A counterexample to Pixton completeness in Chow

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Constructs a tautological relation on a moduli space of stable pointed curves that vanishes in rational Chow, hence in rational cohomology, but lies outside Pixton’s original relation span. This disproves the Chow and rational-cohomological forms of his original completeness conjecture.

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Theorem 1.1

In A high-arity counterexample to Pixton completeness in Chow

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Set

g=10^{60},\qquad D_*=\binom g3,\qquad n=3D_*.

The explicitly defined class Y\in \Salg_{g,n}^{2D_*} of Definition 2.3 satisfies

q_{g,n}(Y)=0,\qquad Y\notin\Pspan_{g,n}.

Consequently Pixton’s original relation span does not exhaust the tautological Chow relations: the equality \ker q_{g,n}=\Pspan_{g,n} fails for this stable pair.

Corollary 1.2 (Rational-cohomological incompleteness)

In A high-arity counterexample to Pixton completeness in Chow

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For the stable pair and class Y of Theorem 1.1,

q^H_{g,n}(Y)=0 \quad\text{in }H^{4D_*}\!\left(\Mbar_{g,n}(\C);\Q\right), \qquad Y\notin\Pspan_{g,n}.

In particular, \Pspan_{g,n}\subsetneq\ker q^H_{g,n}.

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A high-arity counterexample to Pixton completeness in Chow

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We disprove the Chow and rational-cohomological forms of Pixton’s completeness conjecture for his original relation system. We construct a formal tautological class outside the original Pixton relation span whose image is zero in the Chow ring and in rational cohomology. The example has genus \(10^{60}\) and \(3\binom{10^{60}}3\) markings.

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An unsigned public tier list, the OpenAI Math Tier List, classes it as a counterexample and rates its significance, if it holds up, as “Major math journal”. The page says that nothing on it is verified.
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