Access the full text.
Sign up today, get DeepDyve free for 14 days.
J. Martínez-Martínez, V. Muñoz (2014)
E-polynomials of the SL(2,C)-character varieties of surface groupsInternational Mathematics Research Notices, 2016
M. Groechenig, Dimitri Wyss, Paul Ziegler (2017)
Mirror symmetry for moduli spaces of Higgs bundles via p-adic integrationInventiones mathematicae, 221
Camilla Felisetti, Mirko Mauri (2020)
P=W conjectures for character varieties with symplectic resolutionJournal de l’École polytechnique — Mathématiques
C. Simpson (1994)
Moduli of representations of the fundamental group of a smooth projective variety IPublications Mathématiques de l'Institut des Hautes Études Scientifiques, 79
David Baraglia, Pedram Hekmati (2016)
Arithmetic of singular character varieties and their E ‐polynomialsProceedings of the London Mathematical Society, 114
M. Cataldo, L. Migliorini (2003)
The Hodge theory of algebraic mapsAnnales Scientifiques De L Ecole Normale Superieure, 38
M. Cataldo, D. Maulik (2018)
The perverse filtration for the Hitchin fibration is locally constantPure and Applied Mathematics Quarterly
Mirko Mauri (2021)
INTERSECTION COHOMOLOGY OF RANK 2 CHARACTER VARIETIES OF SURFACE GROUPSJournal of the Institute of Mathematics of Jussieu, 22
Tamás Hausel (2011)
Global topology of the Hitchin systemarXiv: Algebraic Geometry
L. Göttsche, W. Soergel (1993)
Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfacesMathematische Annalen, 296
D. Maulik, Junliang Shen (2020)
Endoscopic decompositions and the Hausel–Thaddeus conjectureForum of Mathematics, Pi, 9
M. Cataldo, L. Migliorini (2008)
The perverse filtration and the Lefschetz Hyperplane TheoremarXiv: Algebraic Geometry
D. Maulik, Junliang Shen (2020)
Cohomological χ–independence for moduli of one-dimensional sheaves and moduli of Higgs bundlesGeometry & Topology
Tamás Hausel, M. Thaddeus (2002)
Mirror symmetry, Langlands duality, and the Hitchin systemInventiones mathematicae, 153
The moduli spaces of flat SL - and PGL -connections are known to be singular SYZ-mir- 2 2 ror partners. We establish the equality of Hodge numbers of their intersection (stringy) cohomology. In rank two, this answers a question raised by Tamás Hausel in Remark 3.30 of “Global topology of the Hitchin system”. Keywords Intersection cohomology · Mirror symmetry · Character variety · E-polynomial · Perverse filtration Mathematics Subject Classification 14J33 · 14D20 Let C be a compact Riemann surface of genus g with base point c ∈ C , and G be either SL or PGL . We study the following moduli spaces (cf [14]): 2id∕r • the de Rham moduli space of principal flat G-bundles on C ⧵ c with holonomy e around c; • the Dolbeault moduli space of semistable G-Higgs bundles of degree d, i.e. semi- stable pairs (E, ) consisting of a principal G-bundle E of degree d and a section 𝜙 ∈ H (C, ad(E) ⊗ K ) , where K is the canonical bundle; C C the Betti moduli space parametrising G-representations of the fundamental group of 2id∕r C ⧵ c with monodromy e around c. d d d These moduli spaces are denoted respectively M (C,G) , M (C,G) and M (C,G) . DR Dol B For convenience, we