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A numerical criterion for generalised Monge-Ampère equations on projective manifolds. (English) Zbl 1505.32040

Let’s consider a projective complex manifold \(M\) of complex dimension \(n\) and the following 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 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 J. Song [“Nakai-Moishezon criterions for complex Hessian equations”, Preprint, arXiv:2012.07956]). Consequently, it confirms conjectures of M. Lejmi and 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.

MSC:

32Q15 Kähler manifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds
35J96 Monge-Ampère equations
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References:

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