A formula for the Voevodsky motive of the moduli stack of vector bundles on a curve

Motives of moduli spaces of rank 3 vector bundles and Higgs bundles on a curve

“…The deformation theory and wall-crossing for chains were studied in [1], as well as the relationship between stability for chains and Higgs bundles. In particular, the connected components of the fixed point set of the G m -action on M are moduli spaces of α H -semistable chains for different discrete invariants, where α H is a Higgs stability parameter satisfying α H,i − α H,i+1 = 2g − 2 for all i (see [21] and [14,Corollary 2.6]). Provided n and d are coprime, the Higgs stability parameter is generic for the discrete invariants for chains appearing as fixed loci components (i.e semistability and stability coincide and these chain moduli spaces are smooth projective varieties).…”

“…This scaling action is also used in the study the class of M in the Grothendieck ring of varieties in [9] and the Voevodsky motive of M in [14], where wall-crossing for chains plays an important role. In rank n = 2 and odd degree, we obtained a formula for the integral motive of M in [8, Theorem 1.4] in terms of N .…”

“…Motives of Higgs moduli spaces with fixed determinant. The rational Chow motive of M(n, d) for any rank n and coprime degree d lies in in the thick tensor subcategory generated by the motive of C [14]. As observed in the rank 2 case in [8, Proposition 6.3], for a line bundle L of odd degree d on a general complex curve C, the rational Chow motive of M L (2, d) is not in the thick tensor subcategory generated by the motive of C. We have a similar phenomenon in any rank n. Proposition 5.7.…”

“…An algorithm for computing the class of the Higgs moduli space M in the Grothendieck ring of varieties was described by García-Prada, Heinloth and Schmitt [9] using the Bia lynicki-Birula decomposition associated to the natural scaling action on M considered by Hitchin [13] in rank 2, Gothen [11] in rank 3, and Simpson [21], together with variation of stability for chains of vector bundle homomorphisms. This was upgraded in [14] to a motivic argument in Voevodsky's triangulated category of motives with rational coefficients and, by [14,Corollary 6.9], the motive of M is pure and lies in the tensor subcategory generated by the motive of C.…”

On the motive of O'Grady's ten-dimensional hyper-Kähler varieties

“…Such relations can be made precise in terms of cohomology groups (enriched with Hodge structures and Galois actions for instance) or more fundamentally, at the level of motives 1 . The prototype of such interplay we have in mind is del Baño's result [26], which says that the Chow motive of the moduli space M r,d (C) of stable vector bundles of coprime rank and degree on a smooth projective curve C is a direct summand of the Chow motive of some power of the curve; in other words, the Chow motive of M r,d (C) is in the pseudo-abelian tensor subcategory generated by the Chow motive of C. In [26], a precise formula for the virtual motive of M r,d (C) in terms of the virtual motive of C was obtained, a result which has been recently lifted to the level of motives in a greater generality by Hoskins and Pepin-Lehalleur [37].…”