simply write M(C, G) when we refer indifferently to M (C,G) , Dol 0 0 M (C,G) or M (C,G). DR B In [9], Hausel and Thaddeus showed that the de Rham moduli spaces M (C, SL ) DR and M (C, PGL ) are mirror partners in the sense of Strominger–Yau–Zaslow mirror DR Communicated by Daniel Greb. * Mirko Mauri mauri@mpim-bonn.mpg.de Max Planck Institute for Mathematics, Vivatgasse 7, 53111 Bonn, Germany 1 3 Vol.:(0123456789) 298 M. Mauri symmetry. According to the general mirror symmetric framework, it is reasonable to expect a symmetry between their Hodge numbers. Hausel and Thaddeus conjectured the equality of the stringy E-polynomials e d B d B e E (M (C, SL )) = E (M (C, PGL )), (1) r r st DR st DR for (d, r)=(e, r)= 1 , and they prove it for r = 2, 3 . The conjecture is now a theorem due to [7] or [12]. In [8, Remark 3.30] Hausel asked what cohomology theory we should compute on M (C, SL ) , with (d, r) ≠ 1 , to accomplish the agreement (1). DR We propose to use intersection cohomology. As first piece of evidence, we show the topological mirror symmetry conjecture in rank two, and degree zero, i.e. when we turn off the B-fields B and B. Theorem 0.1 (Topological mirror symmetry in rank two and degree zero) The inter- section E-polynomial of M(C, SL ) equals the stringy intersection E-polynomial of M(C, PGL ) IE(M(C, SL )) = IE (M(C, PGL )). 2 st 2 The refinements of the Hausel–Thaddeus conjecture postulated in [8, Conjecture 3.27] and [8, Conjecture 5.9] also hold true in rank two and degree zero, as long as we consider their intersection cohomology analogues; see Theorems 2.2 and 3.2. 1 Intersection stringy E‑polynomial The intersection cohomology of a complex variety X with compact support, middle perver- ∗ ∗ sity and rational coefficients is denoted by IH (X) . Recall that IH (X) carries a canonical c c mixed Hodge structure, and so we can define the intersection E-polynomial of X as d W d r,s r s IE(X) ∶= (−1) dim(Gr IH (X,ℂ)) u v . r+s c (2) r,s,d Suppose that X is endowed with the action of a finite abelian group Γ , and denote the group of characters of Γ by Γ . The intersection cohomology of X decomposes under the action of Γ into isotypic components: ∗ ∗ IH (X)= IH (X) . 𝜅 ∈Γ Then, if we pose d W d r,s r s IE(X) ∶= (−1) dim(Gr IH (X,ℂ) ) u v , r+s c r,s,d we obtain IE(X)= IE(X) . ̂ 𝜅 𝜅 ∈Γ Define also the intersection stringy E-polynomial by See [9, §4] for a definition of gerbe or B-field. 1 3 Topological mirror symmetry for rank two character varieties… 299 F() IE (X) ∶= IE(X ∕Γ;u, v)(uv) , st ∈Γ where • X is the fixed-point set of ∈Γ. • F() is the Fermionic shift, defined as F()= w , where acts on the normal bundle j j 2iw of X in X with eigenvalues e with w ∈(0, 1). 