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Semiorthogonal decompositions in families. (English) Zbl 07821736

Beliaev, Dmitry (ed.) et al., International congress of mathematicians 2022, ICM 2022, Helsinki, Finland, virtual, July 6–14, 2022. Volume 2. Plenary lectures. Berlin: European Mathematical Society (EMS). 1154-1200 (2023).
Summary: We discuss recent developments in the study of semiorthogonal decompositions of algebraic varieties with an emphasis on their behavior in families.
First, we overview new results concerning homological projective duality.
Then we introduce residual categories, discuss their relation to small quantum cohomology, and compute Serre dimensions of residual categories of complete intersections. After that we define simultaneous resolutions of singularities and describe a construction that works in particular for nodal degenerations of even-dimensional varieties.
Finally, we introduce the concept of absorption of singularities which works under appropriate assumptions for nodal degenerations of odd-dimensional varieties.
For the entire collection see [Zbl 1532.00036].

MSC:

18G80 Derived categories, triangulated categories
14D06 Fibrations, degenerations in algebraic geometry
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)

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