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.
We provide a geometric approach to constructing Lefschetz collections and Landau-Ginzburg Homological Projective Duals from a variation of Geometric Invariant Theory quotients. This approach yields homological projective duals for Veronese embeddings in the setting of Landau Ginzburg models. Our results also extend to a relative Homological Projective Duality framework.
We generalize Kuznetsov's theory of homological projective duality to the setting of noncommutative algebraic geometry. Simultaneously, we develop the theory over general base schemes, and remove the usual smoothness, properness, and transversality hypotheses.
We show that over an algebraically closed field of characteristic not equal to 2, homological projective duality for smooth quadric hypersurfaces and for double covers of projective spaces branched over smooth quadric hypersurfaces is a combination of two operations: one interchanges a quadric hypersurface with its classical projective dual and the other interchanges a quadric hypersurface with the double cover branched along it.
We introduce the notion of a categorical cone, which provides a categorification of the classical cone over a projective variety, and use our work on categorical joins to describe its behavior under homological projective duality. In particular, our construction provides well-behaved categorical resolutions of singular quadrics, which we use to obtain an explicit quadratic version of the main theorem of homological projective duality. As applications, we prove the duality conjecture for Gushel-Mukai varieties, and produce interesting examples of conifold transitions between noncommutative and …
We provide homological foundations to establish conjectural homological projective dualities between 1) S^2 P^3 and the double cover of the projective 9-space branched along the symmetric determinantal quartic, and 2) S^2 P^4 and the double cover of the symmetric determinantal quintic in the projective 14-space branched along the symmetric determinantal locus of rank at most 3.
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.
Homological Projective duality (HP-duality) theory, introduced by Kuznetsov [42], is one of the most powerful frameworks in the homological study of algebraic geometry. The main result (HP-duality theorem) of the theory gives complete descriptions of bounded derived categories of coherent sheaves of (dual) linear sections of HP-dual varieties. We show the theorem also holds for more general intersections beyond linear sections. More explicitly, for a given HP-dual pair $(X,Y)$, then analogue of HP-duality theorem holds for their intersections with another HP-dual pair $(S,T)$, provided that th…
The Segre cubic and Castelnuovo-Richmond quartic are two projectively dual hypersurfaces in $\mathbb{P}^4$, with a long and rich history starting in the 19th century. We will explain how Kuznetsov's theory of homological projective duality lifts this projective duality to a relationship between the derived category of a small resolution of the Segre cubic and a small resolution of the Coble fourfold, the double cover of $\mathbb{P}^4$ ramified along the Castelnuovo-Richmond quartic. Homological projective duality then provides a description of the derived categories of linear sections, which w…
We introduce the notion of a categorical join, which can be thought of as a categorification of the classical join of two projective varieties. This notion is in the spirit of homological projective duality, which categorifies classical projective duality. Our main theorem says that the homological projective dual category of the categorical join is naturally equivalent to the categorical join of the homological projective dual categories. This categorifies the classical version of this assertion and has many applications, including a nonlinear version of the main theorem of homological projec…
In this paper, we first introduce geometric operations for linear categories, and as a consequence generalize Orlov's blow up formula [O04] to possibly singular local complete intersection centres. Second, we introduce refined blowing up of linear category along base--locus, and show that this operation is dual to taking linear section. Finally, as an application we produce examples of Calabi--Yau manifolds which admits Calabi--Yau categories fibrations over projective spaces.
We show that homologically projectively dual varieties for Grassmannians Gr(2,6) and Gr(2,7) are given by certain noncommutative resolutions of singularities of the corresponding Pfaffian varieties. As an application we describe the derived categories of linear sections of these Grassmannians and Pfaffians. In particular, we show that (1) the derived category of a Pfaffian cubic 4-fold admits a semiorthogonal decompositions consisting of 3 exceptional line bundles, and of the derived category of a K3-surface; (2) mutually orthogonal Calabi-Yau linear sections of Gr(2,7) and of the correspondin…
We show how blowing up varieties in base loci of linear systems gives a procedure for creating new homological projective duals from old. Starting with a HP dual pair $X,Y$ and smooth orthogonal linear sections $X_L,Y_L$, we prove that the blowup of $X$ in $X_L$ is naturally HP dual to $Y_L$. The result does not need $Y$ to exist as a variety, i.e. it may be "noncommutative". We extend the result to the case where the base locus $X_L$ is a multiple of a smooth variety and the universal hyperplane has rational singularities; here the HP dual is a categorical resolution of singularities of $Y_L$…
We study the derived category of a complete intersection X of bilinear divisors in the orbifold Sym^2 P(V). Our results are in the spirit of Kuznetsov's theory of homological projective duality, and we describe a homological projective duality relation between Sym^2 P(V) and a category of modules over a sheaf of Clifford algebras on P(Sym^2 V^vee). The proof follows a recently developed strategy combining variation of GIT stability and categories of global matrix factorisations. We begin by translating D^b(X) into a derived category of factorisations on an LG model, and then apply VGIT to obta…
Homological mirror symmetry for crepant resolutions of Gorenstein toric singularities leads to a pair of conjectures on certain hypergeometric systems of PDEs. We explain these conjectures and verify them in some cases.