2 Topological mirror symmetry 0 2g Let Γ ∶= Pic (C)[r]≃(ℤ∕rℤ) be the group of r-torsion line bundles over the com- pact Riemann surface C of genus g, endowed with the canonical flat connection. The d d group Γ acts by tensorisation on M (C, SL ) and M (C, SL ) . Via the non-abelian r r Dol DR Hodge correspondence, the action corresponds to the algebraic action of the characters Γ≃ Hom( (C),ℤ∕rℤ) which acts on M (C, SL ) by multiplication. The quotient of 1 r d d M (C, SL ) by the action of Γ is isomorphic to M (C, PGL ). r r We identify w ∶Γ → Γ through Poincaré duality (also known as Weil pairing) Γ× Γ ≃ H (C,ℤ∕rℤ)× H (C,ℤ∕rℤ) → ℤ∕rℤ. 1 1 Conjecture 2.1 (Topological mirror symmetry in degree zero) For 𝜅 ∈ Γ we have F() IE(M(C, SL )) = IE(M(C, SL ) ∕Γ;u, v)(uv) (3) r r where = w() . In particular, we obtain IE(M(C, SL )) = IE (M(C, PGL )). r st r Theorem 2.2 (Theorem 0.1) Conjecture 2.1 holds for r = 2. Proof Without loss of generality we can suppose ≠ 0 , or equivalently ≠ 1 . Indeed, IE(M(C, SL )) = IE(M(C, SL )∕Γ;u, v); r 1 r see for instance the proof of [6, Proposition 3]. For any ∈Γ ⧵ {0} , we have an associated 2-torsion line bundle L . Consider the étale ⊗2 double cover ∶ C → C consisting of the square root of a non-zero section of L ≃ O in the total space of L , and let be its deck transformation. For any L ∈ M(C , GL ) , the rank-two vector bundle L ⊕𝜄 L is a -invariant object in M(C , GL ) , which descends to an object L ∈ M(C, GL ) . Hence, the pushforward 2 2 morphism ∶ M(C , GL ) → M(C, GL ), L ↦ L , ,∗ 1 2 descends to a Γ-invariant embedding j ∶ M(C , GL )∕ℤ∕2ℤ ↪ M(C, GL ). 1 2 The determinant map det can be identified with the norm map ,∗ 1 3 300 M. Mauri Nm ∶ M(C , GL )∕ℤ∕2ℤ → M(C, GL ), L ↦ L ⊗𝜄 L, C ∕C 𝛾 1 1 Therefore, the fixed-point set M(C, SL ) admits the following geometric characterization: M(C, SL ) = Imj ∩ M(C, SL )≃ ker Nm , 2 2 C ∕C −1 where the last term is the connected component of Nm (O ) containing O . C C C ∕C On the Dolbeault side, M (C, SL ) is isomorphic to the quotient by ℤ∕2ℤ of the Dol 2 cotangent bundle of an abelian variety of dimension g − 1 , as M (C , GL ) is isomorphic Dol 1 to T Pic (C ) ; see also the proof of Theorem 3.2. On the Betti side, we have ∗ 4g−2 ∗ 2g−2 M (C , GL )≃(ℂ ) , and so M (C, SL ) ≃(ℂ ) ∕ℤ∕2ℤ. B 1 B 2 The involution defining the ℤ∕2ℤ-quotient is the inverse of the group law. Since the Γ-module IH (M(C, SL )) is a direct sums of copies of the trivial and regular representations by [13, Remark 4.4], we have IE(M(C, SL )) = IE(M(C, SL )) � 2 2 for any 𝜅 , 𝜅 ∈ Γ ⧵ {1} . Thanks to [13, Corollary 1.11, Equations (23) and (25)] we have 3g−3 g−1 g−1 g−1 g−1 IE(M (C, SL )) = (uv) ((u + 1) (v + 1) +(u − 1) (v − 1) ) Dol 2 2g−2 = IE(M (C, SL ) ∕Γ)(uv) , Dol 2 2g−2 2g−2 2g−2 (4) IE(M (C, SL )) = (uv) ((uv + 1) +(uv − 1) ) B 2 ∗ 2g−2 2g−2 = IE((ℂ ) ∕ℤ∕2ℤ)(uv) 2g−2 = IE(M (C, SL ) ∕Γ)(uv) . B 2 Note that the Fermionic shift F() equals half of the codimension of M (C, SL ) in Dol 2 M (C, SL ) , since is an involution. Hence, for ≠ 0 we have indeed Dol 2 F()= codimM (C, SL ) = 2g − 2. Dol 2 Finally, the same argument of [9, §6], together with [5, Theorem 3.2], implies that Conjec- ture (3) for the Dolbeault moduli spaces yields (3) for the de Rham moduli spaces. ◻ Remark 2.3 (Failure of topological mirror symmetry for ordinary cohomology) In general the equality (3) fails for ordinary cohomology. For instance, for ≠ 1 , = w() and q = uv we have 2g−2 2g−2 2g−2 E(M (C, SL )) = q ((q + 1) +(q − 1) − 2) B 2 2g−2 2g−2 2g−2 F() ≠ q ((q + 1) +(q − 1) )= E(M (C, SL ) ∕Γ)q , B 2 where the first equality follows from [10, Theorem 2] or [1, Theorem 1.3], together with [13, Remark 4.3], while the last equality comes from (4), since M (C, SL ) ∕Γ has only B 2 quotient singularities. This shows that there is a non-negligible contribution of the singu- larity of M(C, SL ) to the agreement (3) of Hodge numbers. 