Moment map for coupled equations of Kähler forms and curvature

In this paper we introduce two new systems of equations in Kähler geometry: The coupled $\mathrm{p}$ equation and the generalized coupled cscK equation. We motivate the equations from the moment map pictures, prove the uniqueness of solutions and find out the obstructions to the solutions for the second equation. We also point out the connections between the coupled cscK equation, the coupled Kähler Yang–Mills equations and the deformed Hermitian Yang–Mills equation.

Moreover, using this moment map, we can show the Mabuchi functional for the generalized coupled cscK equation, and a special case of the coupled Kähler Yang–Mills equations and the deformed Hermitian Yang–Mills equation are convex along the smooth geodesic, which is different from the one using the moment map picture from the gauge group. In our case, the geodesic is given by the natural metric on the product of smooth Kähler potential$\mathcal{K}(X, \omega_0) \times \dotsm \times \mathcal{K} (X, \omega_k)$.

Keywords

Kähler geometry, moment map, differential geometry, partial differential equations

2010 Mathematics Subject Classification

32Q15, 35A30, 53D20

Full Text (PDF format)

Received 27 February 2021

Accepted 9 February 2023

Published 12 October 2023