[Paper Review] Motivic decompositions of moduli spaces of vector bundles on curves

Tomás L. Gómez, Kyoung-Seog Lee|arXiv (Cornell University)|Jul 12, 2020

Algebraic Geometry and Number Theory2 citations

TL;DR

This paper provides a new motivic decomposition of the moduli space M(r, L) of stable vector bundles of rank r = 2, 3 and fixed determinant L of degree d = 1 on a smooth projective curve C of genus g ≥ 2. Using Harder-Narasimhan filtrations and motivic zeta functions in the Grothendieck ring of varieties and Voevodsky motives, the authors derive explicit motivic Poincaré polynomial formulas involving symmetric products of C, confirming a motivic version of Narasimhan's conjecture for r = 2 and extending it to r = 3.

ABSTRACT

Let $r \geq 2, d$ be two integers which are coprime to each other. Let $C$ be a smooth projective curve of genus $g \geq 2$ and $M(r,L)$ be the moduli space of rank $r$ stable vector bundles on $C$ whose determinants are isomorphic to a fixed line bundle $L$ of degree $d$ on $C.$ In this paper, we study motivic decomposition of $M(r,L)$ for $r=2, 3$ cases. We give a new proof of a version of the main result of arXiv:1806.11101. We also found a new motivic decomposition of $M(3,L).$

Motivation & Objective

• To provide a new motivic decomposition of the moduli space M(r, L) of stable vector bundles on a curve C of genus g ≥ 2 with fixed determinant L of degree d = 1.
• To offer a uniform, simplified proof of the motivic Poincaré polynomial formula for M(2, L), confirming a motivic version of Narasimhan's conjecture.
• To extend the motivic decomposition to the rank 3 case, providing a new formula for χ(M(3, L)) in terms of symmetric products of C and Jacobians.
• To establish that these motivic decompositions hold not only in the Grothendieck ring of varieties but also in Voevodsky's category of mixed motives.

Proposed method

• Use the Harder-Narasimhan filtration to decompose the moduli stack Bunr,d into stable and unstable parts in the Grothendieck ring of varieties.
• Compute the motivic class of the unstable part via parametrization of extensions of line bundles, using Riemann-Roch to determine dimensions of Ext and Hom spaces.
• Apply the motivic zeta function identity Z(C, t) = (1 + t)^h1(C) / ((1 - t)(1 - Lt)) in the completed Grothendieck ring of Chow motives.
• Express motives of symmetric products Ck in terms of Jacobians and line bundles using motivic decompositions from [1, 10, 11, 13, 15].
• Derive the motivic Poincaré polynomial by subtracting the unstable part from the total class of Bunr,d.
• Verify the decomposition by comparing the resulting expression with the conjectured form in the Grothendieck ring and in K0(dDMgm).

Experimental results

Research questions

• RQ1 Can the motivic Poincaré polynomial of M(2, L) be re-derived using a uniform, simplified method based on Harder-Narasimhan filtrations and motivic zeta functions?
• RQ2 Does a motivic decomposition of M(3, L) exist that mirrors the conjectured semiorthogonal decomposition of its derived category?
• RQ3 How do the motivic classes of symmetric products Ck and Jacobians J(C) interact in the motivic decomposition of M(r, L) for r = 2, 3?
• RQ4 To what extent do these motivic decompositions lift to the category of Voevodsky motives and the Grothendieck ring of varieties?
• RQ5 Are the motivic decompositions compatible with derived category decompositions and conjectural Fukaya category decompositions?

Key findings

• For r = 2, d = 1, the motivic Poincaré polynomial of M(2, L) is given by ∑_{k=0}^{g-2} χ(Ck)(L^k + L^{3g-3-2k}) + χ(C^{g-1})L^{g-1}, confirming a motivic version of Narasimhan's conjecture.
• For r = 3, d = 1, the motivic Poincaré polynomial is ∑_{k1+k2<2(g-1)} χ(C^{k1} × C^{k2})(L^{k1+2k2} + L^{8g-8-2k1-3k2}) + ∑_{k1+k2=2(g-1), k1<g-1} χ(C^{k1} × C^{k2})(L^{k1+2k2} + L^{8g-8-2k1-3k2}) + χ(C^{g-1} × C^{g-1})L^{3(g-1)}
• The motivic class [M(r, L)] in cK0(Var) admits a decomposition into sums of products of symmetric powers of C and powers of L, with coefficients in the Grothendieck ring.
• The same motivic decomposition holds in K0(dDMgm), the Grothendieck group of Voevodsky's category of mixed motives.
• The unstable part of Bun2,L is computed as [J(C)]L^g / ((L-1)(L^2-1)), which is subtracted from the total class to recover [M(2, L)].
• The derived category conjecture of Narasimhan is supported by the motivic decomposition, suggesting compatibility between motivic and semiorthogonal decompositions.