Motivic integrals and functional equations

Motivic Integrals and Functional Equations

The functional equation for the motivic integral of Milnor number of a plane curve is derived using the Denef-Loeser formula for the change of variables. Its solution is a function of five auxiliary parameters, it is unique up to the multiplication by the constant and there is a simple algorithm to find its coefficients. The method is universal enough and gives, for example, the equations for t...

Motivic Exponential Integrals

Orbital Integrals Are Motivic

This article shows that under general conditions, p-adic orbital integrals of definable functions are represented by virtual Chow motives. This gives an explicit example of the philosophy of Denef and Loeser, which predicts that all “naturally occurring” p-adic integrals are motivic.

Motivic Exponential Integrals and a Motivic Thom-sebastiani Theorem

1.1. Let f and f ′ be germs of analytic functions on smooth complex analytic varieties X and X ′ and consider the function f ⊕ f ′ on X × X ′ given by f ⊕ f (x, x) = f(x) + f (x). The Thom-Sebastiani Theorem classically states that the monodromy of f ⊕ f ′ on the nearby cycles is isomorphic to the product of the monodromy of f and the monodromy of f ′ (in the original form of the Theorem [18] t...

On Motivic Principal Value Integrals

Inspired by p-adic (and real) principal value integrals, we introduce motivic principal value integrals associated to multi-valued rational differential forms on smooth algebraic varieties. We investigate the natural question whether (for complete varieties) this notion is a birational invariant. The answer seems to be related to the dichotomy of the Minimal Model Program.