1 3 Topological mirror symmetry for rank two character varieties… 301 Remark 2.4 The proof of Theorem 2.2 relies on the computation of IE(M(C, SL )) ≠1 in [13, Corollary 1.11], and ultimately on the explicit construction of a desingularization of M(C, SL ) in [13, §3]. To the best of the author’s knowledge, this is not available in higher rank, and so it is unclear if the arguments above extend in higher rank. However, remark- able progress in this direction have been made in [12] and [11]. 3 Perverse topological mirror symmetry The intersection cohomology of M (C, SL ) and M (C, SL ) are filtered by the perverse Dol r Dol 2 filtration P associated to the Hitchin fibrations 0 ⊗i 𝜒 ∶ M (C, SL ) → Λ ∶= H (C, K ) Dol r i=2 𝜒 ∶= 𝜒 ∶ M (C, SL ) → Λ ∶= Im(𝜒 ) ⊆ Λ, 𝛾 M (C,SL ) Dol r 𝛾 𝛾 M (C,SL ) Dol r 𝛾 Dol r 𝛾 which assigns to the Higgs bundle (E, ) the characteristic polynomial of ; see [13, §2.2] for a brief account on the perverse filtration. Recall that the graded pieces of the perverse filtration carries a canonical mixed Hodge structure; see for instance [4, §7]. Thus we can define the perverse intersection E-polynomial PIE(M (C, SL );u, v, q) Dol r d W P d r,s r s k ∶= (−1) dim(Gr Gr IH (M (C, SL ),ℂ)) u v q r+s k c Dol r r,s,d and the stringy perverse intersection E-polynomial F() PIE (M (C, PGL );u, v, q) ∶= PIE(M (C, SL ) ∕Γ;u, v)(uvq) . st Dol r Dol r ∈Γ By Definition (2) and the last paragraph of the proof of Theorem 2.2, we have PIE(M (C, SL );u, v,1)= IE(M (C, SL );u, v)= IE(M (C, SL );u, v). Dol r Dol r DR r Further, Relative Hard Lefschetz [3, Theorem 2.1.1] implies dim PIE(M (C, SL );u, v, q) =(uvq) PIE M (C, SL );u, v, , Dol r Dol r uvq where dim = 2(r − 1)(g − 1). We conjecture the exchange of the perverse Hodge numbers. Conjecture 3.1 (Perverse topological mirror symmetry in degree zero) dim PIE(M (C, SL );u, v, q) =(uvq) PIE M (C, PGL );u, v, . Dol r str Dol r uvq For q = 1 , Conjecture 3.1 specialises to IE(M (C, SL )) = IE (M (C, PGL )). Dol r st Dol r 1 3 302 M. Mauri Further, the PI=WI conjecture [2, Question 4.1.7] would imply PIE(M (C, SL );1, 1, q)= IE(M (C, SL );q), Dol r B r and together with Conjecture 3.1 it would give IE(M (C, SL )) = IE (M (C, PGL )). B r st B r Theorem 3.2 Conjecture 3.1 holds for r = 2. Proof By Relative Hard Lefschetz, it is enough to show F() PIE(M (C, SL )) = PIE(M (C, SL ) ∕Γ;u, v)(uvq) Dol 2 Dol 2 for any 𝜅 ∈ Γ and = w() . As in Theorem 2.2, the case = 1 , alias = 0 , is trivial. Sup- pose then ≠ 1 and ≠ 0 . The perverse filtration on IH (M (C, SL )) is concentrated in Dol 2 degree d − 2g + 2 by [13, Theorem 5.5]. Moreover, the Hitchin map is a quotient by the inverse of the group law of the projection ∗ g−1 g−1 M (C , GL ) ⊃ T Prym = Prym × ℂ → ℂ , Dol 𝛾 1 where Prym is the connected