Geometry of Algebraic Varieties

PTDC/MAT-GEO/2823/2014 (01/06/2016-30/11/2019)

Research project funded by FCT (Portugal) based at the Centre of Mathematics of the University of Porto.

Summary

Algebraic Geometry is an old subject in mathematics and, at the same time, a vibrant area of current research with close connections to other areas. Its main objects of study are algebraic varieties which means, roughly speaking, zero sets of of polynomials. For example, an algebraic curve is a 1-dimensional algebraic variety, and an algebraic surface is a 2-dimensional algebraic variety.

Algebraic invariants of varieties provide us with a way of understanding how these geometric objects look, in a sense. These invariants can be of both topological and geometric nature and, generally, associate numbers to geometric phenomena.

From this point of view the the main objectives of the project are two-fold: on the one hand, we try to understand if algebraic varieties with given numerical invariants exist and, on the other hand, given an algebraic variety, we wish to understand its numerical invariants.

With respect to the first objective, our focus is mainly on algebraic surfaces. A classification of these does exist but, nevertheless, there are many open problems concerning both existence of certain types of surface and also their geometric properties, which we shall throw light on.

With respect to the second objective we focus on moduli spaces of Higgs bundles and quiver bundles. These are central objects in current geometry: indeed Higgs bundle moduli play an important role in hot topics like the geometric Langlands programme and mirror symmetry. Moreover, they are related to character varieties for surface groups through the Non-abelian Hodge Theorem. We shall improve the understanding of the geometry and topology of these moduli spaces and explore their relevance in the aforementioned contexts.

Final report

The main objectives described above have been fully realised. A detailed description of the scientific results of the project is avaliable here.

