Infinite generation for rings of symmetric tensors (Q1902207)

M. Gromov has introduced a Kähler hyperbolic manifold as Kähler manifold \((X, \omega)\) with its universal cover \(\widetilde {X}\) such that the pullback of \(\omega\) can be written as \(d\alpha\), where \(\alpha\) is a 1-form bounded in the \(L_\infty\)-norm with respect to the induced Kähler metric on \(\widetilde {X}\). Let \(\Delta\) be the unit disc and \(X\) an irreducible compact quotient of \(\Delta\times \Delta\). The author proves that \(X\) is Kähler hyperbolic and the algebra \(\bigoplus_{n\geq 0} H^0 (X, \text{Symm}^n \Omega^1_X)\) of holomorphic symmetric tensors on \(X\) is not finitely generated.