About J-flow, J-balanced metrics, uniform J-stability and K-stability

From the work of Dervan-Keller, there exists a quantization of the critical equation for the J-flow. This leads to the notion of J-balanced metrics. We prove that the existence of J-balanced metrics has a purely algebro-geometric characterization in terms of Chow stability, complementing the result of Dervan-Keller. We also obtain various criteria that imply uniform J-stability and uniform K-stability. Eventually, we discuss the case of K\"ahler classes that may not be integral over a compact manifold.Comment: 23 pages; In honor of Ngaiming Mok's 60th birthday. To appear in Asian J. Mat

Anticanonically balanced metrics and the Hilbert-Mumford criterion for the δm\delta_m-invariant of Fujita-Odaka

We prove that the stability condition for Fano manifolds defined by Saito-Takahashi, given in terms of the sum of the Ding invariant and the Chow weight, is equivalent to the existence of anticanonically balanced metrics. Combined with the result by Rubinstein-Tian-Zhang, we obtain the following algebro-geometric corollary: the $\delta_m$-invariant of Fujita-Odaka satisfies $\delta_m >1$ if and only if the Fano manifold is stable in the sense of Saito-Takahashi, establishing a Hilbert-Mumford type criterion for $\delta_m >1$. We also extend this result to the K\"ahler-Ricci $g$-solitons and the coupled K\"ahler-Einstein metrics, and as a by-product we obtain a formula for the asymptotic slope of the coupled Ding functional in terms of multiple test configurations.Comment: v2: 36 pages, details adde

Balanced metrics for extremal K\"ahler metrics and Fano manifolds

The first three sections of this paper are a survey of the author's work on balanced metrics and stability notions in algebraic geometry. The last section is devoted to proving the well-known result that a geodesically convex function on a complete Riemannian manifold admits a critical point if and only if its asymptotic slope at infinity is positive, where we present a proof which relies only on the Hopf--Rinow theorem and extends to locally compact complete length metric spaces.Comment: 14 pages, to appear in Proceedings of Hayama Symposium 2022 on Complex Analysis in Several Variable