Log Abundance Conjecture, Characteristic p: OpenAI Claims a Proof

About this video

The abundance conjecture says that when the canonical class K of a variety is nef, some multiple of it gives a map, splitting the variety into fibres over a smaller base. OpenAI claims a proof of log abundance for log canonical threefold pairs in characteristic p greater than 3 when K + B has numerical dimension one, so the threefold maps onto a curve; these are OpenAI's preprints, not yet peer reviewed.

The animation shows a threefold where this can be seen exactly: X = C × A, with C the genus-2 curve y² = -(x+3)(x+2)x(x-1)(x-3)(x-4) and A an abelian surface. Here K³ = 0, K²·H = 0 and K·H² = 4, and the map given by mK sends every fibre to a point of C. The grids show the same curve over the finite fields F_11, F_13, F_17 and F_19.

Papers (OpenAI, catalogue result 035):
- Log abundance in numerical dimension one for threefolds in positive characteristic: github.com/openai/math/blob/main/preprints/Log-abundance-in-n…
- Abundance in numerical dimension one for terminal threefolds in positive characteristic: github.com/openai/math/blob/main/preprints/Abundance-in-numer…

The animation illustrates the result; it is not a proof.

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Every result explained: gradientdamage.com/p/035