25.1.7 Differential forms, currents and holomorphic divisorsCE9C93DDA94A

For positive forms and currents, the characterization of weak positivity by restriction to complex subspaces and the fact that positive currents have order zero, compare [Chapter III, Section 1.A–B, pp. 129–134, especially (1.4), (1.6) and (1.14), DemBook]. In addition, proposition 5.3.2 shows that $\beta ^ p$ lies in the interior of the cone of strongly positive forms, which gives the trace bounds for positive currents; the proof averages over sign changes and permutations of the coordinates. Our currents are indexed by bidegree, whereas Demailly's subscripts there refer to bidimension. Pushforward and pullback of currents (proposition 5.6.1, proposition 5.6.2) correspond to [Chapter III, (1.17), p. 134, DemBook]; for the pushforward the map need only be proper on the support of the current.

The mixed Hodge–Riemann theorem of linear algebra, due to Timorin, is recorded in [Proposition 2.1, p. 4, DN2005mixedv2] (second arXiv version). Lemma 5.3.4 is its case of real $(1,1)$-forms, written with mixed discriminants of Hermitian matrices; we prove it by induction on principal minors and deformation to identity matrices, where the form is given by an explicit trace formula.

For local potentials of closed positive $(1,1)$-currents compare [Chapter III, Section 1.18 and (1.19), p. 135, DemBook]. Our proof of theorem 5.4.3 is different: we subtract the complex Hessian of the potential of the trace measure, and the remainder, which has trace zero and therefore harmonic coefficients, is $\mathrm{dd}^{\mathrm{c}}$-exact by the smooth local result lemma 5.4.2. In proposition 5.4.4 the composition with a holomorphic map is taken pointwise, for the canonical representative; it is proved by radial regularization.

Our proof of the Weierstrass preparation theorem, theorem 5.5.2, is the proof by contour integrals in [Chapter II, Section 2.A, pp. 79–80, Theorem (2.1), DemBook], where it is attributed to Siegel. The Poincaré–Lelong formula, theorem 5.5.6, is [Chapter III, (2.15), pp. 143–144, DemBook]. The expression $(\mathrm{i}/\pi )\partial \bar\partial \log |f|$ appearing there equals our $\mathrm{dd}^{\mathrm{c}}\log |f|^2$, while the operator $\mathrm{dd^ c}$ used subsequently in that book is twice ours. In our proof, slicing by coordinate lines and lemma 5.5.5 show that $\mathrm{dd}^{\mathrm{c}}\log |f|^2$ puts no mass on the singular locus of the divisor, so that it equals the current of integration over the regular part, weighted by multiplicities. Example 5.5.8 gives the corresponding identity in $L^1_{\mathrm{loc}}$ for quotients of holomorphic functions.