Homological Projective Duality Research Articles

Derived equivalence and homological projective duality in GLSM

Brane transport provides a way to realize functors between the categories of B-branes of different phases in gauged linear sigma model (GLSM). When appropriately designed, these functors induce derived equivalence between Calabi–Yau manifolds. In some cases, brane transport can also be used to extract the homological projective dual (HPD) of certain projective embeddings. We describe the GLSMs realizing derived equivalence between different geometries via brane transport with emphasis on the nonabelian models, and how to construct the GLSMs where brane transport gives rise to the embedding of HPD category into the derived category of universal hyperplane section. The latter gives rise to an explanation of the relationship between GLSM and HPD, and also a noncommutative geometric interpretation to HPD.

HPD-invariance of the Tate conjecture(s)

We prove that the Tate conjecture (and its variants) is invariant under homological projective duality . As an application, we obtain a proof, resp. an alternative proof, of the Tate conjecture (and of its variants) in the new case of linear sections of determinantal varieties, resp. in the old cases of Pfaffian cubic fourfolds and complete intersections of quadrics. In addition, we generalize the Tate conjecture (and its variants) from schemes to stacks and prove this generalized conjecture(s) for low-dimensional root stacks and low-dimensional (twisted) orbifolds.

Categorical cones and quadratic homological projective duality

Categorical cones and quadratic homological projective duality

A GLSM View on Homological Projective Duality

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 geometric point, we propose an 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/Lefschetz decompositions of \({\mathcal {C}}\) and \(D^{b}Coh(X)\). We also study the models \({\mathcal {T}}_{X_{L}}\) and \({\mathcal {T}}_{{\mathcal {X}}_{L}}\) that correspond to homological projective duality of linear sections \(X_{L}\) of X. This explains why, in many cases, two phases of a GLSM are related by homological projective duality. We study mostly abelian examples: linear and Veronese embeddings of \({\mathbb {P}}^{n}\) and Fano complete intersections in \({\mathbb {P}}^{n}\). In such cases, we are able to reproduce known results as well as produce some new conjectures. In addition, we comment on the construction of the HPD to a nonabelian GLSM for the Plücker embedding of the Grassmannian G(k, N).

Noncommutative Riemann hypothesis

In this note, making use of noncommutative l l -adic cohomology, we extend the generalized Riemann hypothesis from the realm of algebraic geometry to the broad setting of geometric noncommutative schemes in the sense of Orlov. As a first application, we prove that the generalized Riemann hypothesis is invariant under derived equivalences and homological projective duality. As a second application, we prove the noncommutative generalized Riemann hypothesis in some new cases.

Categorical joins

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 projective duality.

Categorical Plücker Formula and Homological Projective Duality

Kuznetsov’s homological projective duality (HPD) theory [K4] is one of the most active and powerful recent developments in the homological study of algebraic geometry. The fundamental theorem of HPD systematically compares derived categories of dual linear sections of a pair of HP-dual varieties (X,X^{\natural}) .In this paper we generalize the fundamental theorem of HPD beyond linear sections. More precisely, we show that for any two pairs of HP-duals (X,X^{\natural}) and (T,T^{\natural}) which intersect properly, there exist semiorthogonal decompositions of the derived categories D(X \cap T) and D(X^{\natural} \cap T^{\natural}) into primitive and ambient parts, and that there is an equivalence of primitive parts ^\mathrm {prim} D(X \cap T) \simeq D(X^{\natural} \cap T^{\natural})^{\mathrm prim} .

Homological projective duality for quadrics

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.

Derived categories of quintic del Pezzo fibrations

We provide a semiorthogonal decomposition for the derived category of fibrations of quintic del Pezzo surfaces with rational Gorenstein singularities. There are three components, two of which are equivalent to the derived categories of the base and the remaining non-trivial component is equivalent to the derived category of a flat and finite of degree 5 scheme over the base. We introduce two methods for the construction of the decomposition. One is the moduli space approach following the work of Kuznetsov on the sextic del Pezzo fibrations and the components are given by the derived categories of fine relative moduli spaces. The other approach is that one can realize the fibration as a linear section of a Grassmannian bundle and apply homological projective duality.

L‐equivalence for degree five elliptic curves, elliptic fibrations and K3 surfaces

We construct non-trivial L-equivalence between curves of genus one and degree five, and between elliptic surfaces of multisection index five. These results give the first examples of L-equivalence for curves (necessarily over non-algebraically closed fields) and provide a new bit of evidence for the conjectural relationship between L-equivalence and derived equivalence. The proof of the L-equivalence for curves is based on Kuznetsov's Homological Projective Duality for Gr ( 2 , 5 ) , and L-equivalence is extended from genus one curves to elliptic surfaces using the Ogg–Shafarevich theory of twisting for elliptic surfaces. Finally, we apply our results to K3 surfaces and investigate when the two elliptic L-equivalent K3 surfaces we construct are isomorphic, using Neron–Severi lattices, moduli spaces of sheaves and derived equivalence. The most interesting case is that of elliptic K3 surfaces of polarization degree ten and multisection index five, where the resulting L-equivalence is new.

