Motivic local density of isolated surface singularities

The goal of this paper is to compute the motivic local density of an isolated algebraic surface singularity, in order to explain its link with algebraic multiplicity. In this context, we can use an additional data: the inner rate related to the bilipschitz geoemtry of the singularity, as studied by A. Belotto da Silva, L. Fantini and A. Pichon.

💡 Research Summary

This paper, titled “Motivic local density of isolated surface singularities,” presents a detailed study aimed at computing the motivic local density for isolated singularities of algebraic surfaces and elucidating its relationship with algebraic multiplicity. The work innovatively incorporates data from the bilipschitz geometry of the singularity—specifically, the inner rates—to achieve an explicit formula.

The introduction contextualizes the concept of local density, tracing its history from Lelong’s work in complex analysis, where it equals a positive integer identified by Draper as the algebraic multiplicity of the local ring. The study extends to real subanalytic and non-Archimedean settings (p-adic and Laurent series fields). In these non-Archimedean contexts, where classical measures fail and limits may not converge, Cluckers, Comte, and Loeser defined the motivic local density using motivic integration. It is obtained as the mean value of the limits of convergent subsequences of normalized local volumes, which exhibit periodic behavior. Prior work by Forey showed that for curve singularities, this motivic density is the sum of the reciprocals of the multiplicities of its irreducible components, contrasting with the classical complex density (the sum of multiplicities).

The core of the paper is structured into several technical sections. It begins with a recap of Cluckers-Loeser motivic integration, which provides a framework to define volumes for definable sets over fields like C((t)) as elements of the Grothendieck ring of varieties M_C. The crucial change-of-variables formula for birational morphisms is highlighted, wherein the Jacobian factor involves Mather discrepancies.

The paper then introduces the inner rates of a complex surface singularity, based on work by Belotto da Silva, Fantini, and Pichon. An inner rate q is a rational number measuring the contact order between two families of curve germs on the singularity with respect to the inner distance. These rates are bilipschitz invariants associated with irreducible components E_i of the exceptional divisor in a good resolution.

A key technical achievement is Proposition 1.3, which establishes a profound link between geometry and algebra: for a good resolution factoring through the Nash transform, the inner rate q_v associated with a divisor E_v equals the normalized Mather logarithmic discrepancy minus one (q_v = ˆk_log_v / m_v - 1). This connection allows the translation of a metric invariant into an algebro-geometric one, which is essential for computations.

The authors then define the motivic local density formally and discuss its properties, notably its independence of embedding and its nature as a 1-bilipschitz invariant for the non-Archimedean norm.

The main result, Theorem 1.2, provides the explicit formula for the motivic local density Θ_mot_2(X, O) of an isolated surface singularity. Let π: Y → X be a good resolution factoring through the blowup of the maximal ideal and the Nash transform, with simple normal crossing exceptional divisor. Let E_i be its irreducible components, m_i their multiplicities, and q_i their associated inner rates. Let E^0_i = E_i \ ∪{j≠i} E_j. Then the density is given by a sum over the components: Θ_mot_2(X, O) = Σ{i | q_i=1} (1/m_i) * (1/(L+1)