Summary:

The focus of this project is to understand the geometry of mirror symmetry in the Strominger-Yau-Zaslow (SYZ) setup, in the particular case where the Calabi-Yau manifolds in question are hyperkähler. The hyperkähler structure allows one to study the geometry of mirror symmetry via complex algebraic geometry, a crucial idea in this project.

Our main guiding example is the Hitchin system for complex Lie groups G, given by the moduli space of G-Higgs bundles over a smooth curve, together with the Hitchin fibration, whose generic fibers are abelian subvarieties of Jacobians of so-called spectral curves. This moduli is hyperkähler and fits in the SYZ setup, the mirror being the Hitchin system for the Langlands dual group of G. Indeed, mirror symmetry on Hitchin systems is deeply related with the Geometric Langlands Program. 

In a hyperkähler manifold it makes sense to define BBB- and BAA-branes and Kapustin-Witten conjectured that such branes should be exchanged under mirror symmetry, via (extensions of) a Fourier-Mukai transform. Non-trivial examples of these branes are not easily found, but some of them have been, and evidence supporting the KW conjecture was obtained. Most examples were, however, over the smooth locus of the Hitchin fibration. One of the main problems (but not the only one) we intend to address is to develop machinery to deal with mirror symmetry on the singular locus of the Hitchin fibration.

Project's new preprints/publications:

• D. Alfaya, A. Oliveira, Lie algebroid connections, twisted Higgs bundles and motives of moduli spaces, Journal of Geometry and Physics, 201 (2024) 105195.

•  G. Barajas, O. García-Prada, P. B. Gothen, I. Mundet i Riera, Non-connected Lie groups, twisted equivariant bundles and coverings, Non-connected Lie groups, twisted equivariant bundles and coverings, Geometriae Dedicata, 217 (2023), Paper No. 27, 41 pp.

• V. Bertini, Rational curves on O'Grady's tenfolds, Kyoto Journal of Mathematics, in press. 

• V. Bertini, A. Grossi, Rational curves on primitive symplectic varieties of OG6 singular type, Math. Z. 304, 36 (2023).

• V. Bertini, A. Grossi, M. Mauri, E. Mazzon, Terminalization of quotients of compact hyperkähler manifolds by induced symplectic automoprhisms. Preprint arXiv:2401.13632.

• V. Bertini, F. Giovenzana, Hodge structure of O'Grady's singular moduli spaces, accepted at Annali della Scuola Normale Superiore di Pisa.

• S. Bradlow, B. Collier, O. García-Prada, P. Gothen, A. Oliveira, A general Cayley correspondence and higher Teichmüller spaces. Accepted for publication in Annals of Mathematics, 200 (2024).

• M. Brandt, J. Bruce, M. Chan, M. Melo, G. Moreland, C. Wolfe, On the top-weight rational cohomology of A_g. Accepted for publication in Geometry & Topology. 

• L. Caporaso, M. Melo, M. Pacini, Theta-characteristics on tropical curves. Journal of Pure and Applied Algebra, Volume 227, Issue 5, May 2023.

• G. Codogni, L. Tasin, F. Viviani, Slope inequalities for KSB-stable and K-stable families, Proc. Lond. Math. Soc. (3) 126 (2023), no. 4, 1394--1465.

• G. Codogni, L. Tasin, F. Viviani, On the first steps of the minimal model program for the moduli space of stable pointed curves, J. Inst. Math. Jussieu 22 (2023), 145–211.

• M. Melo, S. Molcho, M. Ulirsch, F. Viviani, Tropicalization of the universal Jacobian, Épijournal de Géom. Algébr., EPIGA 6, Article 15, 51 p. (2022).

• E. Franco, P. Gothen, A. Oliveira, A. Peón-Nieto, Narasimhan--Ramanan branes and wobbly Higgs bundles. Accepted for publication in International Journal of Mathematics.

• E. Franco, R. Hanson, J. Horn, A. Oliveira, Fourier-Mukai transforms and normalization of nodal curves, Preprint: arXiv:2405.11860.

• R. Fringuelli, F. Viviani, The Picard group of the universal moduli stack of principal bundles on pointed smooth curves, J. of Topology 15 (2022), 2065--2142.

• R. Fringuelli, F. Viviani, The Picard group of the universal moduli stack of principal bundles on pointed smooth curves II. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) Vol. XXIV (2023), 367-447. 