We introduce the notion of a tropical coamoeba which gives a combinatorial description of the Fukaya category of the mirror of a toric Fano stack. We show that the polyhedral decomposition of a real n-torus into (n + 1) permutohedra gives a tropical coamoeba for the mirror of the projective space, and prove a torus-equivariant version of homological mirror symmetry for the projective space. As a corollary, we obtain homological mirror symmetry for toric orbifolds of the projective space.
In this paper we prove Homological Projective Duality for crepant categorical resolutions of several classes of linear determinantal varieties. By this we mean varieties that are cut out by the minors of a given rank of a n x m matrix of linear forms on a given projective space. As applications, we obtain pairs of derived-equivalent Calabi-Yau manifolds, and address a question by A. Bondal asking whether the derived category of any smooth projective variety can be fully faithfully embedded in the derived category of a smooth Fano variety. Moreover we discuss the relation between rationality an…
We obtain homological properties of the second symmetric product of P^4 and the double cover of the symmetric determinantal quintic hypersurface in P^{14} (the double quintic symmetroids), which indicate the homological projective duality between (suitable noncommutative resolutions of) them. Among other things, we construct their good desingularizations and also (dual) Lefschetz collections in the derived categories of the desingularizations. These are expected to give (dual) Lefschetz decompositions of suitable noncommutative resolutions. The desingularization of the double quintic symmetroi…
In this paper, we apply the idea of T-duality to projective spaces. From a connection on a line bundle on $\mathbb P^n$, a Lagrangian in the mirror Landau-Ginzburg model is constructed. Under this correspondence, the full strong exceptional collection $\mathcal O_{\mathbb P^n}(-n-1),...,\mathcal O_{\mathbb P^n}(-1)$ is mapped to standard Lagrangians in the sense of \cite{nz}. Passing to constructible sheaves, we explicitly compute the quiver structure of these Lagrangians, and find that they match the quiver structure of this exceptional collection of $\mathbb P^n$. In this way, T-duality prov…
Given a gauged linear sigma model (GLSM) $\mathcal{T}_{X}$ realizing a projective variety $X$ in one of its phases, i.e. its quantum Kähler moduli has a maximally unipotent point, we propose an \emph{extended} GLSM $\mathcal{T}_{\mathcal{X}}$ realizing the homological projective dual category $\mathcal{C}$ to $D^{b}Coh(X)$ as the category of B-branes of the Higgs branch of one of its phases. In most of the cases, the models $\mathcal{T}_{X}$ and $\mathcal{T}_{\mathcal{X}}$ are anomalous and the analysis of their Coulomb and mixed Coulomb-Higgs branches gives information on the semiorthogonal/L…
Classically, the projective duality between joins of varieties and the intersections of varieties only holds in good cases. In this paper, we show that categorically, the duality between joins and intersections holds in the framework of homological projective duality (HPD) [K07], as long as the intersections have expected dimensions. This result together with its various applications provide further evidences for the proposal of homological projective geometry of Kuznetsov and Perry [KP18]. When the varieties are inside disjoint linear subspaces, our approach also provides a new proof of the m…
We study hybrid models arising as homological projective duals (HPD) of certain projective embeddings $f:X\rightarrow\mathbb{P}(V)$ of Fano manifolds $X$. More precisely, the category of B-branes of such hybrid models corresponds to the HPD category of the embedding $f$. B-branes on these hybrid models can be seen as global matrix factorizations over some compact space $B$ or, equivalently, as the derived category of the sheaf of $\mathcal{A}$-modules on $B$, where $\mathcal{A}$ is an $A_{\infty}$ algebra. This latter interpretation corresponds to a noncommutative resolution of $B$. We compute…
We study Homological Mirror Symmetry (HMS) for $A_n$-resolutions from the SYZ viewpoint. Let $X\to\bC^2/\bZ_{n+1}$ be the crepant resolution of the $A_n$-singularity. The mirror of $X$ is given by a smoothing $\check{X}$ of $\bC^2/\bZ_{n+1}$. Using SYZ transformations, we construct a geometric functor from a derived Fukaya category of $\check{X}$ to the derived category of coherent sheaves on $X$. We show that this is an equivalence of triangulated categories onto a full triangulated subcategory of $D^b(X)$, thus realizing Kontsevich's HMS conjecture by SYZ.
We describe the Kuznetsov component of the Plücker embedding of the Grassmannian as a category of matrix factorizations on an noncommutative crepant resolution (NCCR) of the affine cone of the Grassmannian. We also extend this to a full homological projective dual (HPD) statement for the Plücker embedding. The first part is finding and describing the NCCR, which is also of independent interest. We extend results of Špenko and Van den Bergh to prove the existence of an NCCR for the affine cone of the Grassmannian. We then relate this NCCR to a categorical resolution of Kuznetsov. Deforming thes…
It is shown that projectivized irreducible components of nilpotent cones of complex symmetric spaces are projective self-dual algebraic varieties. Other properties equivalent to their projective self-duality are found.
