Mirror symmetry for moduli spaces of Higgs bundles via p-adic integration

Abstract

We prove the Topological Mirror Symmetry Conjecture by Hausel-Thaddeus for smooth moduli spaces of Higgs bundles of type SLn and P GLn. More precisely, we establish an equality of stringy Hodge numbers for certain pairs of algebraic orbifolds generically fibred into dual abelian varieties. Our proof utilises p-adic integration relative to the fibres, and interprets canonical gerbes present on these moduli spaces as characters on the Hitchin fibres using Tate duality. Furthermore we prove for d coprime to n, that the number of rank n Higgs bundles of degree d over a fixed curve defined over a finite field, is independent of d. This proves a conjecture by Mozgovoy-Schiffman in the coprime case.Contents arXiv:1707.06417v2 [math.AG] 8 Jun 2018Proof. We begin the proof by fixing notation. The relative Jacobian of the trivial family of curves X × S A / A will be denoted by J. The relative Jacobian of the universal spectral curve Y / A will be denoted by J. Similarly we denote by J 1 and J 1 the relative moduli spaces of degree 1 line bundles.Henceforth we restrict every A-scheme to the open subset A good . To avoid awkward notation we will omit the corresponding superscript.The relative norm map induces a morphism of abelian A-schemes J Nm / / J. Similarly, pullback of line bundles yields π * : J / / J. We claim that these two morphisms are dual to each other with respect to the canonical isomorphism J ∨ J induced by the Poincaré bundle (and similarly for J). To see this we observe that we have a commutative diagram (the horizontal arrows represent the Abel-Jacobi map)to which we can apply the contravariant Pic 0 (?/ A) functor to obtain the commutative diagram of abelian schemes Furthermore, the top row of the first diagram is the fibre of the vertical arrows and hence the top row of the second diagram is the corresponding cofibre. By explicitly computing the fibres in the second diagram

Geometric stabilisation via $p$-adic integration

“…In order to prove Theorems 1.3 and 1.1 we need twisted versions of (3), which we develop in Section 6. Similar to the proof of the refined topological mirror symmetry conjecture [GZW17,Theorem 5.21], the idea is to consider for an a ∈ AG(k) and t ∈ π0( PG,a) the stack M t G,a = MG,a × P G,a Tt, where Tt is a PG,a-torsor representing t via the isomorphisms H 1 (k, PG,a) ∼ = H 1 (k, π0( PG,a)) ∼ = π0( PG,a). To compensate this twist on the G-side we introduce a natural function on χt : I µ M G,a (k) → C and show that for every b ∈ A ♦ G (F ) ∩ AG(O) restricting to a over Spec(k) the function χs • e on M G,b (F ) can be interpreted as the Hasse invariant of a certain Gm-gerbe on M G,b .…”

“…Using the non-degeneracy of the Tate-duality pairing for abelian varieties we are able to derive in Theorem 6.11 the twisted versions of (3) and (4) needed to prove first Theorem 1.3 and then 1.1. This part relies on the results of Section 3 which gives a stack-theoretic interpretation of the Hasse invariant (generalising [GZW17,3.3]).…”

“…It was already observed by Hausel [Hau11,Section 5.4], that their conjecture was closely related to the Geometric Stabilisation Theorem for SLn. In [GZW17] we established their conjecture via p-adic integration relying heavily on the aforementioned duality, which by work of Donagi-Pantev [DP12] and Chen-Zhu [CZ17] also holds for general pair of Langlands dual groups (G, G). In the the present work we thus extend our methods from [GZW17] to general (G, G) and infer geometric stabilisation for G. This sheds new light on the algebro-geometric origins of Geometric Stabilisation, and thus the Fundamental Lemma.…”

Higgs Bundles—Recent Applications

“…Hence, there is no obvious reason for the Betti numbers to be invariant with regards to the choice of deg(E), which would normally follow from non-abelian Hodge theory. However, the independence holds in direct calculations of the Betti numbers in low rank, and was recently shown for GL(n, C) and SL(n, C)-Higgs bundles by Groechenig-Wyss-Ziegler in [GWZ17]. This suggests that some topological properties of Hitchin systems are independent of the hyperkähler geometry (see references in [Hau13,Ray18] for more details).…”

“…Very recently, the first proof of this conjecture was established for the moduli spaces of SL(n, C) and P GL(n, C)-Higgs bundles by Groechenig-Wyss-Ziegler in [GWZ17], where they established the equality of stringy Hodge numbers using p-adic integration relative to the fibres of the Hitchin fibration, and interpreted canonical gerbes present on these moduli spaces as characters on the Hitchin fibres. Further combinatorial properties of M G C can be glimpsed through their twisted version, consisting of Higgs bundles (E, Φ) on Σ with Φ : E → E ⊗ L, where Σ now has any genus, L is a line bundle with deg(L) > deg(K), but without any punctures or residues being fixed.…”