The deformed Hermitian–Yang–Mills equation, the Positivstellensatz, and the solvability

Abstract

Let (M,ω) be a compact connected Kähler manifold of complex dimension four and let [χ]∈H1,1(M;R). We confirm the conjecture by Collins–Jacob–Yau [8] of the solvability of the deformed Hermitian–Yang–Mills equation, which is given by the following nonlinear elliptic equation ∑iarctan⁡(λi)=θˆ, where λi are the eigenvalues of χ with respect to ω and θˆ is a topological constant. This conjecture was stated in [8], wherein they proved that the existence of a supercritical C-subsolution or the existence of a C-subsolution when θˆ∈[((n−2)+2/n)π/2,nπ/2) will give the solvability of the deformed Hermitian–Yang–Mills equation. Collins–Jacob–Yau conjectured that their existence theorem can be improved to θˆ>(n−2)π/2, where n is the complex dimension of the manifold. In this paper, we confirm their conjecture that when the complex dimension equals four and θˆ is close to the supercritical phase π from the right, then the existence of a C-subsolution implies the solvability of the deformed Hermitian–Yang–Mills equation.