On the Positivity of the Products of Positive Primitive Forms

Abstract

Let $(V_i,ω_i)$ be real symplectic vector spaces and let $Ω_i\in\mathcal U^+(V_i)$ in the sense of Haiden. We prove that $p_1^*Ω_1\wedge p_2^*Ω_2$ belongs to $\mathcal U^+(V_1\oplus V_2)$ whenever one factor has real dimension at most six. There are forms $Ω\in\mathcal{U}^+(\mathbb{R}^6)\setminus\mathcal{U}_{\mathrm{ag}}(\mathbb{R}^6)$ for which $p_1^*Ω\wedge\cdots\wedge p_k^*Ω$ belongs to $\mathcal{U}^+$ for every $k\geq 1$; 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 $N\geq 26$, 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.