Geometry of Algebraic Varieties

PTDC/MAT-GEO/2823/2014

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.

Publications and preprints

  1. A. Araújo, The moduli space of generalized quivers, International Journal of Mathematics, to appear. 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, 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. 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, to appear in the proceedings for the conference Hitchin 70. arXiv:1703.06168 [math.AG].
  5. A. Castorena, M. Mendes Lopes and G. P. Pirola, Semistable fibrations over an elliptic curve with only one singular fibre arXiv:1707.08671 [math.AG].
  6. C. Florentino, P. B. Gothen and A. Nozad, Homotopy type of moduli spaces of G-Higgs bundles and reducibility of the nilpotent cone, arXiv: 1805.10081[math.AG].
  7. E. Franco and M. Jardim, Mirror symmetry for Nahm branes, arXiv:1709.01314 [math.AG].
  8. E. Franco, P. B. Gothen, A. Oliveira and A. Peón-Nieto, Torsion line bundles and branes on the Hitchin system, arXiv:1802.05237 [math.AG].
  9. E. Franco and A. Peón-Nieto, The Borel subgroup and branes on the Higgs moduli space, arXiv:1709.03549 [math.AG].
  10. O. Garcia-Prada and A. Oliveira, Maximal Higgs bundles for adjoint forms via Cayley correspondence, Geometriae Dedicata 190 (2017), 1–22. 10.1007/s10711-017-0224-2. arXiv:1612.06621 [math.AG].
  11. P. B. Gothen and A. Nozad, Quiver Bundles and Wall Crossing for Chains, Geometriae Dedicata (in press, 2018). doi:10.1007/s10711-018-0341-6. arXiv:1709.09581 [math.AG].
  12. P. B. Gothen and A. Oliveira, Topological mirror symmetry for parabolic Higgs bundles, arXiv:1707.08536 [math.AG].
  13. 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].
  14. 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].
  15. 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].
  16. 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].
  17. A. Oliveira, Quadric bundles and their moduli spaces, arXiv:1610.05568 [math.AG].
  18. 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, Int. Math. Res. Not. IMRN (accepted). arXiv:1703.10646 [math.AG].
  19. C. Rito, X. Roulleau, A. Sarti, Explicit Schoen surfaces, Algebraic Geometry (accepted). arXiv:1609.02235 [math.AG].

Project members

Grant holders (concluded)

Outreach activity


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Last modified: Mon Jun 25 09:14:19 WEST 2018