[Paper Review] Projective Geometry II: Cones and Complete Classifications

Stuart Armstrong|ArXiv.org|Feb 27, 2006

Geometric Analysis and Curvature Flows13 references3 citations

TL;DR

This paper introduces the projective cone construction—a Ricci-flat, torsion-free affine manifold of dimension n+1 whose holonomy matches that of the Tractor connection on a projective manifold. By leveraging this construction and results from Ricci-flat holonomy classification, the authors complete the full classification of irreducible projective Tractor holonomy algebras, proving the existence of all listed holonomy types, including complex and symplectic cases, via explicit geometric constructions and patching arguments.

ABSTRACT

The aim of this paper and its prequel is to introduce and classify the irreducible holonomy algebras of the projective Tractor connection. This is achieved through the construction of a `projective cone', a Ricci-flat manifold one dimension higher whose affine holonomy is equal to the Tractor holonomy of the underlying manifold. This paper uses the result to enable the construction of manifolds with each possible holonomy algebra.

Motivation & Objective

• To complete the classification of irreducible holonomy algebras for the projective Tractor connection.
• To establish the existence of all possible holonomy algebras listed in prior work by constructing explicit model manifolds.
• To resolve the gap in existence proofs for holonomy types not realizable via standard Ricci-flat cone constructions.
• To extend the classification to complex and symplectic holonomy types using holomorphic and symplectic patching techniques.
• To demonstrate that all reductive holonomy algebras from the Ricci-flat affine holonomy classification can be realized as projective Tractor holonomy algebras.

Proposed method

• Construct the projective cone over a projective manifold M, a Ricci-flat, torsion-free affine manifold of dimension n+1 with isomorphic holonomy to the Tractor connection on M.
• Use the cone construction to lift Tractor holonomy algebras to Ricci-flat affine holonomy, enabling application of the classification results from [Arm3] and [MeSc1].
• Apply partition-of-unity patching techniques to conjugate local holonomy algebras and achieve full global holonomy, particularly for non-Ricci-flat cases.
• Utilize holomorphic and symplectic analogues of the cone construction to realize complex and symplectic holonomy algebras such as $\mathfrak{sl}(n,\mathbb{C})$, $\mathfrak{sp}(2n,\mathbb{C})$, and $\mathfrak{sp}(4,\mathbb{R})$.
• Leverage the curvature decomposition of the Tractor connection, including the Weyl tensor and Cotton-York tensor, to analyze holonomy reductions and preserve structure under deformation.
• Apply the splitting $\mathcal{T} = T[\mu] \oplus L^\mu$ and the Tractor connection formula $\overrightarrow{\nabla}_X = \nabla_X + X + \mathsf{P}(X)$ to analyze holonomy behavior under preferred connections.

Experimental results

Research questions

• RQ1 Which irreducible holonomy algebras can arise as Tractor holonomy algebras for projective structures on smooth manifolds?
• RQ2 Can all reductive holonomy algebras from the Ricci-flat affine classification be realized as projective Tractor holonomy algebras?
• RQ3 How can holonomy algebras such as $\mathfrak{sl}(n,\mathbb{C})$, $\mathfrak{sp}(2n,\mathbb{C})$, and $\mathfrak{spin}(7)$ be realized geometrically in the projective setting?
• RQ4 What role does the projective cone construction play in linking projective Tractor holonomy to Ricci-flat affine holonomy?
• RQ5 Can non-Ricci-flat holonomy types be constructed via patching techniques when the cone method fails?

Key findings

• The projective cone construction yields a Ricci-flat, torsion-free affine connection on a manifold of dimension n+1 whose holonomy is isomorphic to the Tractor holonomy of the original n-dimensional projective manifold.
• All irreducible holonomy algebras listed in Table 1 and Table 2—such as $\mathfrak{so}(p,q)$, $\mathfrak{su}(p,q)$, $\mathfrak{g}_2$, $\mathfrak{spin}(7)$, and $\mathfrak{sl}(n,\mathbb{C})$—are realized as projective Tractor holonomy algebras via explicit constructions.
• The existence of $\mathfrak{sl}(3,\mathbb{C})$ and $\mathfrak{sl}(4,\mathbb{C})$ holonomy is established using holomorphic patching and cone constructions, with the latter arising from the cone over a $\mathfrak{sl}(3,\mathbb{R})$-holonomy manifold.
• The algebras $\mathfrak{sp}(4,\mathbb{R})$ and $\mathfrak{sp}(4,\mathbb{C})$ are shown to arise as Tractor holonomy algebras through symplectic projective structures and patching of non-flat symplectic connections.
• For holonomy types not realizable via Ricci-flat cones, the paper constructs models using partition-of-unity patching of local connections, preserving the desired holonomy algebra.
• The anti-holomorphic part of curvature in complex constructions does not increase the holonomy algebra, ensuring that full holonomy is achieved without overgeneration.