The Monge-Ampère equation for (𝑛-1)-plurisubharmonic functions on a compact Kähler manifold

2013/05/31 by Valentino Tosatti, Ben Weinkove
Mathematics · #Algebraic Geometry and Number Theory #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Kähler manifold #Manifold (fluid mechanics) #Mathematical analysis #Mathematics #Pure mathematics #math.CV #math.DG #msc:32Q15 #msc:32U05 #msc:32W20 #msc:53C55

paper · pdf · doi:10.1090/jams/875

published as J. Amer. Math. Soc. 30 (2017), no.2, 311-346 · 40 pages, final version to appear in JAMS

openalex created_date 2016/06/24 · arxiv created 2016/11/15 · openalex publication_date 2016/11/23 · arxiv updated 2017/01/25 · openalex updated_date 2026/08/05

Abstract

A <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C squared"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">C²</mml:annotation> </mml:semantics> </mml:math> </inline-formula> function on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C Superscript n"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb Cⁿ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is called <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis n minus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(n-1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -plurisubharmonic in the sense of Harvey-Lawson if the sum of any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n minus 1"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> eigenvalues of its complex Hessian is non-negative. We show that the associated Monge-Ampère equation can be solved on any compact Kähler manifold. As a consequence we prove the existence of solutions to an equation of Fu-Wang-Wu, giving Calabi-Yau theorems for balanced, Gauduchon, and strongly Gauduchon metrics on compact Kähler manifolds.

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