Chern forms of Hermitian metrics with analytic singularities on vector bundles

On Lelong Numbers of Generalized Monge-Ampère Products

“…(Lem. 6.3 in [LRSW22]) Let Y be a complex manifold. For i = 1,..,t, let X i be complex manifolds, let π i : X i → Y be proper holomorphic submersions, and let α i be forms on…”

“…Using the idea of [LRSW22] to define the wedge product of pushforwards currents with the fibre product, Theorem 3.5 implies a monotone continuity result for the recursively defined mixed products: Corollary 3.6. For i = 1,..,t, let π i : X i → Y be proper holomorphic submersions between complex manifolds X i and Y with m i -dimensional fibres, let θ i be closed γ i -positive (1, 1)currents on X i for (positive) (1, 1)-forms γ i on X i such that for any small enough open V ⊂ Y , there are closed positive (1, 1)-forms γ i,+ and α i,± with γ i ≤ γ i,+ and α i,+ − α i,− ∈ {θ i } ∂∂ on π −1 i (V ) (for example, the fibres of π i are Kähler).…”

“…Since there is no restriction on k, this definition of (mixed) Monge-Ampère products is an extension of BTD MA, see [LRSW22,3.4]). It can be defined for quasipsh potentials (i. p. for γ-psh ones for a (pos) form γ), see [B lo19, LSW20] and also [LRSW22, Lem 3.5].…”

“…Loosely said, the AW MA product removes too much. Inspired by [ABW19, Thm 1.2], Lärkäng, Raufi, Wulcan and the author give the following definition in [LRSW22,LSW20] which keeps track of the removed part by adding an extra term to the AW MA product. Definition 4.6.…”

“…Remark 4.9. Using the idea from [LRSW22] (motivated by Lemma 2.1), we can use the fibre product to define the wedge product of pushforwards of currents with analytic singularities in the following setting. For i = 1,..,t, let π i : X i → Y be proper holomorphic submersions with m i -dimensional fibres, let θ i be closed γ i -pos (1, 1)-currents with analytic singularities in Z i , and let α i ∈ {θ i } ∂∂ be closed γ i -pos (1, 1)-forms where γ i are (pos) (1, 1)-forms, each on X i .…”