[Paper Review] Categorical duality between joins and intersections

Qingyuan Jiang, Naichung Conan Leung|arXiv (Cornell University)|Nov 13, 2018

Algebraic Geometry and Number Theory16 references4 citations

TL;DR

This paper establishes a categorical duality between joins and intersections in the framework of homological projective duality (HPD), proving that the HPD dual of a categorical join $X_1 \star X_2$ is isomorphic to the fiber product of their HPD duals, $X_1^\natural \times_{\mathbb{P}(V^\vee)} X_2^\natural$, provided the intersections have expected dimensions. This result provides a categorical refinement of classical projective duality and confirms a key prediction of homological projective geometry.

ABSTRACT

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 main result "formation of categorical joins commutes with HPD" of [KP18]. We also introduce the concept of an $n$-HPD category, and study its properties and connections with joins and HPDs.

Motivation & Objective

• To establish a categorical duality between joins and intersections in homological projective duality (HPD), extending classical projective duality beyond its limited scope.
• To resolve the failure of classical projective duality between joins and intersections by introducing a homological framework where duality holds categorically.
• To provide a new proof of the 'formation of categorical joins commutes with HPD' result from [KP18], using the duality framework developed in this paper.
• To introduce and study the concept of $n$-HPD categories and their connections with categorical joins and HPD.
• To demonstrate that the categorical join construction is associative up to $\mathbb{P}(V)$-linear equivalence under the HPD framework.

Proposed method

• Define the categorical join $X_1 \star X_2$ as a subcategory of the derived category of a projective bundle over $X_1 \times X_2$, using the tautological line bundle and Lefschetz structures.
• Use the framework of homological projective duality (HPD) to replace classical projective duality, where $X \mapsto X^\natural$ is a reflexive correspondence on derived categories.
• Establish a $\mathbb{P}(V)$-linear equivalence between the HPD dual of the categorical join $(X_1 \star X_2)^\natural$ and the fiber product $X_1^\natural \times_{\mathbb{P}(V^\vee)} X_2^\natural$.
• Apply Lefschetz decomposition techniques to analyze the structure of categorical joins and their duals, showing compatibility of $\mathbb{P}(V)$-linear structures.
• Use base change and adjunction formalism for derived categories to prove that the functors $\pi_{(12)3}^* \circ \pi_{1,23,*}$ and their adjoints induce $\mathbb{P}(V)$-linear equivalences between different associative groupings of categorical joins.
• Generalize the associativity and duality results to $n$-fold categorical joins by extending the $\mathbb{P}(V)$-linear equivalence argument to higher $n$.

Experimental results

Research questions

• RQ1 Does the duality between joins and intersections hold categorically in the framework of homological projective duality?
• RQ2 Under what conditions does the HPD dual of a categorical join $X_1 \star X_2$ coincide with the fiber product $X_1^\natural \times_{\mathbb{P}(V^\vee)} X_2^\natural$?
• RQ3 Can the result that 'categorical joins commute with HPD' be recovered as a consequence of a more general duality principle?
• RQ4 What is the structure of the $n$-fold categorical join, and how does it relate to $n$-HPD categories?
• RQ5 How do $\mathbb{P}(V)$-linear structures on derived categories interact under categorical joins and their duals?

Key findings

• The main result establishes a $\mathbb{P}(V)$-linear equivalence $(X_1 \star X_2)^\natural \simeq X_1^\natural \times_{\mathbb{P}(V^\vee)} X_2^\natural$, valid when the intersection $X_1^\natural \cap X_2^\natural$ has expected dimension.
• The proof provides a new derivation of the 'formation of categorical joins commutes with HPD' result from [KP18], now as a consequence of the duality between joins and intersections.
• The associativity of categorical joins is established up to $\mathbb{P}(V)$-linear equivalence, with all groupings of $n$-fold joins being equivalent via $\mathbb{P}(V)$-linear functors.
• The $n$-fold categorical join $\mathcal{J}$ admits a canonical $\mathbb{P}(V)$-linear structure induced from the $\mathbb{P}(V)$-linear structures on the individual categories $\mathcal{A}^{(k)}$.
• The Lefschetz components of the categorical join and its dual are explicitly described via pullbacks and tensor products, with the zeroth component given by $p_{(12)3}^*(p_{12}^*(\mathcal{A}_0^{(1)} \otimes \mathcal{A}_0^{(2)}) \otimes \mathcal{A}_0^{(3)})$.
• The equivalence between different associative groupings of categorical joins is induced by $\mathbb{P}(V)$-linear functors and their adjoints, preserving the $\mathbb{P}(V)$-linear structure throughout.