Moment map for coupled equations of K\"{a}hler forms and curvature

Abstract

In this paper we introduce two new systems of equations in K\"ahler geometry: The coupled 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 K\"{a}hler 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 K\"ahler 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 K\"{a}hler potential $\mathcal{K}(X,\omega_0)\times \cdots \times \mathcal{K}(X,\omega_k)$.