The J-equation and the supercritical deformed Hermitian–Yang–Mills equation

A numerical criterion for generalised Monge-Ampère equations on projective manifolds

“…For certain values of the coefficients and phase angles, as explained in [PIN19], the dHYM equation is a generalised Monge-Ampère equation with non-negative coefficients. Hence, for such phase angles we recover the results of [CHEN21] when G is trivial and those of [PIN19] when G = (S 1 ) n . As far as we know, our results on generalised Monge-Ampère equations on projective manifolds are new in the case of G = (S 1 ) k (where 1 ≤ k ≤ n − 1).…”

“…We recover many other existing results when they are specialised to the projective case. If G is taken to be trivial, and c n−1 > 0, then this result reduces to the main theorem in [CHEN21]. If G = (S 1 ) n , we recover the main theorem in [CS17].…”

“…To remedy this problem, numerical criteria akin to the Nakai-Moizeshon criterion for projective manifolds (and the Demailly-Paun [DP01] criterion in the Kähler case) have been conjectured [SZE18,CJY20] to be sufficient for the existence of smooth solutions. Results supporting such conjectures had been proven earlier under some symmetry assumptions (i.e., the toric case) [CS17,PIN19], but recently, building on the work of Demailly and Paun [DP01], Chen [CHEN21] proved far more general results in the case of the J-equation and the dHYM equation. The aim of this paper is to prove similar results on projective manifolds for generalised Monge-Ampère equations [PIN14] -a family of PDE which includes the Monge-Ampère equations, the inverse Hessian equations, and special cases of the dHYM equations.…”

“…The statement 1 ⇔ 2 is known in the case f ≥ 0 in much greater generality. (However, very few cases where f is allowed to be negative appear to be studied [CHEN21,ZHE15,PIN19].) The case of the J-equation was studied originally in [CHEN00, SW08,WEI06].…”

“…Remark 1.3. (A comparison with Chen's work [CHEN21]) The argument of Chen in [CHEN21] relies on the "diagonal trick" of Demailly and Paun. This involves solving a certain J-equation on the product M × M and obtaining currents that concentrate along the diagonal Δ ⊂ M ×M .…”