[Paper Review] Homological projective duality for Grassmannians of lines

Alexander Kuznetsov|ArXiv.org|Oct 31, 2006

Algebraic Geometry and Number Theory12 references99 citations

TL;DR

This paper establishes homological projective duality (HP-duality) for Grassmannians Gr(2,6) and Gr(2,7) by constructing noncommutative resolutions of their dual Pfaffian varieties. It proves that derived categories of linear sections—especially Pfaffian cubic 4-folds—admit semiorthogonal decompositions into exceptional collections and derived categories of K3 surfaces, revealing deep connections between cubic 4-folds and K3 surfaces via derived equivalence.

ABSTRACT

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 corresponding Pfaffian variety are derived equivalent. We also conjecture a rationality criterion for cubic 4-folds in terms of their derived categories.

Motivation & Objective

• To establish homological projective duality (HP-duality) for Grassmannians Gr(2,6) and Gr(2,7), which extends classical projective duality to derived categories.
• To resolve the singularities of Pfaffian varieties—classically dual to Grassmannians—using noncommutative resolutions, as commutative resolutions are too large.
• To describe the derived categories of linear sections of Gr(2,6), Gr(2,7), and their dual Pfaffian varieties via semiorthogonal decompositions.
• To demonstrate derived equivalence between mutually orthogonal Calabi-Yau linear sections of Gr(2,7) and its HP-dual Pfaffian variety.
• To propose a rationality criterion for cubic 4-folds based on their derived categories, linking rationality to the structure of their derived categories.

Proposed method

• Construct noncommutative resolutions of singularities for Pfaffian varieties Pf(4,6) and Pf(4,7) using sheaves of algebras that are matrix algebras generically and have finite homological dimension.
• Use the framework of homological projective duality (HP-duality) to relate the derived category of Gr(2,n) to that of its noncommutative dual, given a Lefschetz decomposition of the derived category.
• Define convolution of kernels via fiber products and projections to construct integral kernels for the semiorthogonal decompositions of linear sections.
• Apply the machinery of semiorthogonal decompositions to linear sections of Gr(2,n) and Pf(4,n), showing that derived categories decompose into exceptional collections and derived categories of K3 surfaces.
• Use exact triangles and derived category techniques to analyze the homological properties of the convolution kernels, ensuring compatibility with the HP-duality framework.
• Leverage known geometric properties: Pfaffian cubic 4-folds are intersections of Pf(4,6) with P^5, and their dual K3 surfaces arise as intersections of orthogonal P^8 with Gr(2,6).

Experimental results

Research questions

• RQ1 What is the homological projective dual of the Grassmannian Gr(2,6), and how does it relate to its classical projective dual, the Pfaffian variety Pf(4,6)?
• RQ2 Can noncommutative resolutions of singular Pfaffian varieties serve as homologically projectively dual varieties to Grassmannians?
• RQ3 How do the derived categories of linear sections of Gr(2,6) and its dual Pfaffian variety decompose, and what structures do they reveal?
• RQ4 Are mutually orthogonal Calabi-Yau linear sections of Gr(2,7) and its HP-dual Pfaffian variety derived equivalent?
• RQ5 Can the derived category of a cubic 4-fold determine its rationality, and if so, under what conditions?

Key findings

• The derived category of a Pfaffian cubic 4-fold admits a semiorthogonal decomposition into three exceptional line bundles and the derived category of a K3 surface of degree 14.
• Mutually orthogonal Calabi-Yau linear sections of Gr(2,7) and its homologically projectively dual Pfaffian variety are derived equivalent.
• The Fano variety of lines on a Pfaffian cubic 4-fold is isomorphic to the Hilbert scheme of length 2 subschemes on the associated K3 surface.
• The primitive Hodge structure of the K3 surface is a substructure of the primitive Hodge structure of the Pfaffian cubic 4-fold.
• The noncommutative resolution (Y,R) of the Pfaffian variety Pf(4,6) is homologically projectively dual to Gr(2,6), and similarly for Pf(4,7) and Gr(2,7).
• A rationality criterion for cubic 4-folds is conjectured based on the structure of their derived categories, particularly the presence of a K3 component in their semiorthogonal decomposition.