• R. Fringuelli, F. Viviani, On the Picard group scheme of the moduli stack of stable pointed curves. Preprint: arXiv: 2005.06920v3.

• R. Fringuelli, F. Viviani, The line bundles on the moduli stack of principal bundles on families of curves. Preprint arXiv:2401.02204.

• P. Gothen, Geometry on surfaces and Higgs bundles, CIM Bulletin 45 (2023), 25--34.

• P. Silva, P. Gothen, The conformal limit and projective structures, Preprint arXiv:2401.07759. 

• F. Viviani, On the classification of fine compactified Jacobians of nodal curves. Preprint  arXiv:2310.20317

Talks:

• V. Bertini: Terminalization of quotients of hyperkähler manifolds via symplectic actions. Stuttgart University. Workshop: Young perspectives on irreducible holomorphic symplectic manifolds, 13 Feb 2023.

• V. Bertini: The Hodge structure of singular O’Grady’s moduli spaces, Symplectic singularities in geometry and representation theory, CIRM Luminy, 07 April 2022.

• V. Bertini: Introduction on singular symplectic varieties, Hyperkaehler varieties and related topics, Università La Sapienza, Rome, 08 September 2022.

• V. Bertini: Terminalization of quotients of hyperkähler manifolds via symplectic actions}, Seminario di Natale 2022, Milano University, 20 December 2022.

• V. Bertini: Compact Hyperkaehler varieties: from smooth to singular, Geometry and Topology seminar, CMUP Porto, 23 September 2022.

• V. Bertini: Singular symplectic varieties as symplectic quotients of hyperkähler manifolds, VBAC webinar, 21 November 2022.

• V. Bertini: The Hodge structure of O’Grady’s singular moduli spaces, Algebra and Geometry seminar, Bologna University, 29 November 2022.

• P. Gothen: Components of G-Higgs bundle moduli and the Cayley correspondence, Moduli spaces and geometric structures, Conference in honour of Oscar García-Prada on the occasion of his 60th birthday, ICMAT (Madrid), 16 September 2022 (online talk).

• P. Gothen: Lecture Series: Moduli Spaces of Higgs bundles. Workshop on Moduli Spaces and Vertex Algebras, University of Auckland, New Zealand. 20 Feb. 2023.

• P. Gothen: Twisted equivariant bundles on coverings and the Hitchin-Kobayashi correspondence for non-connected groups. Higgs bundles, character varieties and higher Teichmüller spaces, ICMAT. 22 May 2023.

• M. Melo: On the top weight rational cohomology of the moduli space of abelian varieties, Geometry and Algebra Seminar of the University of Roma Sapienza, June 22, 2022.

• M. Melo: Mini-course on Compactified Jacobians and Fourier-Mukai transforms, School on Mirror Symmetry and Moduli Spaces (SMSMS), Instituto Superior Técnico, 27 June -- 1 July 2022, Lisbon, Portugal.

• M. Melo: Tropicalizing Universal Jacobians, Workshop in Combinatorial Algebraic Geometry, Bath, UK, August 1-5 2022.

• M. Melo: Tropical Universal Jacobians, Mini-Workshop on Complex and Tropical Geometry, October 26, 2022, Università di Padova.

• M. Melo: Tropicalization of the universal Jacobian. Non-commutative geometry meets topological recursion, Erwin Schrödinger Institute for Mathematics and Physics, Wien, Austria. 24 April 2023.

• M. Melo: Tropicalization of the universal Jacobian: logarithmic and non-archimedian view points. Refined invariants in Moduli Theory, SISSA, Trieste, Italy. 5 May 2023.

• M. Melo: Tropicalizing moduli spaces and applications. Global Portuguese Mathematicians, Coimbra, Portugal. 26 July 2023.

• M. Melo: Tropical Universal Jacobians. Curves: algebraic, tropical and logarithmic, Ban  International Research station, Canada. 6 Aug 2023.

• M. Melo: Compactified Jacobians. VBAC 2023: Recent applications to the geometry of moduli spaces. Essen, Germany. 28 Aug 2023.

• M. Melo: Tropicalizing moduli spaces and applications. Algebraic Geometry session of the conference of the Italian Mathematical Union (UMI), Pisa, Italy. 4 Sep 2023.

