Homological projective duality

Abstract

Abstract. 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.

GLSMs for partial flag manifolds

“…In this paper and [9], we would like to propose that these non-birational phases are related by Kuznetsov's "homological projective duality" [6,7,8]. Put another way, here and in [9] we propose that gauged linear sigma models implicitly give a physical realization of Kuznetsov's homological projective duality.…”

“…The correct dual [6,28] [8,24]), so there was no 'noncommutative space' to consider. However, for G(2, N) [1 N ] for N even, the derived category of the original space matches only a subcategory of that of the hypersurface in the Pfaffian.…”

Triangulated categories of singularities and equivalences between Landau-Ginzburg models

“…This kind of semiorthogonal decomposition is a particular case of more general situation which will be described in the paper [15].…”

Some recent advances in gauged linear sigma models

“…Instead of being related by a birational transformation, these phases are instead related by a newer notion, namely Kuznetsov's homological projective duality [21,22,23]. Describing homological projective duality in detail is beyond the scope of these lectures, but we can observe briefly that not only the novel examples discussed above but also more traditional GLSM phases [17,18,19,20] naturally fit into that framework, and so many now believe that all GLSM phases are related by homological projective duality.…”

“…[2,3,4,5,6,7,8,9,10] and references therein) to new developments in Calabi-Yau compactifications, including realizations of non-complete-intersection target spaces [11,12,13,14,15], non-birational phases [11,12,15,16,17,18,19,20] (leading to modern conjectures that all phases of GLSM's are related by a new notion known as "homological projective duality" [17,18,19,20,21,22,23]), nonperturbative realizations of geometry [11,12,15,16,24,25,26], realizations of noncommutative resolutions [12,16,25,26], and localization techniques [27,28]. Those localization techniques have been applied to deduce new methods for computing Gromov-Witten invariants [24,29], computations of elliptic genera directly in GLSM's [30,31,…”