The Motive of the Stack of Vector Bundles Over a Curve

The motive of the stack of vector bundles over a curve

Simon Pepin Lehalleur (Freie Universität Berlin) February 8, 2018 Summary

• Joint work with Victoria Hoskins (Freie Universität Berlin).

• Bunn,d(C) moduli stack of vector bundles of rank n and degree d over a smooth projective curve C over a field.

• The study of the topology of Bunn,d(C) and related moduli spaces of stable vector bundles is a classical topic (Atiyah-Bott, Harder-Narashiman, Bifet-Ghione-Letizia, Behrend-Dhillon, etc.).

• Goal: define and study the motive M(Bunn,d(C)) in Voevodsky’s triangulated category of mixed motives.

1 Motives of schemes and stacks Voevodsky motives

• k field, R coefficient ring. Assume characteristic of k invertible in R. • DM(k, R) tensor triangulated category of mixed motives over k with coefficients in R. • Motive functor M : Sm /k → DM(k, R) “universal” for functors to R-linear tensor triangulated categories satisfying Mayer-Vietoris for the Nisnevich topology M(U ∩ V) → M(U) ⊕ M(V) → M(X) → M(U ∩ V)[1], A1-homotopy invariance M(X × A1) ≃ M(X) and ⊗-invertible Tate twist R(1) := Cone(M(Spec(k)) → M(P1))[−2]. • DM(k, R) “large” category, compactly generated by geometric

subcategory DMgm(k, R).

2 Knowns and Unknowns

• Realisation functors to (enriched) cohomology theories. e.g. Hodge realisation pol RHdg : DM(C, Q) → D(Ind(MHSQ )). i ∨ ≃ i C Q such that H (RHdg(M(X)) ) Hsing(X( ), ) as graded-polarizable mixed Hodge structure. • Voevodsky: many classical properties of cohomology theories lift to DM(k, R), e.g., Künneth formula (built-in), localisation triangle, Gysin triangle, projective bundle formula, Poincaré duality, etc. • Relation to algebraic cycles: for X smooth,

n CH (X, R) ≃ HomDM(k,R)(M(X), R(n)[2n]).

• Missing piece of the puzzle: for R a Q-algebra, conjectural motivic t-structure on DM(k, R) compatible with realisations.

3 Algebraic stacks and quotient stacks

• Bunn,d(C) moduli stack of vector bundles: (lax 2-)functor which to a scheme S associates the groupoid of vector bundles of rank n and degree d over C × S. 2 • Bunn,d(C) algebraic stack, smooth of dimension n (g(C) − 1), not quasi-compact. ≤µ • Bunn,d(C) admits a filtration by open quotient substacksn Bun ,d parametrizing families of vector bundles with maximal slope ≤ µ. • Quotient stacks [X/G] are basic examples of stacks. • For S a scheme, [X/G](S) groupoid of pairs (P, f : P → X) with P a G-torsor over S and f a G-equivariant morphism. • For G algebraic group over k, classifying stack BG := [Spec(k)/G].

4 Cohomological invariants of stacks

• Extending cohomological invariants from schemes to (algebraic) stacks, including stacks which are not of finite type, can be delicate. • A smooth covering of a scheme can be refined by an étale covering ⇒ any cohomological invariant satisfying étale descent extends naturally to algebraic stacks. • Examples: coherent sheaf cohomology, étale cohomology, singular cohomology, de Rham cohomology... • Non-example: Chow groups. • DM(k, R) has an étale variant DMet (k, R) with DM(k, R ⊗ Q) ≃ DMet (k, R ⊗ Q). • For X algebraic stack, this idea leads to a definition of the étale motive Met (X ) ∈ DMet (k, R) and M(X ) ∈ DM(k, R ⊗ Q). 5 Motives of quotient stacks

• Slogan: Cohomology of [X/G] = G-equivariant cohomology of X. • In topology, can work with contractible free G-space EG and define ∗ ∗ H ([X/G]) := H (X ×G EG). • Totaro, Edidin-Graham, Morel-Voevodsky: work with algebraic approximations of EG. • Assume X smooth quasi-projective k-variety and G smooth linear algebraic group.

