8.1 Complex Hessian equationsCFD38285654D

CFD38285654D

Let $X$ be a connected compact complex manifold of dimension $n$, let $\omega $ be a smooth positive Hermitian form, and fix $2\leq m<n$. The complex Hessian operator $(\omega +\mathrm{dd}^{\mathrm{c}}u)^ m\wedge \omega ^{n-m}$ replaces the determinant by an intermediate symmetric polynomial. Its admissible cone allows negative eigenvalues, so the product calculus for plurisubharmonic potentials from section 5.21 no longer applies directly.

For a continuous admissible weak solution with smooth positive density depending on the potential, the difficulty is to obtain derivative estimates before smoothness is known. Proper exponential equations associate smooth solutions to its smooth admissible approximations. The complex-Hessian estimate is independent of derivatives of the prescribed approximation, so it survives passage to the limit. Uniform convergence defines the continuous mixed products, while obstacle envelopes give the domination principle that identifies the limit with the original solution. A concave uniformly elliptic extension then supplies classical interior regularity. The determinant case is treated in theorem 14.19.32; the argument here concerns the intermediate Hessian degrees. Uniform stability also constructs solutions for strictly increasing exponents, by contraction between constant barriers.

Bounded admissible potentials require decreasing approximation. The recursion first gives currents with measure coefficients below degree $m$; local Hessian capacities control multiplication by their discontinuous potentials. Quasicontinuity and weighted convergence make the products symmetric and allow all approximation indices to tend to their limits together. At degree $m$, the positive product after wedging with $\omega ^{n-m}$ is a Radon measure.

These weighted limits pass locality and the maximum inequality to bounded potentials. Truncation above the infimum then proves that the Hessian measure of each potential is nonzero on an arbitrary Hermitian background. In global comparison, the maximum inequality leaves a signed total-mass difference. The complex Hessians of the background powers determine when this difference vanishes for every bounded admissible list. On small sublevel sets, a modified comparison estimate absorbs the torsion error on arbitrary Hermitian backgrounds. It yields strict domination, bounded stability for proper equations and smoothness for their solutions with smooth data between constant barriers.