We give a survey on results related to the Berglund-Hübsch duality of invertible polynomials and the homological mirror symmetry conjecture for singularities.
We prove Homological Mirror Symmetry for a smooth d-dimensional Calabi-Yau hypersurface in projective space, for any d > 2 (for example, d = 3 is the quintic three-fold). The main techniques involved in the proof are: the construction of an immersed Lagrangian sphere in the `d-dimensional pair of pants'; the introduction of the `relative Fukaya category', and an understanding of its grading structure; a description of the behaviour of this category with respect to branched covers (via an `orbifold' Fukaya category); a Morse-Bott model for the relative Fukaya category that allows one to make ex…
We introduce a natural symplectic structure on the moduli space of quadratic differentials with simple zeros and describe its Darboux coordinate systems in terms of so-called homological coordinates. We then show that this structure coincides with the canonical Poisson structure on the cotangent bundle of the moduli space of Riemann surfaces, and therefore the homological coordinates provide a new system of Darboux coordinates. We define a natural family of commuting "homological flows" on the moduli space of quadratic differentials and find the corresponding action-angle variables. The space …
A new section on projections of coherent sheaves from a projective space to a lower-dimensional projective space has been added. Also some of the notation has been altered to bring it into line with the joint paper with Eisenbud and Schreyer.
In this largely expository paper we extend properties of the homological duality functor $RHom_{\mathcal H}(-,{\mathcal H})$ where ${\mathcal H}$ is the Hecke algebra of a reductive $p$-adic group, to the case where it is the Hecke algebra of a finite central extension of a reductive $p$-adic group. The most important properties being that $RHom_{\mathcal H}(-,{\mathcal H})$ is concentrated in a single degree for irreducible representations and that it gives rise to Schneider--Stuhler duality for Ext groups (a Serre functor like property). Along the way we also study Grothendieck--Serre dualit…
We observe that linear relations among Chern-Mather classes of projective varieties are preserved by projective duality. We deduce the existence of an explicit involution on a part of the Chow group of projective space, encoding the effect of duality on Chern-Mather classes. Applications include Plücker formulae, constraints on self-dual varieties, generalizations to singular varieties of classical formulas for the degree of the dual and the dual defect, formulas for the Euclidean distance degree, and computations of Chern-Mather classes and local Euler obstructions for cones.
We use a generalization of Horrocks monads for arithmetic Cohen-Macaulay (ACM) varieties to establish a cohomological characterization of linear and Steiner bundles over projective spaces and quadric hypersurfaces. We also study resolutions of bundles on ACM varieties by line bundles, and characterize linear homological dimension in the case of quadric hypersurfaces.
Let $X_Σ$ be a complete toric variety. The coherent-constructible correspondence $κ$ of \cite{FLTZ} equates $\Perf_T(X_Σ)$ with a subcategory $Sh_{cc}(M_\bR;\LS)$ of constructible sheaves on a vector space $M_\bR.$ The microlocalization equivalence $μ$ of \cite{NZ,N} relates these sheaves to a subcategory $Fuk(T^*M_\bR;\LS)$ of the Fukaya category of the cotangent $T^*M_\bR$. When $X_\Si$ is nonsingular, taking the derived category yields an equivariant version of homological mirror symmetry, $DCoh_T(X_\Si)\cong DFuk(T^*M_\bR;\LS)$, which is an equivalence of triangulated tensor categories. Th…
The Beilinson-Bloch type conjectures predict that the low degree rational Chow groups of intersections of quadrics are one dimensional. This conjecture was proved by Otwinowska. Making use of homological projective duality and the recent theory of (Jacobians of) noncommutative Chow motives, we give an alternative proof of this conjecture in the case of a complete intersection of either two quadrics or three odd-dimensional quadrics. Moreover, without the use of the powerful Lefschetz theorem, we prove that in these cases the unique non-trivial algebraic Jacobian is the middle one. As an applic…
Given a smooth projective toric variety X, we construct an A-infinity category of Lagrangians with boundary on a level set of the Landau-Ginzburg mirror of X. We prove that this category is quasi-equivalent to the DG category of line bundles on X. This establishes part of the Homological Mirror Conjecture for toric varieties.
Let X be a smooth, linearly normal n dimensional complex projective variety. Assume that the projective dual of X has codimension one with defining polynomial D(X). In this paper the log of the norm of D(X) is expressed as the restriction to the Bergman metrics of an energy functional on X. We show how, for smooth plane curves, this energy functional reduces to the standard action functionals of Kahler geometry.
The present paper is an extension of a previous paper written in collaboration with Markus Reineke dealing with quiver representations. The aim of the paper is to generalize the theory and to provide a comprehensive theory of Donaldson-Thomas invariants for abelian categories of homological dimension one (without potential) satisfying some technical conditions. The theory will apply for instance to representations of quivers, coherent sheaves on smooth projective curves, and some coherent sheaves on smooth projective surfaces. We show that the (motivic) Donaldson-Thomas invariants satisfy the …
