Intersection theory from the perspectives of the generalized Lelong numbers (Viet-Anh Nguyen - Université de Lille)

Séminaire « Analyse complexe et équations différentielles »
Salle Kampé de Fériet

We address the following problem:

Let T1 and T2 be two positive closed currents on a compact Kähler manifold X. When
are they wedgeable? And what is their intersection? More concretely, we want to know
when and how one can define a reasonable wedge-product T1 ∧ T2 ?

In 2018 Tien-Cuong Dinh and Nessim Sibony introduced a notion of wedge-product
which is based on their theory of tangent currents for positive closed currents. This seems
to be the most general notion of intersection of positive closed currents up to now. On the
other hand, our recent work in 2021 introduced a new concept of the generalized Lelong
numbers νj (T, V ), where T is a positive plurisubharmonic current in a complex manifold
X, and V is a submanifold in X. In general, we have dim V + 1 generalized Lelong numbers
associated to T along V. The classical case where V is a single point x ∈ X corresponds
to dim V = 0.

In this talk we give an effective sufficient condition (in terms of the generalized Lelong
numbers) ensuring that T1 and T2 are wedgeable in the sense of Dinh-Sibony.

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