The homological projective dual of

We study the derived category of a complete intersection $X$ of bilinear divisors in the orbifold $\operatorname{Sym}^{2}\mathbb{P}(V)$. Our results are in the spirit of Kuznetsov’s theory of homological projective duality, and we describe a homological projective duality relation between $\operatorname{Sym}^{2}\mathbb{P}(V)$ and a category of modules over a sheaf of Clifford algebras on $\mathbb{P}(\operatorname{Sym}^{2}V^{\vee })$. The proof follows a recently developed strategy combining variation of geometric invariant theory (VGIT) stability and categories of global matrix factorisations. We begin by translating $D^{b}(X)$ into a derived category of factorisations on a Landau–Ginzburg (LG) model, and then apply VGIT to obtain a birational LG model. Finally, we interpret the derived factorisation category of the new LG model as a Clifford module category. In some cases we can compute this Clifford module category as the derived category of a variety. As a corollary we get a new proof of a result of Hosono and Takagi, which says that a certain pair of non-birational Calabi–Yau 3-folds have equivalent derived categories.

Hori-mological projective duality

Kuznetsov has conjectured that Pfaffian varieties should admit noncommutative crepant resolutions which satisfy his Homological Projective Duality. We prove half the cases of this conjecture by interpreting and proving a duality of nonabelian gauged linear sigma models proposed by Hori.

Derived Categories of Families of Sextic del Pezzo Surfaces

AbstractWe construct a natural semiorthogonal decomposition for the derived category of an arbitrary flat family of sextic del Pezzo surfaces with at worst du Val singularities. This decomposition has three components equivalent to twisted derived categories of finite flat schemes of degrees 1, 3, and 2 over the base of the family. We provide a modular interpretation for these schemes and compute them explicitly in a number of standard families. For two such families the computation is based on a symmetric version of homological projective duality for ${{\mathbb{P}}}^2 \times{{\mathbb{P}}}^2$ and ${{\mathbb{P}}}^1 \times{{\mathbb{P}}}^1 \times{{\mathbb{P}}}^1$, which we explain in an appendix.

Noncommutative homological projective duality

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.

On stringy Euler characteristics of Clifford non-commutative varieties

It was shown by Kuznetsov that complete intersections of n generic quadrics in P2n−1 are related by Homological Projective Duality to certain non-commutative (Clifford) varieties which are in some sense birational to double covers of Pn−1 ramified over symmetric determinantal hypersurfaces. Mirror symmetry predicts that the Hodge numbers of the complete intersections of quadrics must coincide with the appropriately defined Hodge numbers of these double covers. We observe that these numbers must be different from the well-known Batyrev's stringy Hodge numbers, else the equality fails already at the level of Euler characteristics. We define a natural modification of stringy Hodge numbers for the particular class of Clifford varieties, and prove the corresponding equality of Euler characteristics in arbitrary dimension.

Homological Projective Duality for Linear Systems with Base Locus

Abstract 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 homological projective (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 also holds true when $Y$ is a noncommutative variety or just a category. 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 weakly crepant categorical resolution of singularities of $Y_L$. Finally we give examples where, starting with a noncommutative $Y$, the above process nevertheless gives geometric HP duals.

A note on Grothendieck’s standard conjectures of type $C^+$ and $D$

Grothendieck conjectured in the sixties that the even Kunneth projector (with respect to a Weil cohomology theory) is algebraic and that the homological equivalence relation on algebraic cycles coincides with the numerical equivalence relation. In this note we extend these celebrated conjectures from smooth projective schemes to the broad setting of smooth proper dg categories. As an application, we prove that Grothendieck's original conjectures are invariant under homological projective duality. This leads to a proof of Grothendieck's conjectures in the case of intersections of quadrics, linear sections of determinantal varieties, and intersections of bilinear divisors. Along the way, we prove also the case of quadric fibrations.

Towards homological projective duality for [formula omitted] and [formula omitted

We provide homological foundations to establish conjectural homological projective dualities in two cases; 1) the duality between S2P3 and the double cover of the projective 9-space branched along the symmetric determinantal quartic, and 2) the duality between S2P4 and the double cover of the symmetric determinantal quintic in P14 branched along the symmetric determinantal locus of rank at most 3.

Homological projective duality via variation of geometric invariant theory quotients

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.

Chow groups of intersections of quadrics via homological projective duality and (Jacobians of) non-commutative motives

Conjectures of Beilinson–Bloch type predict that the low- degree rational Chow groups of intersections of quadrics are one- dimensional. This conjecture was proved by Otwinowska in [20]. By making use of homological projective duality and the recent theory of (Jacobians of) non- commutative 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, we prove that in these cases the unique non-trivial algebraic Jacobian is the middle one. As an application, we make use of Vial's work [26], [27] to describe the rational Chow motives of these complete intersections and show that smooth fibrations into such complete intersections over bases of small dimension satisfy Murre's conjecture (when ), Grothendieck's standard conjecture of Lefschetz type (when ), and Hodge's conjecture (when ).