Pure motives, mixed motives and extensions of motives associated to singular surfaces

We first recall the construction of the Chow motive modelling intersection cohomology of a proper surface and study its fundamental properties. Using Voevodsky’s category of effective geometrical motives, we then study the motive of the exceptional divisor in a non-singular blow-up. If all geometric irreducible components of the divisor are of genus zero, then Voevodsky’s formalism allows us to construct certain one-extensions of Chow motives, as canonical sub-quotients of the motive with compact support of the smooth part of the surface. Specializing to Hilbert–Blumenthal surfaces, we recover a motivic interpretation of a recent construction of A. Caspar.

💡 Research Summary

The paper is divided into two main parts. In the first part the authors revisit the construction of a Chow motive that models the intersection cohomology of a proper surface S. By using the normalization of S and the natural maps between Chow groups and intersection cohomology groups, they define a Chow motive M_IC(S) which reproduces the usual topological properties of intersection cohomology—Poincaré duality, Künneth formula, and the compatibility with the Borel–Moore homology of S—inside the category of Chow motives. This motive is shown to be self‑dual and to carry the same weight filtration as the underlying mixed Hodge structure.

The second part deals with a non‑singular blow‑up π : \tilde S → S, focusing on the exceptional divisor E = π⁻¹(Sing S). The authors work in Voevodsky’s triangulated category of effective geometric motives DM^eff_gm(k) and study the motive M(E) of the divisor. When every geometric irreducible component of E is a rational curve (genus 0), they prove that M(E) fits into a canonical short exact triangle of Chow motives
 0 → M₀ → M(E) → M₁ → 0,
where M₀ and M₁ are respectively the degree‑0 and degree‑1 Chow motives associated with the components of E. This decomposition follows from the weight‑filtration formalism in Voevodsky’s category and from the fact that rational curves have trivial higher Chow groups.

Next the authors consider the motive with compact support M_c(S°) of the smooth locus S° = S \ Sing S. They show that the above short exact triangle appears naturally as a sub‑quotient of M_c(S°). Consequently, M_c(S°) contains a mixed motive of weight 1 sandwiched between pure weight‑0 and weight‑2 Chow motives. This mixed piece cannot be seen in the pure Chow motive framework alone, illustrating how Voevodsky’s triangulated category captures extensions that are invisible to classical Chow theory.

Finally, the theory is applied to Hilbert–Blumenthal surfaces. These surfaces have normal crossing singularities and an exceptional divisor that is a disjoint union of ℙ¹’s. By inserting the previously constructed extension into the motive of the smooth part of such a surface, the authors recover the motivic construction recently announced by A. Caspar. While Caspar’s work was formulated using Hodge‑theoretic and automorphic methods, the present paper provides a purely motivic interpretation, showing that the same extension arises from the interaction of Chow motives with Voevodsky’s effective motives.

Overall, the article demonstrates a powerful synthesis of classical Chow motives and Voevodsky’s triangulated motives. It establishes that, under the genus‑zero hypothesis for the exceptional divisor, one can canonically construct one‑extensions of Chow motives that sit inside the compact‑support motive of the smooth part of a singular surface. This not only clarifies the motivic nature of recent constructions on Hilbert–Blumenthal surfaces but also opens the door to analogous extensions for more complicated singularities, suggesting a promising direction for future research in the theory of mixed motives and their applications to arithmetic geometry.