Projective Geometry II: Cones and Complete Classifications

Core claim

The central claim is Theorem 3.1 and its corollary: for any projective manifold $(M,\vec\nabla)$, there is a torsion-free Ricci-flat affine connection $\hat\nabla$ on the projective cone $C(M)$ whose holonomy equals the holonomy of the Tractor connection $\vec\nabla$. The paper then uses the list of irreducible holonomy algebras of torsion-free Ricci-flat affine connections to obtain a complete classification. The possible projective Tractor holonomy algebras are $\mathfrak{so}(p,q)$ with $p+q\ge 5$, $\mathfrak{so}(n,\mathbb{C})$ with $n\ge 5$, $\mathfrak{su}(p,q)$, $\mathfrak{sp}(p,q)$, the exceptional algebras $\mathfrak{g}_2$, $\tilde{\mathfrak{g}}_2$, $\mathfrak{g}_2(\mathbb{C})$, $\mathfrak{spin}(7)$, $\mathfrak{spin}(7,\mathbb{C})$, and the full algebras $\mathfrak{sl}(n,\mathbb{R})$, $\mathfrak{sl}(n,\mathbb{C})$, $\mathfrak{sl}(n,\mathbb{H})$, $\mathfrak{sp}(2n,\mathbb{R})$, $\mathfrak{sp}(2n,\mathbb{C})$, with low-dimensional exceptions explicitly settled. The proof proceeds by constructing cones (real, complex, quaternionic, and symplectic) that realize each algebra, using direct products of projectively flat or Einstein pieces to force full holonomy, and by using gluing-and-patching arguments for the low-dimensional cases that resist the general constructions.