The J-flow and stability

Abstract

We study the J-flow from the point of view of an algebrogeometric stability condition. In terms of this we give a lower bound for the natural associated energy functional, and we show that the blowup behavior found by Fang and Lai [11] is reflected by the optimal destabilizer. Finally we prove a general existence result on complex tori.

On a Class of Fully Nonlinear Elliptic Equations on Closed Hermitian Manifolds II: L∞ Estimate

“…However, it is not easy to directly find a cone condition in applications. For the Donaldson equation, Lejmi and Székelyhidi [14] proposed a numerical criterion for the existence of a cone condition. Lejmi and Székelyhidi [14] verified the criterion in dimension 2, while Collins and Székelyhidi [6] verified it on toric manifolds.…”

“…For the Donaldson equation, Lejmi and Székelyhidi [14] proposed a numerical criterion for the existence of a cone condition. Lejmi and Székelyhidi [14] verified the criterion in dimension 2, while Collins and Székelyhidi [6] verified it on toric manifolds. Later Székelyhidi [17] posed the following extension for general complex quotient equations.…”

Relative K-stability for Kähler manifolds

“…Analogues of Theorem 1.4 can be proven using similar methods for twisted cscK metrics and the J-flow. Here one replaces the left hand side with S(ω) − α ω − c 1 p for twisted cscK metrics, and with α ω − c 2 p for the J-flow (where c 1 , c 2 are the appropriate topological constants and α is an auxiliary Kähler metric in an arbitrary Kähler class), and replaces the right hand side with the corresponding numerical invariants [22,31] (for the J-flow one should use the numerical invariants as formulated in [23,Section 4.2] rather than the original formulation in [31], as in [24,Section 6]). In the projective case, these results were proven in [22,31] for p = q = 2.…”

Kähler–Einstein metrics along the smooth continuity method

“…Recall here that (W, β) is the limit of the pair (M, α) under the C * -action generated by w. The following result builds on work in [32] and Dervan [24].…”

“…Let us write w k for the total weight of the action λ on R k , and w ′ k for the weight of the action on S k . From the equivariant Riemann-Roch theorem we have (32) dim…”