2018 Intercity Geometry Seminar

The spring 2018 Intercity Geometry Seminar will be held around Mirror Symmetry and moduli spaces of Higgs bundles. We will be following a recent preprint of Groechenig, Wyss and Ziegler in which they prove a conjecture of Hausel and Thaddeus on the Hodge numbers of these moduli spaces. Announcements will be sent out via the am-l mailing list. We will be posting an outline of the seminar below, which will be updated and adjusted as we progress. If you want to volunteer a presentation, please contact David Holmes, Arne Smeets or Chris Lazda.

March 9 - Nijmegen

Lectures all take place in Room HG00.068 of the Huygens Building.

 13:00 - 14:00 Andrey Soldatenkov (Bonn) Introduction to the Hausel-Thaddeus conjecture Abstract: Calabi-Yau varieties, special Lagrangian subvariaties and the SYZ fibration. Mirror partners via torus fibrations and hyperkähler rotations. Expected relationship between Hodge numbers, outline of the goal of the seminar - produce a non-trivial example of an SYZ mirror pair, and show that the Hodge numbers behave as expected. References: [HT03].
 14:30 - 15:30 Arne Smeets (Nijmegen) $$p$$-adic integration and point counting Abstract: $$p$$-adic manifolds, gauge forms, induced measures and $$p$$-adic integration. Structure of residue discs for analytic spaces with good reduction, and relationship with point-counting over the residue field. References: [Igu00] §2,7 [Wei82].
 16:00 - 17:00 Johan Commelin (Utrecht) Birational Calabi-Yau varieties have the same Betti numbers Abstract: The Weil conjectures and the link between point-counting and Betti numbers. Batyrev's proof that birational Calabi-Yau's have equal Betti numbers via $$p$$-adic integration. Ito's refinement via $$p$$-adic Hodge theory to show an equality of Hodge numbers. References: [Bat99a], [Ito04], [HRV08] Appendix.

April 13 - Leiden

Lectures all take place in Room 407-409 of the Snellius Building. Videos for some of the talks can be found here.

 13:00 - 14:00 Wessel Bindt (Amsterdam) Higgs bundles on curves Abstract: Definition of a Higgs bundle, Higgs bundles with structure groups $$\mathrm{SL_n}$$ and $$\mathrm{PGL}_n$$. Families of Higgs bundles, the moduli stack and coarse moduli space of Higgs bundles. Relationship between the $$\mathrm{SL_n}$$ and $$\mathrm{PGL}_n$$ moduli spaces. References: [Sim94a], [Sim94b], [ACV03] §5.
 14:30 - 15:30 Mingmin Shen (Amsterdam) Non-abelian Hodge theory Abstract: De Rham, Betti and Dolbeault moduli spaces for smooth projective varieties over $$\mathbb{C}$$. Complex analytic isomorphism between the de Rham and Betti moduli spaces, and homeomorphism between the Betti and Dolbeailt moduli spaces. Hyperkähler structure on the moduli space. References: [Sim94a], [Sim94b], [Sim97].
 16:00 - 17:00 Wim Veys (Leuven) Stringification of Hodge Numbers Abstract: Hodge numbers and E-polynomials, string theoretic E-polynomials of toric and finite quotient singularities. Stringy E-polynomials via resolutions and log discrepancies. Properties of stringy E-polynomials and stringy Hodge numbres. References: [BD96], [Bat99b], [Bat99c].

May 4 - Amsterdam

Lectures all take place in Room A1.04 at Science Park 904. Videos for the talks can be found here here.

 13:00 - 14:00 Bas Edixhoven (Leiden) Gerbes and stringy point counting Abstract: Finite quotient stacks and their inertia stacks, gerbes and transgression. Stringy point counts and the relationship with stringy Hodge numbers. References: [GWZ17] §2.
 14:30 - 15:30 Peter Bruin (Leiden) Arithmetic duality for abelian varieties Abstract: Brauer groups of local fields and the Hasse invaraint. Local duality for abelian varieties. References: [GWZ17] §2, [Mil86] §I.3.
 16:00 - 17:00 Chris Peters (Grenoble / Eindhoven) The Hitchin fibration Abstract: The Hitchin base and the map from the moduli space of Higgs bundles. Spectral curves and the good locus of the Hitchin base. Prym varieties and description of the fibres over the good locus. References: [Hau13], [Hit87], [Nit91].

May 25 - Utrecht

Lectures all take place in Room MIN 2.02 of the Minnaert Building. Videos for the talks can be found here here.

 13:00 - 14:00 David Holmes (Leiden) The Hitchin fibration revisited and abstract dual Hitchin systems Abstract: The $$\mu_n$$-gerbes $$\alpha_{\mathrm{SL}_n}$$ and $$\alpha_L$$ living on the $$\mathrm{SL}_n$$ and $$\mathrm{PGL}_n$$ moduli spaces. Splitting varieties of gerbes and the pair $$\left( \mathrm{M}^L_{\mathrm{SL}_n} ,\alpha_{\mathrm{SL}_n}^e \right)$$,$$\left(\underline{\mathrm{M}}_{\mathrm{PGL}_n}^e,\alpha_L^d \right)$$ as a dual pair of abstract Hitchin systems. References: [HT03] §3, [GWZ17] §§4-5.
 14:30 - 15:30 Chris Lazda (Amsterdam) $$p$$-adic integration of the Hasse invariant Abstract: Transgression and $$p$$-adic integration on finite quotient stacks. Stringy point counting via $$p$$-adic integration, and description in terms of the Hasse invariant and the arithmetic duality pairing for abelian varieties. References: [GWZ17] §3, Appendix A.
 16:00 - 17:00 Dimitri Wyss (Jussieu) Completion of the proof Abstract: Fibrewise equality of $$p$$-adic integrals over the good locus on dual pairs of abstract Hitchin systems. Equality of stringy Hodge numbers $$h^{p,q}(\mathrm{M}^L_{\mathrm{SL}_n}) = h^{p,q}_\mathrm{st}(\underline{\mathrm{M}}^e_{\mathrm{PGL}_n},\alpha_L^d)$$. References: [GWZ17] §4.

References

[ACV03] Abramovich, Corti, Vistoli - Twisted Bundles and Admissible Covers.

[Bat99a] Batyrev - Birational Calabi-Yau $$n$$-folds have equal Betti numbers.

[Bat99b] Batyrev - Stringy Hodge numbers of varieties with Gorenstein canonical singularities.

[Bat99c] Batyrev - Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs.

[BD96] Batyrev, Dais - Strong McKay correspondence, string-theoretic Hodge numbers and mirror symmetry.

[GWZ17] Groechenig, Wyss, Ziegler - Mirror symmetry for moduli spaces of Higgs bundles via $$p$$-adic integration.

[Hau13] Hausel - Global topology of the Hitchin system.

[HT03] Hausel, Thaddeus - Mirror symmetry, Langlands duality, and the Hitchin system.

[Hit87] Hitchin - Stable bundles and integrable systems.

[Igu00] Igusa - An introduction to the Theory of Local Zeta Functions.

[Mil86] Milne - Arithmetic Duality Theorems.

[Nit91] Nitsure - Moduli Space of Semistable Pairs on a Curve.

[Sim94a] Simpson - Moduli of representations of the fundamental group of a smooth projective variety I.

[Sim94b] Simpson - Moduli of representations of the fundamental group of a smooth projective variety II.

[Sim97] Simpson - The Hodge filtration on nonabelian cohomology.

[Wei82] Weil - Adeles and Algebraic Groups.