A tropical motivic Fubini theorem with applications to Donaldson–Thomas theory

Abstract

We present a new tool for the calculation of Denef and Loeser's motivic nearby fiber and motivic Milnor fiber: a motivic Fubini theorem for the tropicalization map, based on Hrushovski and Kazhdan's theory of motivic volumes of semi-algebraic sets. As applications, we prove a conjecture of Davison and Meinhardt on motivic nearby fibers of weighted homogeneous polynomials, and give a very short and conceptual new proof of the integral identity conjecture of Kontsevich and Soibelman, first proved by Lê Quy Thuong. Both of these conjectures emerged in the context of motivic Donaldson-Thomas theory.

Bounded integral and motivic Milnor fiber

“…One recovers in particular the fact that Vol(F an f,0 ) is the motivic Milnor fiber, without using resolution of singularities. This fact was already proven by Nicaise, Payne and Schroeter in [7] (without taking the μ-action into account), Nicaise and Payne in [6] and the first author in [3]. Those proofs rely on explicit computations using resolution of singularities.…”

“…The morphism Z, whose construction is inspired by [5], associate to a definable set a power-series in T which turns out to be rational, lim associate to this rational function its limit when T goes to +∞, and Θ • E b is built from Euler characteristic applied to the Γ-part of a definable set in RV. The composition Θ • E b • , usually denoted Vol and already constructed by Hrushovski and Kazhdan, has been used in many applications, see [10], [7], [6], [3], [9], [5]. The diagram 1.2 is particularly interesting when applied to the following object.…”

Virtual rigid motives of semi-algebraic sets

“…, we can see it either by a direct computation using resolution of singularities as in [28] and [27] or by adapting results by Hrushovski and Loeser [20]. Now Theorem 1.2 shows that…”

Deformed dimensional reduction

“…Note that, in contrast to (3), the monodromy here may be nontrivial; our main application, detailed in the following section, is such a case. Theorem 2.3 turns out to be a consequence of a theorem of Nicaise and Payne [27], stating that for certain G m -equivariant functions g, the "naive" motivic nearby fiber [g −1 (1)] and the motivic nearby fiber defined by Denef and Loeser agree.…”