Complex Hessian-type equations on projective manifolds admit solutions if and only if a uniform integral inequality holds over every subvariety, under a root-separation condition on the associated polynomial.
A Numerical Criterion for the 2-Hessian Equation on Compact K\"ahler Manifolds
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abstract
We show that a Nakai--Moishezon-type criterion associated with the complex $2$-Hessian equation produces a Gauduchon class. In complex dimension three, this numerical criterion is equivalent to the existence of a smooth $2$-admissible representative and hence to the solvability of the $2$-Hessian equation. As consequences of these results, we prove the corresponding conjectures of Murakami for the complex Hessian equation and of Sz\'ekelyhidi for the Hessian quotient equation in dimension three. We also establish a boundary version of the above results.
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A numerical criterion for complex Hessian type equations on projective manifolds
Complex Hessian-type equations on projective manifolds admit solutions if and only if a uniform integral inequality holds over every subvariety, under a root-separation condition on the associated polynomial.