Griffiths positivity on the quadric
Griffiths asked whether every ample vector bundle carries a Hermitian metric with positive curvature in the sense of Griffiths. The answer is no, already in rank two on P¹ × P¹: an explicit bundle glued along the graphs of three projective transformations gives a family of ample bundles of which all but finitely many carry no such metric. The proof: symmetric-power filtrations for ampleness; for the metrics, a plurisubharmonic quadratic limit of dual norms along power maps, and an invariant measure that two transformations of order three with a hyperbolic product cannot share. Proved in full in the programme, with no named hypothesis. The course is an exposition of OpenAI's proof (2026).
Five lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.
- Hermitian metrics and Griffiths positivity
- A bundle glued along three graphs
- Ampleness of the power family
- A quadratic limit of dual norms
- An invariant measure and the proof
Theorems and assumptions · Reader and editable sources · Course record
Prerequisites
- Ample invertible sheaves · Morphisms of schemes
- Very ample invertible sheaves, Segre and Veronese embeddings · Morphisms of schemes
- Cohomology of affine schemes and Serre's criterion · Cohomology of quasi-coherent sheaves
- Čech cohomology · Cohomology of quasi-coherent sheaves
- Cohomology of projective space · Cohomology of quasi-coherent sheaves
- Projective space, relative Proj and maps to projective space · Sheaves and schemes
- Order, positivity and distributional limits (Distributions, kernels and analytic singularities)
- Positive derivatives and canonical representatives (Distributions, kernels and analytic singularities)
- Compactness and complex-line positivity (Distributions, kernels and analytic singularities)
- Weak topologies: Tychonoff, Banach–Alaoglu, Mazur, bipolars, Krein–Milman and Eberlein–Šmulian · Foundations of von Neumann algebras
- Plurisubharmonic functions (course Distributions, kernels and analytic singularities); projective bundles and embeddings (courses Sheaves and schemes, Morphisms of schemes); weak compactness in Hilbert space (course Foundations of von Neumann algebras).