component of the identity of the kernel of the norm map 0 0 ∗ Nm ∶ Pic (C ) → Pic (C) , given by Nm(L)= L ⊗𝜄 L . Hence, the perverse filtration on IH (M (C, SL ) ∕Γ) , with ≠ 0 , is concentrated in degree d; cf proof of [5, Theo- Dol 2 rem 6.6]. Then one easily see that Conjecture 3.1 for r = 2 is equivalent to Theorem 2.2. Acknowledgements This work have been supported by the Max Planck Institute for Mathematics. We thank an anonymous referee for helpful comments. Funding Open Access funding enabled and organized by Projekt DEAL. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Com- mons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. References 1. Baraglia, D., Hekmati, P.: Arithmetic of singular character varieties and their E-polynomials. Proc. Lond. Math. Soc. (3) 114(2), 293–332 (2017) 2. de Cataldo, M.A., Maulik, D.: The perverse filtration for the Hitchin fibration is locally constant. Pure Appl. Math. Q. 16(5),1441–1464 (2020) 3. de Cataldo, M.A., Migliorini, L.: The Hodge theory of algebraic maps. Ann. Sci. École Norm. Sup. (4) 38(5), 693–750 (2005) 4. de Cataldo, M.A., Migliorini, L.: The perverse filtration and the Lefschetz hyperplane theorem. Ann. Math. (2) 171(3), 2089–2113 (2010) 5. Felisetti, C., Mauri, M.: P=W conjectures for character varieties with symplectic resolution. arXiv: 2006. 08752 (2020) 1 3 Topological mirror symmetry for rank two character varieties… 303 6. Göttsche, L., Soergel, W.: Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic sur- faces. Math. Ann. 296(2), 235–245 (1993) 7. Groechenig, M., Wyss, D., Ziegler, P.: Mirror symmetry for moduli spaces of Higgs bundles via p-adic inte- gration. Invent. Math. 221(2), 505–596 (2020) 8. Hausel, T.: Global topology of the Hitchin system. In: Handbook of moduli. Vol. II, volume 25 of Adv. Lect. Math. (ALM), pp. 29–69. Int. Press, Somerville (2013) 9. Hausel, T., Thaddeus, M.: Mirror symmetry, Langlands duality, and the Hitchin system. Invent. Math. 153(1), 197–229 (2003) 10. Martínez, J., Muñoz, V.: E-polynomials of the SL(2,ℂ)-character varieties of surface groups. Int. Math. Res. Not. IMRN 3, 926–961 (2016) 11. Maulik, D., Shen, J.: Cohomological -independence for moduli of one-dimensional sheaves and moduli of Higgs bundles. arXiv: 2012. 06627 (2020) 12. Maulik, D., Shen, J.: Endoscopic decompositions and the Hausel–Thaddeus conjecture. to appear at Forum Math. Pi. arXiv: 2008. 08520 (2020) 13. Mauri, M.: Intersection cohomology of rank two character varieties of surface groups. arXiv: 2101. 04628 (2021) 14. Simpson, C.T.: Moduli of representations of the fundamental group of a smooth projective variety. II. Inst. Hautes Études Sci. Publ. Math. 80, 5–79 (1995) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 1 3
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg – Springer Journals
Published: Oct 1, 2021
Keywords: Intersection cohomology; Mirror symmetry; Character variety; E-polynomial; Perverse filtration; 14J33; 14D20
You can share this free article with as many people as you like with the url below! We hope you enjoy this feature!
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.