic approach to Kähler—Einstein metrics on log Fano manifolds X. In this framework, the Kähler--Einstein geometry of X is encoded by a canonical random point process admitting a statistical-mechanical interpretation in terms of a family of Gibbs measures on the products X^N, whose associated partition functions define Archimedean zeta functions. The main contribution of the paper is an extension of this framework to log Fano manifolds with non-discrete automorphism groups. To this end, we propose a symmetry-breaking procedure based on a moment-map constraint for the Gibbs measures, and introduce an algebraic notion of Gibbs polystability, conjecturally equivalent to the existence of a Kähler—Einstein metric on X. Moreover, we conjecture that if X is Gibbs polystable, then the unique Kähler—Einstein metric with vanishing moment emerges when sampling N points on X subject to the moment-map constraint as N tends to infinity.
Inter alia, we verify several of our conjectures for log Fano curves.