Publications and preprints

  1. A. Araújo, Generalized quivers, orthogonal and symplectic representations, and Hitchin-Kobayashi correspondences, International Journal of Mathematics 30 (2019), no. 3, 1850085, 46 pp. doi:10.1142/S0129167X18500854. arXiv:1703.10386 [math.AG].
  2. M. Aparicio-Arroyo, S. Bradlow, B. Collier, O. Garcia-Prada, P. B. Gothen and A. Oliveira, SO(p,q)-Higgs bundles and higher Teichmüller components, Inventiones Mathematicae, 218 (2019), 197-299. doi:10.1007/s00222-019-00885-2. arXiv:1802.08093 [math.AG].
  3. M. Aparicio-Arroyo, S. Bradlow, B. Collier, O. Garcia-Prada, P. B. Gothen and A. Oliveira, Exotic components of SO(p,q) surface group representations, and their Higgs bundle avatars, C. R. Acad. Sci. Paris, Ser. I 356 (2018), 666-673. doi:10.1016/j.crma.2018.04.024. arXiv:1801.08561 [math.AG].
  4. I. Biswas, L. A. Calvo, E. Franco and O. García-Prada, Involutions of Higgs moduli spaces over elliptic curves and pseudo-real Higgs bundles, J. Geom. Phys. 142 (2019), 47-65. doi:10.1016/j.geomphys.2019.03.014. arXiv:1612.08364 [math.AG].
  5. S. B. Bradlow, O. Garcia-Prada, P. B. Gothen and J. Heinloth, Irreducibility of moduli of semistable Chains and applications to U(p,q)-Higgs bundles, Geometry and Physics: Volume 2, A Festschrift in honour of Nigel Hitchin, Oxford University Press, 2018, pp. 455-470. doi:10.1093/oso/9780198802020.003.0018. arXiv:1703.06168 [math.AG].
  6. A. Castorena, M. Mendes Lopes and G. P. Pirola, Semistable fibrations over an elliptic curve with only one singular fibre, Osaka J. Math 57 (2020), 9-15. arXiv:1707.08671 [math.AG].
  7. E. Dias, On the Graded Equations of (1,3)-Abelian Surfaces, arXiv:1911.02315 [math.AG].
  8. E. Dias, C. Rito and G. Urzúa, On degenerations of Z/2-Godeaux surfaces, arXiv:2002.08836 [math.AG].
  9. C. Florentino, P. B. Gothen and A. Nozad, Homotopy type of moduli spaces of G-Higgs bundles and reducibility of the nilpotent cone, Bulletin des Sciences Mathématiques, 150 (2019), 84-101. doi:10.1016/j.bulsci.2018.10.002. arXiv: 1805.10081[math.AG].
  10. E. Franco and M. Jardim, Mirror symmetry for Nahm branes, arXiv:1709.01314 [math.AG].
  11. E. Franco, P. B. Gothen, A. Oliveira and A. Peón-Nieto, Unramified covers and branes on the Hitchin system, arXiv:1802.05237 [math.AG].
  12. E. Franco and A. Peón-Nieto, The Borel subgroup and branes on the Higgs moduli space, arXiv:1709.03549 [math.AG].
  13. O. Garcia-Prada and A. Oliveira, Maximal Higgs bundles for adjoint forms via Cayley correspondence, Geometriae Dedicata 190 (2017), 1-22. doi:10.1007/s10711-017-0224-2. arXiv:1612.06621 [math.AG].
  14. C. Gleissner, R. Pignatelli and Carlos Rito, New surfaces with canonical map of high degree, Commun. Anal. Geom., to appear. arXiv:1807.11854 [math.AG].
  15. P. B. Gothen and A. Nozad, Quiver Bundles and Wall Crossing for Chains, Geometriae Dedicata, 199 (2019), 137-146. doi:10.1007/s10711-018-0341-6. arXiv:1709.09581 [math.AG].
  16. P. B. Gothen and A. Oliveira, Topological mirror symmetry for parabolic Higgs bundles, Journal of Geometry and Physics, 137 (2019), 7-34. doi:10.1016/j.geomphys.2018.08.020. arXiv:1707.08536 [math.AG].
  17. P. B. Gothen and R. A. Zúñiga-Rojas, Stratifications on the Moduli Space of Higgs Bundles, Portugaliae Mathematica 74 (2017), 127-148. doi:10.4171/PM/1996. arXiv:1511.03985 [math.AG].
  18. P. B. Gothen and A. Nozad, Birationality of moduli spaces of twisted U(p,q)-Higgs bundles, Revista Matemática Complutense 30 (2017), 91-128. doi:10.1007/s13163-016-0207-0. arXiv:1602.02712 [math.AG].
  19. P. B. Gothen, Hitchin Pairs for non-compact real Lie groups, Travaux Mathématiques 24 (2016), 183-200, Special issue based on School GEOQUANT at the ICMAT Madrid, Spain, September 2015. arXiv:1607.08150 [math.AG].
  20. V. Koziarz, C. Rito and X. Roulleau, The Bolza curve and some orbifold ball quotient surfaces, Michigan Mathematical Journalc. arXiv:1904.00793 [math.AG].
  21. A. Oliveira, Quadric bundles applied to non-maximal Higgs bundles, Travaux Mathématiques 24 (2016), 201-220, Special issue based on School GEOQUANT at the ICMAT Madrid, Spain, September 2015. arXiv:1608.00423 [math.AG].
  22. A. Oliveira, Quadric bundles and their moduli spaces, arXiv:1610.05568 [math.AG].
  23. F. Polizzi, C. Rito, X. Roulleau, A pair of rigid surfaces with p_g=q=2 and K^2=8 whose universal cover is not the bidisk, International Mathematics Research Notices, rny107 (published online: 05 June 2018). arXiv:1703.10646 [math.AG].
  24. C. Rito, On the construction of complex algebraic surfaces, CIM Bulletin 38-39 (2017), 3-7.
  25. C. Rito, Surfaces with canonical map of maximum degree, Journal of Algebraic Geometry (accepted). arXiv:1903.03017 [math.AG]
  26. C. Rito, X. Roulleau, A. Sarti, Explicit Schoen surfaces, Algebraic Geometry 6 (2019), 410-426. arXiv:1609.02235 [math.AG].
  27. J. A. M. Silva, Hodge-Deligne Polynomials of Symmetric Products of Linear Algebraic Groups, arXiv:1812.05871 [math.AG].

Project members (including grant holders)

Outreach activity

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Last modified: Fri May 15 18:17:48 WEST 2020