On a fully nonlinear elliptic equation with differential forms (Q6592058)

Recently, owing to the constant scalar curvature Kähler (cscK) problem and the study of the deformed Hermitian-Yang-Mills (dHYM) equation motivated from mirror symmetry, the problem of the equivalence of some numerical criteria with the solvability of partial differential equations involving various symmetric polynomials of the Kähler form has been at the fore-front of research.\N\NMotivated by prior work due to many researchers, this paper aims to prove the aforementioned type of equivalence theorem for a broad class of equations involving forms \(\omega^n=\sum \Lambda_k \omega^{n-k}+ f \omega^n,\) where \(\omega\) is the Kähler form and \(\Lambda_k\) is required to be uniformly strongly positive.\N\NThe method of proof follows the basic strategy of G. Chen augmented with an argument by \textit{J. Song} [``Nakai-Moishezon criterions for complex Hessian equations'', Preprint, \url{arXiv:2012.07956}]. The strategy is to first prove a pure partial differential equations result which states that these partial differential equation have solutions iff a certain type of a subsolution exists. The proof for such a result relies on standard estimates going all the way to \textit{S.-T. Yau}'s work [Commun. Pure Appl. Math. 31, 339--411 (1978; Zbl 0369.53059)] on the Calabi conjecture. Then one proves a concentration of mass result which produces a Kähler current that is also a subsolution (in the sense of G.Chen). Lastly, one regularises the current, and here induction and a degenerate concentration of mass, see [\textit{V. V. Datar} and \textit{V. P. Pingali}, Geom. Funct. Anal. 31, No. 4, 767--814 (2021; Zbl 1505.32040)], is used.\N\NThe key innovation in this paper is a beautiful observation that allows one to write the ``correct'' equation on a product manifold (involving general forms \(\Lambda_k\)) to prove concentration of mass. Prior to this paper, it was not known as to what equation one needs to write on the product manifold. In fact, [{V. V. Datar} and {V. P. Pingali}, loc. cit.] completely avoided the product manifold construction (at the expense of assuming projectivity) precisely for this reason.