The fundamental theorem of homological projective duality via variation of GIT stability

Abstract

We reprove Kuznetsov's "fundamental theorem of homological projective duality" using LG models and variation of GIT stability. This extends the validity of the theorem from smooth varieties to nice subcategories of smooth quotient stacks, and moreover shows that the line bundles polarising the varieties in HP duality do not have to be globally generated.The proof combines ideas from Kuznetsov's original proof with ideas from work of Ballard-Deliu-Favero-Isik-Katzarkov.

Hybrid models for homological projective duals and noncommutative resolutions

“…In the following we assume that the aforementioned geometric phase is a pure Higgs phase 4 and its category of B-branes will be denoted by D(X ζ L ≫1 ), if this phase is located at ζ L ≫ 1. As we vary the parameter ζ L we find, in general that the phase at ζ L ≪ −1 has a Higgs branch whose category of B-branes we denote 5…”

“…Recently, it is found that homological projective dual (HPD) [3] of certain projective embeddings can be described by hybrid models. This was found in mathematics [4,5] and a physics formulation is presented in 1 [7]: if a GLSM T X for a projective morphism f : X → P(V ) is known, one can build up an extended GLSM T X such that the Higgs branch of one of its phases gives rise to the HPD of f : X → P(V ). In the abelian cases, this Higgs branch is a hybrid model.…”

“…This is always possible in the case X is Fano or Calabi-Yau (CY), wich are essentially the cases we will cover in this work. The generalization is straightforward 5. The spaceD(Y ζ L ≪−1 , W ζ L ≪−1 )denotes the category of B-branes of a hybrid Landau-Ginzburg (LG) model with target space Y ζ L ≪−1 and superpotential W ζ L ≪−1 .…”

Noncommutative homological projective duality

“…Finally, in [23] Rennemo builds on [2] to develop a version of HPD for categories C that are admissible subcategories in the derived category of a smooth quotient stack.…”

Derived category of projectivization and generalized linear duality

“…If L is locally free, then the "correction terms" A i = ∅, and the theorem reduces to the usual fundamental theorem of HPD (see [K07,JLX17,R17,P18]).…”