Hypercritical deformed Hermitian-Yang-Mills equation

In this work, we study the hypercritical deformed Hermitian-Yang-Mills equation and the space of almost calibrated ( 1 , 1 ) (1,1) -forms H \mathcal {H} on compact Kähler manifold. Motivated by Tian’s properness conjecture and Collins-Székelyhidi’s work on J J -equation, we introduce the d p d_{p} -distance on H \mathcal {H} and prove that ( H , d p ) (\mathcal {H},d_{p}) is metric space. Then we propose the notions of properness and coerciveness of the Kempf-Ness functional (i.e. J \mathcal {J} -functional) on the space H \mathcal {H} and show that they are both equivalent to the existence of solutions to the hypercritical deformed Hermitian-Yang-Mills equation. This gives an equivalent characterization of the solvability of the hypercritical deformed Hermitian-Yang-Mills equation from the viewpoint of Geometric Invariant Theory.