Endoscopic decompositions and the Hausel-Thaddeus conjecture

Abstract

We construct natural operators connecting the cohomology of the moduli spaces of stable Higgs bundles with different ranks and genera which, after numerical specialization, recover the topological mirror symmetry conjecture of Hausel-Thaddeus concerning SLn-and PGLn-Higgs bundles. This provides a complete description of the cohomology of the moduli space of stable SLn-Higgs bundles in terms of the tautological classes, and gives a new proof of the Hausel-Thaddeus conjecture, proven recently by Gröchenig-Wyss-Ziegler via p-adic integration.Our method is to relate the decomposition theorem for the Hitchin fibration, using vanishing cycle functors, to the decomposition theorem for the twisted Hitchin fibration whose supports are simpler.

On the intersection cohomology of the moduli of $\mathrm{SL}_n$-Higgs bundles on a curve

“…The purpose of this paper is to explore cohomological structures for the moduli space of degree d semistable SL n -Higgs bundles on C with respect to an effective divisor D of degree deg(D) > 2g − 2. More precisely, we show that the support theorem [4] and the topological mirror symmetry conjecture [14,10,22], which were proven in the case gcd(n, d) = 1, actually hold for arbitrary d.…”

“…In fact, by combining the techniques of [8,4] and [23], we prove in Sections 1 and 2 a more general support theorem (Theorem 1.1) for certain relative moduli space of Higgs bundles associated with a cyclic étale Galois cover π : C ′ → C. These moduli spaces are tightly connected to the endoscopic theory for SL n [26,27] and the topological mirror symmetry for Hitchin systems [14,10,22]. 0.4.…”

“…Motivated by the Strominger-Yau-Zaslow mirror symmetry, Hausel-Thaddeus [14] conjectured that the moduli of semistable SL n -and PGL n -Higgs bundles should have identical (properly interpreted) Hodge numbers. In the case of gcd(n, d) = 1, the match of the Hodge numbers for the SL n -and PGL n -Higgs moduli spaces was formulated precisely in [14] using singular cohomology, and was proven recently in [10,21,22] by different methods. From the viewpoint of S-duality [15,Section 5.4] and the approach of [22], the Hausel-Thaddeus conjecture is closely connected to the endoscopy theory and the fundamental lemma for SL n .…”

“…In the case of gcd(n, d) = 1, the match of the Hodge numbers for the SL n -and PGL n -Higgs moduli spaces was formulated precisely in [14] using singular cohomology, and was proven recently in [10,21,22] by different methods. From the viewpoint of S-duality [15,Section 5.4] and the approach of [22], the Hausel-Thaddeus conjecture is closely connected to the endoscopy theory and the fundamental lemma for SL n .…”

“…In this paper, we explore the Hausel-Thaddeus conjecture for arbitrary degree d. Under the assumption deg(D) > 2g−2, we prove that an analog of the Hausel-Thaddeus conjecture holds for intersection cohomology and arbitrary degree d. Our approach follows the spirit of [22], that we view the (refined) Hausel-Thaddeus conjecture [15,Conjeture 4.5] as an extension of Ngô's geometric stabilization theorem [27] in his proof of the fundamental lemma of the Langlands program. Our new input is the support theorem for SL n and its endoscopic groups (see Theorem 1.1), relying on the framework of [23].…”