• M. Melo: Tropicalization of moduli spaces of curves, Jacobians and Abelian Varieties. Journées Louis Antoine 2023: Géométries tropicales et applications. Rennes. 9 Nov 2023.

• M. Melo: Tropicalization of the universal Jacobian: logarithmic and non-archimedian view points I, Geometry Seminar of CMUP, Universidade do Porto, January 6, 2023.

• M. Melo: Compactified and Tropicalized Universal Jacobians. STEW: Oslo Stability and Enumerative Geometry Workshop, Oslo, Norway. 11 Dec 2023.

• A. Oliveira: Higgs bundles: at the crossroads of algebra, geometry and physics. Seminar of the Joint PhD Program UC--UP, FCUP, Universidade do Porto.

• A. Oliveira: Components of Higgs bundles moduli spaces: old and new. Workshop on Higgs bundles, character varieties and higher Teichmüller spaces, ICMAT, Miraflores de la Sierra (Madrid), 24 May 2023.

• A. Oliveira: Moduli of twisted Higgs bundles as a degeneration of moduli of Lie algebroid connections. Geometry and Algebra Seminar, CMUC, Universidade de Coimbra, 8 March 2023.

• A. Oliveira: Mirror symmetry on Higgs bundles moduli spaces. Geometry and Topology Seminar, CMUP, Universidade do Porto, 17 Feb 2023.

• A. Oliveira: A pair of dual branes on the singular locus of the Hitchin system -- SIMMAC -- International Symposium on Mathematical Methods Applied to Sciences, Universidad de Costa Rica, Costa Rica, 23 February 2022 (online talk).

• A. Oliveira: Higgs bundles, sl_2-triples and higher Teichmüller spaces, Geometric Structures (re)United -- University of Illinois at Chicago, 14 June 2022.

• A. Oliveira: Mirror symmetry on Higgs bundles moduli spaces, XXX International Fall Workshop on Geometry and Physics -- ICMAT, Madrid, 02 September 2022.

• A. Oliveira: Lagrangians of Hecke cycles, Moduli spaces and geometric structures: Conference in honour of Oscar García-Prada on the occasion of his 60th birthday -- ICMAT, Madrid, 12 September 2022.

• F. Viviani: The Picard group of the universal moduli stack of principal bundles on pointed curves, School on Mirror Symmetry and Moduli Spaces  (SMSMS), Instituto Superior Técnico, 27 June, 1 July 2022, Lisbon, Portugal.

• F. Viviani: Tropicalization of the universal Jacobian: logarithmic and non-archimedian view points II, Geometry Seminar of CMUP, Universidade do Porto, January 6, 2023.

• F. Viviani: On a new class of fine compactified Jacobians of nodal curves. Derived Workshop, Bonn, Germany. 6 Nov. 2023.

• F. Viviani: On a new class of fine compactified Jacobians of nodal curves. Workshop Algebraic geometry in Bologna, Bologna, Italy 26 Sep. 2023.

• F. Viviani: On the Picard group of the stack of G-bundles on families of curves. Workshop Geometric and Arithmetic Frontiers of Orbifolds II - Arithmetic and hyperbolicity, Strasbourg, France. 5 June 2023.

• F. Viviani: On the Picard group of the stack of G-bundles on families of curves. Workshop Moduli and Algebraic Cycles, Mittag-Leffler, Sweden. 29 May 2023.

• F. Viviani: On the Universal Jacobian: algebraic, tropical and logarithmic aspects. Workshop Logarithmic geometry and moduli spaces, Frankfurt, Germany. 22 Feb. 2023.

News:

1. New team members: Pedro Silva as a post-doc (from 1 June 2023); Rodrigo Pereira as a master student (from 1 June 2022).

2. Open call for a Research Grant - Master Degree. Check all information in https://euraxess.ec.europa.eu/jobs/100459. 

3. New team members: Valeria Bertini as a post-doc (from 15 June 2022);

4. Rodrigo Pereira as a master student (from 1 June 2022).

5. Open call for a post-doc position (deadline: 18 February 2022). Check all information in https://www.euraxess.pt/jobs/735076 or https://sigarra.up.pt/fcup/pt/noticias_geral.ver_noticia?p_nr=84333. 

6. Starting date of the Project: 17 January 2022.