About \(J\)-flow, \(J\)-balanced metrics, uniform \(J\)-stability and K-stability (Q1654917)

Let \(X\) be a complex projective manifold of dimension \(n\), endowed with two distinct polarisations \(L_1\) and \(L_2\). From this data, Donaldson has defined a natural flow on the space of Kähler metrics in \(c_1(L_1)\), called the \(J\)-flow. In their recent work [``A finite dimensional approach to Donaldson's J-flow'', Preprint \url{arXiv:1507.03461}], \textit{R. Dervan} and \textit{J. Keller} showed that the existence of a critical point of the \(J\)-flow implies the existence of \(J\)-balanced metrics for an embedding \(M \hookrightarrow \mathbb P (H^0(M, L_1^k)^*)\) with \(k \gg 0\). In this paper the authors prove an algebro-geometric counterpart of this result: the triple \((M, L_1, L_2)\) admits a \(J\)-balanced metric in the Kähler class \(c_1(L_1^k)\) if and only if the linear system \(|L_2|\) is Chow stable for \(M \subset \mathbb P (H^0(M, L_1^k)^*)\). They also obtain sufficient conditions for uniform stability: assume that \(\gamma = \frac{L_2 L_1^{n-1}}{L_1^n}>0\) and \(\gamma L_1-L_2\) is nef. Then the triple \((M, L_1, L_2)\) is uniformly \(J\)-stable.