Homological projective duality

By Alexander Kuznetsov. 2005

237 citations recorded by the source corpus.

We introduce a notion of homological projective duality for smooth algebraic varieties in dual projective spaces, a homological extension of the classical projective duality. If algebraic varieties X and Y in dual projective spaces are homologically projectively dual, then we prove that the orthogonal linear sections of X and Y admit semiorthogonal decompositions with an equivalent nontrivial component. In particular, it follows that triangulated categories of singularities of these sections are equivalent. We also investigate homological projective duality for projectivizations of vector bundles.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0b5ed92ecf29208d26e7fbb7035c70c9a791ae19

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Variation of geometric invariant theory quotients and derived categories (2012, methodology): “rweaving of Isik/Shipman’s result with Theorem 1.0.2, yields both the commutative version of Orlov’s σ-model/LGcorrespondence, [Orl09], and a framework for Kuznetsov’s Homological Projective Duality, [Kuz07]. The class of smooth and projective Deligne-Mumford stacks provides a natural setting where ”
Non-commutative resolutions of quotient singularities for reductive groups (2015, unknown): “Amongother thingsNCRs are a crucial part of theKuznetsov’sHomological Projective Duality program [31,32].”
Aspects of non-abelian gauge dynamics in two-dimensional Script N = (2,2) theories (2006, methodology): “A mathematical proof of the derived equivalence has been given by A. Kuznetsov in [40] as an application of his Homological Projective Duality [41].”
Duality in two-dimensional (2,2) supersymmetric non-Abelian gauge theories (2011, unknown): “A point of view that seems to underlie all of these equivalences is projective duality and its categorical counterpart proposed by A. Kuznetsov, called Homological Projective Duality [41].”
Semiorthogonal decompositions in algebraic geometry (2014, background): “ Y . We will say that XL and YL have expected dimensions if dimXL = dimX −r and dimYL = dimY −(N − r), where N = dimV and r = dimL (so that N −r = dimL⊥). Theorem 2.5 (Homological projective duality, [K07a]). Let (Y,g) be an HP dual variety for (X,f) with respect to (18). Then 1. Y is smooth and Db(”
Floer cohomology and pencils of quadrics (2010, methodology): “There are equivalences between the Fukaya category of a space in the top line, and the nilpotent summand of the Fukaya category of the space below it, in each of the first 3 columns, mirror to a small part of Kuznetsov’s “homological projective duality” [58].”
Hochschild homology and semiorthogonal decompositions (2009, unknown): “t ξ: A → B be an equivalence and consider the functor Φ : Db(X) → Db(Y), Φ = β◦ ξ◦ α∗, where α: A → Db(X) and β: B → Db(Y) are the embedding functors. Note that this is a right splitting functor (see [K2]). In particular, ΦΦ! ∼= ββ! is the right projection onto B, while Φ!Φ ∼= αα∗ is the left projec”
Triangulated categories of singularities and equivalences between Landau-Ginzburg models (2005, background): “of more general situation which will be described in the paper [ 15 ].”
Lefschetz decompositions and categorical resolutions of singularities (2006, background): “decomposition. Lefschetz decompositions were introduced in [ 14 ] in the context of homological projective duality.”
Derived categories of coherent sheaves (2006, background): “n classical algebraic geometry. The theory also leads to some extremely interesting derived equivalences between non-commutative varieties. For more details we refer the reader to the original papers [31, 32, 33, 34]. Acknowledgements. I’m very grateful to Antony Maciocia for teaching me about deriv”