5.1 Complex differential forms and positive currents4DEEEF894245

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This chapter sets up the calculus on which Part II is built. The central object of the following chapters is the complex Hessian $(\partial ^2u/\partial z_ j\partial \bar z_ k)$ of a function $u$ on an open subset of $\mathbb {C}^ n$, or rather the $(1,1)$-form $\mathrm{dd}^{\mathrm{c}}u$ that it defines. Two features distinguish it from the Laplacian of Part I: its positivity is a condition in every complex direction, and it transforms naturally under holomorphic changes of coordinates. Complex differential forms express both features, and currents, that is, differential forms with distribution coefficients, allow us to apply $\mathrm{dd}^{\mathrm{c}}$ to locally integrable functions such as $\log |f|^2$ with $f$ holomorphic.

We begin with the elementary calculus of holomorphic functions, complex manifolds and differential forms of type $(p,q)$. Positivity of a current is defined by duality from positivity of forms, which is a matter of linear algebra (section 5.3); along the way we prove a pointwise form of the Hodge index theorem for mixed discriminants (lemma 5.3.4). The basic analytic fact is theorem 5.3.7: the coefficients of a positive current are measures, all dominated by a single positive measure, its trace measure. Consequently, on a manifold with a countable basis, every sequence of positive currents with locally bounded trace measures has a weakly convergent subsequence.

A closed positive $(1,1)$-current $T$ is locally of the form $\mathrm{dd}^{\mathrm{c}}u$, with $u$ unique up to pluriharmonic functions (theorem 5.4.3), and positivity of $\mathrm{dd}^{\mathrm{c}}u$ survives composition with holomorphic maps. The fundamental example is the Poincaré–Lelong formula $\mathrm{dd}^{\mathrm{c}}\log |f|^2=[\operatorname{div}(f)]$ of theorem 5.5.6, which identifies the complex Hessian of $\log |f|^2$ with the current of integration over the zero set of $f$, counted with multiplicities. The chapter closes with pushforward and pullback of currents and with Hermitian metrics, which give trace measures independent of coordinates.

Historical remarks and references for this chapter are collected in subsection 25.1.7.