On the Positivity of the Products of Positive Primitive Forms

Abstract

Let $(V_i,{\omega }_i)$ be real symplectic vector spaces and let ${\Omega }_i\in {\mathcal{U}}^{+}(V_i)$ in the sense of Haiden. We prove that $p_1^{\ast }{\Omega }_1\wedge p_2^{\ast }{\Omega }_2$ belongs to ${\mathcal{U}}^{+}(V_1\oplus V_2)$ whenever one factor has real dimension at most six. There are forms $\Omega \in {\mathcal{U}}^{+}({\mathbb{R}}^6)\setminus {\mathcal{U}}_{\mathrm{ag}}({\mathbb{R}}^6)$ for which $p_1^{\ast }\Omega \wedge \cdots \wedge p_k^{\ast }\Omega$ belongs to ${\mathcal{U}}^{+}$ for every $k\ge 1$; hence a conjecture of Kontsevich does not hold in complex dimension three. We also give a criterion for a product to belong to $\mathcal{U}$ and show that, for every $N\ge 26$, there are two forms in ${\mathcal{U}}^{+}({\mathbb{C}}^N)$ whose exterior product does not belong to $\mathcal{U}$. The main content of this paper is generated by ChatGPT 5.6 and verified by the author.

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