As the central focus of our group, Poisson structures form an essential class of geometric structures that come into contact with many other fields in mathematics. This semester we shall focus on the study of Poisson structures in the context of complex geometry and algebraic/analytic geometry.
The seminar will be following [Pym18] as the main source, supplemented by other references deemed important for the better understanding of the material. In the seminar, we will assume knowledge of Poisson geometry but we will start the seminar with a small crash-course-type talk on complex geometry following, mainly, [Huy05].
The seminar shall occur every Tuesday from 10h to 11:30h in Übungsraum 1 (-119) with the following tentative schedule:
13.10. (Rodrigo) An introduction to Complex Geometry and Analytic Varieties [Huy05, Ch. 2].
20.10. (Christoph) Holomorphic Poisson structures I: algebraic definitions [Pym13, Ch. 4, Pym18, §4.1–4.2].
27.10. (Jorn) Holomorphic Poisson structures II: the Poisson-Nijenhuis way [LSX08, §2–3].
03.11. (Matheus) Holomorphic Poisson surfaces I: local structure [Pym18, §2.1–2.3].
10.11. (Christoph) Holomorphic Poisson surfaces II: Ruled surfaces and their Poisson structures [Pym18, §2.4].
17.11. (Rodrigo) Holomorphic Poisson surfaces III: Blow-ups [Huy05, §2.5, Pym18, §2.5].
24.11. (Rodrigo) Holomorphic Poisson surfaces IV: Birational classification [Pym18, §2.6].
01.12. (Jorn) Holomorphic Poisson 3-manifolds I: regular Poisson structures and pencils of surfaces [Pym18, §3.1–3.2].
08.12. (Christoph) Holomorphic Poisson 3-manifolds II: the projective case [Pym13, Ch. 8, Pym18, §3.3–3.4].
15.12. (Rodrigo) Holomorphic Poisson structures in higher dimensions I: Poisson subspaces and Degeneracy loci [Pym13, §6.2–6.4, 6.8, Pym18, §4.3].
22.12. Early Christmas break! No talk this week.
12.01. (Rodrigo) Holomorphic Poisson structures in higher dimensions II: Poisson structures on projective spaces [Pym13, Ch. 7].
19.01. (Jorn) Holomorphic Poisson structures in higher dimensions III: Log-symplectic structures [Pym18, §5].
26.01. Talk by Aidan Lindberg (T.B.D.)
References
[Pym18] B. Pym, ‘Constructions and classifications of projective Poisson varieties’, Letters in Mathematical Physics, vol. 108, no. 3, pp. 573–632, Mar. 2018, doi: 10.1007/s11005-017-0984-5.
[Huy05] D. Huybrechts, Complex geometry. in Universitext. Springer-Verlag, Berlin, 2005.
[Pym13] B. Pym, Poisson Structures and Lie Algebroids in Complex Geometry. ProQuest LLC, Ann Arbor, MI, 2013, p. 186. [Online]. Available at: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_ fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:3666078
[LSX08] C. Laurent-Gengoux, M. Stiénon, and P. Xu, ‘Holomorphic Poisson manifolds and holomorphic Lie algebroids’, Int. Math. Res. Not. IMRN, pp. Art. ID rnn 088, 46, 2008, doi: 10.1093/imrn/rnn088.