7.1 Bedford–Taylor products and local capacity2F5912FF9FEA

2F5912FF9FEA

Let $u$ be a plurisubharmonic function on an open set $\Omega \subset \mathbb {C}^ n$. By theorem 5.9.1, $\mathrm{dd}^{\mathrm{c}}u$ is a closed positive $(1,1)$-current, but in general its coefficients are only measures. Since products of measures have no general meaning, the exterior powers of $\mathrm{dd}^{\mathrm{c}}u$ cannot be formed by multiplying coefficients. For $u\in C^2(\Omega )$ the top power is the Monge–Ampère measure $(\mathrm{dd}^{\mathrm{c}}u)^ n=n!\, \pi ^{-n}\det (u_{j\bar k})\, \mathrm{d}V_{2n}$. This fully nonlinear expression in the second derivatives of $u$ plays in pluripotential theory the role that the Laplacian plays in classical potential theory. Bedford and Taylor showed that for locally bounded $u$ these powers can be defined by induction, and so can the mixed products $\mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ q\wedge T$ with a closed positive current $T$. Indeed, since $T$ has measure coefficients and $u$ is a locally bounded Borel function, the product $uT$ makes sense, and one puts $\mathrm{dd}^{\mathrm{c}}u\wedge T=\mathrm{dd}^{\mathrm{c}}(uT)$. This chapter develops the local theory of these products, whose factors are always locally bounded; products of unbounded potentials are treated in section 13.1.

The first two sections establish the basic properties of the products: positivity, the Chern–Levine–Nirenberg mass bounds, continuity along decreasing sequences and symmetry (theorem 7.3.7), and the comparison principle. The products then define the Monge–Ampère capacity, a complex analogue of the relative capacity of section 4.1. Locally, pole sets of plurisubharmonic functions have outer capacity zero; plurisubharmonic functions are quasi-continuous with respect to this capacity, and the products are continuous under convergence in capacity and along increasing sequences. Solving the homogeneous Monge–Ampère equation on a ball, we show that a locally bounded plurisubharmonic function is maximal if and only if its Monge–Ampère measure vanishes. These are the tools with which section 5.47 characterizes pluripolar sets.

The remaining sections, needed for non-pluripolar products and envelopes on compact manifolds in section 13.1, section 16.10, solve the equation $(\mathrm{dd}^{\mathrm{c}}u)^ n=\mu $ on a ball for increasingly singular measures $\mu $: continuous densities, bounded densities, and finally compactly supported measures dominated by the Monge–Ampère measure of a bounded plurisubharmonic function. The main application is a lower bound for mixed products: if $(\mathrm{dd}^{\mathrm{c}}u_ i)^ n\geq f_ i\mu $ for $1\le i\le n$, then $\mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ n\geq (f_1\cdots f_ n)^{1/n}\mu $ (theorem 7.12.1). The chapter closes with the locality of complex Hessians on contact sets and a chain rule for gradients against positive currents.

Historical remarks and references for this chapter are collected in subsection 25.1.9.