A numerical criterion for generalised Monge-Ampère equations on projective manifolds (Q2232155)

Let's consider a projective complex manifold \(M\) of complex dimension \(n\) and the following \textit{generalized Monge-Ampère equation} (the unknown is a Kähler metric \(\Omega\) in a fixed Kähler class \([\Omega_0]\)) \[ \Omega^n = \sum_{k=1}^{n-1} c_k \chi^{n-k} \Omega^{k} + f\chi^n\] where \(\chi\) is a fixed Kähler metric, \(f\) is a smooth function and \(c_k\) are constants. Note that some natural constraints are also required on \(f\) and \(c_k\). Apart from the classical Monge-Ampère equation studied by S. T. Yau, this very general equation covers many PDE including the J-equation studied by S. K. Donaldson and X. X. Chen and some special cases of the deformed Hermtian Yang-Mills equations. The main result of this very nice paper provides an equivalence between the existence of a smooth solution to above equation and the positivity of the intersection numbers \[\int_V \left(\binom{n}{p} [\Omega_0]^{n-p} - \sum_{k=p}^{n-1}c_k \binom{k}{p} [\chi]^{n-k} [\Omega_0]^{k-p}\right)>0\] for all subvarieties \(V\subset M\) of co-dimension \(p\). This improves in the projective setting a recent result of \textit{G. Chen} [Invent. Math. 225, No. 2, 529--602 (2021; Zbl 1481.53093)] who studied the J-equation and needed a uniform condition of positivity on the intersection numbers (see also the paper of \textit{J. Song} [``Nakai-Moishezon criterions for complex Hessian equations'', Preprint, \url{arXiv:2012.07956}]). Consequently, it confirms conjectures of \textit{M. Lejmi} and \textit{G. Székelyhidi} [Adv. Math. 274, 404--431 (2015; Zbl 1370.53051)] about the J-equation and some Hessian equations. The proof is based on a continuity method and the use of degenerate mass concentration result. Eventually, the authors prove an equivariant version of the main result. The paper is well written.