Relative non-pluripolar product of currents
Duc‐Viet Vu
Abstract
Abstract Let X be a compact Kähler manifold. Let $$T_1, \ldots , T_m$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:mrow> </mml:math> be closed positive currents of bi-degree (1, 1) on X and T an arbitrary closed positive current on X . We introduce the non-pluripolar product relative to T of $$T_1, \ldots , T_m$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:mrow> </mml:math> . We recover the well-known non-pluripolar product of $$T_1, \ldots , T_m$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:mrow> </mml:math> when T is the current of integration along X . Our main results are a monotonicity property of relative non-pluripolar products, a necessary condition for currents to be of relative full mass intersection in terms of Lelong numbers, and the convexity of weighted classes of currents of relative full mass intersection. The former two results are new even when T is the current of integration along X .