On Finite Parts of Divergent Complex Geometric Integrals

Licentiatavhandling, 2023

This licentiate thesis is about extracting finite values from divergent integrals. We study finite parts for a large class of divergent integrals over reduced complex analytic spaces. The integrands are given by generically smooth top forms which have certain, possibly non-integrable, singularities along a subvariety defined by a global holomorphic section of a holomorphic vector bundle. The finite part then depends on the choice of section, as well as on a choice of metric on the bundle. The main result of the thesis is a formula for the metric dependence.

The idea of extracting a finite number from a divergent integral turns out to be useful in both mathematics and theoretical physics contexts. In mathematics, Hadamard defined a finite part for non-absolutely convergent integrals in order to give formal solutions to certain differential equations. This idea was one of the precursors to the theory of distributions. In physics, divergent integrals appear in the context of perturbative quantum field theory when computing the probability amplitude for the transition from an initial quantum state to a final state, e.g., the probability of a given outcome of a particle scattering experiment. The amplitude is expressed as a power series in some (necessarily small) coupling constant and approximated by truncating the series. In most cases, the higher order terms are divergent integrals and need to be dealt with in order for the theory to work as intended.

In this thesis we follow two standard methods of regularization of a divergent integral. The first is to consider a modified integral depending on a complex parameter $\lambda$ such that the singularity of the integral is suppressed for $\mathfrak{Re}\lambda \gg 0$, and such that the original divergence is recovered as $\lambda \rightarrow 0$. This integral, as a function of $\lambda$, can then be meromorphically continued to all of $\C$, and we study its Laurent series expansion about $\lambda = 0$. The second method is to integrate only over the complement of some $\epsilon$-neighborhood of the singular set of the integrand, and then study the asymptotic behavior of the integral as $\epsilon \rightarrow 0$. For each method, we study a finite part, depending on a choice of metric, of the divergent integral. We show that these finite parts coincide and, as mentioned above, we provide a formula for their dependence of the choice of metric. Throughout our approach, we make use of the machinery of currents in several complex variables.