Semiorthogonal decompositions in families

  • Alexander Kuznetsov

    Algebraic Geometry Section, Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkin str., Moscow 119991, Russia
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This book chapter is published open access.


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.