Complex Analytic and Differential Geometry 2025, 05 - Positive Currents

A complex vector space V of dim n, and with coordinates (z), corresponding basis (∂_z) and dual basis (dz) in V* with exterior algebra ΛV*_ℂ = ⊕Λ^p,qV*, Λ^p,qV* = (Λ^pV*)² Assume for now that V = T_xX. V has a canonical orientation τ. A (p, p)-form with u in Λ^p,pV* is "positive" if ∀ α_j∈V*, 1≤j≤q=n-p, then the wedge-product of u and the n-th normed exponentials of iα is a positive (n, n)-form. A (q, q)-form v in Λ^{q, q}V* is strongly positive if v is a convex combination. For arbitrary coords on V Λ^p,pV* admits a basis consisting of strongly positive forms. All positive forms u are real and hermitian. A form u is positive iff its restriction to all p-dim. subspaces S is a positive volume form on S. If (u) is a family of positive forms, and all but at most one are strongly positive, then their wedge-product is strongly positive (or positive). A current T in D'_{p, p}(X) is (strongly) positive if ⟨T, u⟩ ≥ 0 ∀ i ∈ D_{p, p}(X) that are strongly positive (positive) at each point. The set of (strongly) positive currents is written D'^{+ (⊕)}_{p, p}. (Strong) positivity is a local property and these sets are closed convex cones wrt weak topology. If such a current T and v in C⁰_{s, s}(X) are positive, and one of both of them are strongly positive, then the their wedge product is also positive (or strongly in the case of both).

For all positive currents, the trace measure of T wrt ω is the positive measure σ_T = 1/(2^^pp!)T∧ω^p. A positive continuous function δ on X implies the set of positive currents so that the integral over δT∧ω^p on X is less or equal to 1, and weakly compact. Assume that the (1, 1)-form ω is exact, and Y, Z ⊂ X is a 2p-dim oriented compact real submanifolds of class C¹ with equal boundary, and Z is complex analytic, then Vol(Y) ≥ Vol(Z).

E ⊂ X is complete pluripolar in X if ∀x₀∈X ∃ a neighborhood Ω and a function u ∈ Psh(Ω)∩L_loc¹(Ω): E∩Ω = {z ∈ Ω: u(z) = -∞}. If Ω is sufficiently small, then there is a v ∈ Psh(Ω)∩C^∞(Ω \ E): v = -∞|_E∩Ω and an increasing sequence v_k∈Psh(Ω)∩C^∞(&Omega) with values between 0 and 1, converging uniformly toward 1 on all compact subsets of Ω \ E, and v_k = 0 on a neighborhood on Ω∩E. For a closed positive current T with finite mass in a neighborhood of all points of E, then the trivial extension of T is obtained by extending its measures by 0, on E is closed on X. If the positive current is closed, and 1_E is the characteristic function on E, then 1_ET and 1_{X \ E}T are closed and positive. For all pure p-dim analytic A ⊂ X the current of integration [A] is a closed, positive current on X. A current Θ is normal, if it, and its derivative are currents of order 0. If Supp[Θ] is contained in a real submanifold M of CR dim < p, then Θ = 0.

If Supp[Θ] us contained in an analytic subset A of dim < p, then Θ = 0. A CR submanifold M of CR dim p with submersion σ:M → Y of class C¹ whose fibers are connected and are integral manifolds of the holomorphic tangent space. Any closed current of order 0 with support in this M can be written as Θ = ∫_Y[F_t]dμ(t) with unique complex measure μ on Y. It's strongly positive iff μ is positive.

A smoothly bounded open set Ω ⋐ X with forms f, g of class C² on the closure of Ω with pure bidegrees (p, p), (q, q) with p + q = n - 1, then

dd^cu∧T is a closed positive current.