• There exists a sequence of vector bundles Vn → [X/G] with Vn ,→ Vn+1 and compatible open subsets Un ⊂ Vn with Un finite − → ∞ type k-scheme and codimVn (Vn Un) . • Can then define

M([X/G]) = hocolim M(Un) ∈ DM(k, R). • Example: ⊕∞ M(BGm) = M([Spec(k)/Gm]) ≃ R(n)[2n]. n=0 6 Motives of exhaustive stacks

Definition An exhaustive stack is a smooth algebraic stack X admitting an

increasing exhaustive filtration (Xn)n∈N by quasi-compact open substacks together with compatible choices of / / Un Vn Xn

/ / Un+1 Vn+1 Xn+1

with Vn → Xn vector bundle, Un → Vn open immersion with Un finite X − → ∞ type k-scheme and codimXn ( n Un) . • Quotient stacks are exhaustive. • Can then also define

M(X ) = hocolim M(Un) ∈ DM(k, R) • For R a Q-algebra, compatible with definition using étale descent. 7 The motive of the stack of vector bundles Moduli spaces of matrix divisors

• Construction due to Weil and Bifet-Ghione-Letizia (beautiful paper “On the Abel-Jacobi map for divisors of higher rank on a curve”). • For D effective divisor on C, define the space of matrix divisors as

{ ⊂ O⊕n | } Divn,d(D) = E C (D) E subsheaf, rk(E) = n, deg(E) = d .

2 • Divn,d(D) smooth projective variety of dimension n deg(D) − d. ′ ′ • For D ≤ D , closed immersion iD,D′ : Divn,d(D) → Divn,d(D ).

8 Matrix divisors approximate Bun

Theorem

Fix D0 > 0 effective divisor. The stack Bunn,d(C) is exhaustive, and we have

M(Bunn,d(C)) ≃ hocolim M(Divn,d(lD0)). l∈N

∪ → ∞ • Idea: D Divn,d(D) Bunn,d(C) “ -dimensional vector bundle”. 1 ∨ ⊕n • µmax(E) ≤ l deg(D0) − 2g + 2 ⇒ H (E ⊗ OC(lD0) ) = 0 by Serre duality and Harder-Narashiman filtration arguments.

• Implies that, for well chosen sequence of slopes (µl)l∈N, ≤ ≤ µl → µl Divn,d (lD0) Bunn,d

≤µl is open in a vector bundle over Bunn,d . Can then estimate codimensions and prove that Bunn,d is exhaustive. • Using another codimension estimate, we show ≤ µl ≃ hocolim M(Divn,d (lD0)) hocolim M(Divn,d(lD0)). l∈N l∈N 9 Motivic Bialynicki-Birula decomposition

⊕n • GLn = Aut OC(D) acts on Divn,d(D). Pick generic Gm ⊂ GLn. • Fixed points of the form ⊕n O(D − F ) ⊂ O⊕n(D), ∪ i=1 i C Gm m1 m2 mn Divn,d(D) = Sym (C) × Sym (C) × ... × Sym (C). m⊣ n deg(D)−d • Bialynicki-Birula decomposition ∪ + Divn,d(D) = Divn,d(D)m m⊣ n deg(D)−d with strata affine bundles over the components of fixed points. • Motivic Bialynicki-Birula decomposition (Karpenko, Brosnan) ( ) ⊕ ⊗n mi M(Divn,d(D)) ≃ M(Sym (C)) (cm)[2cm]. m⊣ n deg(D)−d i=1 Corollary

M(Bunn,d(C)) ∈ DM(k, R ⊗ Q) is a pure motive, contained in the localising tensor triangulated subcategory generated by M(C). 10 A conjectural formula

Conjecture Assume C(k) ≠ ∅. Then ( ) ⊗n−1 ⊕∞ i M(Bunn,d) ≃ M(Jac(C)) ⊗ M(BGm) ⊗ M(Sym (C))(ij)[2ij] . j=1 i=0

• True for n = 1, compatible with known results (Betti numbers, Harder’s stacky point count, Behrend-Dhillon’s formula in Grothendieck group of varieties).

• For R a Q-algebra, would give formula for motives Hi(M(Bunn,d)) realising to individual (co)homology groups. • Need to understand behaviour of motivic Bialynicki-Birula

decompositions with respect to the closed immersion iD,D′′ . • Conjecture would follow from explicit conjecture on transition maps ∗ ∗ ′ → ∗ iD,D′ : CH (Divn,d(D )) CH (Divn,d(D)) (work in progress).