SISSA Algebraic Geometry Seminar 2022–2023
We will start with an introduction to triangulated categories and derived functors following D. Huybrechts' book Fourier–Mukai transforms in Algebraic Geometry, then we will proceed towards Bridgeland stability conditions following (probably a subset of) the following references:
- T. Bridgeland, Stability conditions on triangulated categories, Annals of Mathematics, 166 (2007), 317–345.
- D. Huybrechts, Introduction to stability conditions.
- E. Macrì, Stability conditions on curves, Math. Res. Lett. 14 (2007), no. 4, 657–672.
- A. Bayer, A tour to stability conditions on derived categories.
- Monday, 24 October 2022, Mohamed Aliouane (SISSA): Introduction to abelian and triangulated categories.
- Monday, 7 November 2022, Qiangru Kuang (SISSA): Derived categories and derived functors.
- Monday, 14 November 2022, Elisa Vitale (SISSA): Derived category of coherent sheaves.
- Monday, 21 November 2022, Hayato Morimura (SISSA): Bondal–Orlov Reconstruction Theorem.
- Monday, 28 November 2022, Warren Cattani (SISSA): The derived category of projective space.
Monday, 12 December 2022,
Michele Graffeo (SISSA): Dynamics of some birational maps of the projective 3-space.
Abstract. The standard Cremona transformation of the projective space is a classical object in algebraic geometry. In a joint work with G. Gubbiotti (University of Milan), we studied the algebraic entropy and the invariants of birational maps defined as the composition of the standard Cremona transformation with some special projectivities. More precisely, we consider projectivities acting on 12 points of the 3-dimensional projective space in Reye configuration. These kind of maps appear for instance in the Kahan–Hirota–Kimura discretisation of the Euler top.
- Monday, 16 January 2023, Sohaib Khalid (SISSA): Fourier–Mukai functors.
- Monday, 23 January 2023, Riccardo Ontani (SISSA): Stability conditions on abelian categories. 11–12:30 in Room 136.
- Monday, 30 January 2023, Matteo Montagnani (SISSA): Stability conditions on triangulated categories. 11–12:30 in Room 136.
- Monday, 6 February 2023, Ajay Gautam (SISSA): Bridgeland's Deformation Theorem.
- Monday, 20 February 2023, Qiangru Kuang (SISSA): Bridgeland stability conditions on curves.
Monday, 6 March 2023,
Emanuele Macrì (Université Paris-Saclay): Stability conditions on Hilbert schemes of points on K3 surfaces.
Abstract. I will present joint work with Chuyni Li, Paolo Stellari and Xiaolei Zhao on invariant stability conditions on product varieties. In particular, this provides examples of stability conditions on Hilbert schemes of points on K3 surfaces.
- Monday, 13 March 2023, Dmitrii Rachenkov (SISSA): Examples of wall-crossing.
Monday, 20 March 2023,
Emanuele Pavia (SISSA):
Mixed graded structures on Chevalley Eilenberg complexes.
Abstract. In classical differential geometry, one of the most important algebraic objects associated to the Lie algebra g of a Lie group G is the (cohomological) Chevalley-Eilenberg complex C•(g), which computes the Lie algebra cohomology of g and, under some suitable assumptions, the de Rham cohomology of G as well. In the setting of formal geometry in characteristic 0, the correspondence between Lie groups and Lie algebras can be interpreted in terms of an equivalence of ∞-categories between formal moduli problems and homotopy Lie algebras; in particular, it is natural to expect the Chevalley-Eilenberg complex to be defined for any homotopy Lie algebra, while faithfully retaining the same amount of information as in the classical case. Unfortunately, the standard definition of the Chevalley-Eilenberg complex for a homotopy Lie algebra is too rough, since it does not even distinguish between non-quasi-isomorphic Lie algebras with same Lie algebra cohomology: instead, one needs to consider the Chevalley-Eilenberg complex as an object in the derived filtered category. In this talk, we will see how to interpret the datum of the filtration on a chain complex over a ring of characteristic 0 in terms of a mixed graded structure in the sense of Pantev, Toën, Vaquié and Vezzosi, and describe how the Chevalley-Eilenberg functor can be refined to a mixed graded enhancement. The construction is completely ∞-categorical in nature, and does not require the choice of a presentation of homotopy Lie algebras in terms of differential graded Lie algebras or L∞-algebras.