Limit Behavior of Deformed Hermitian–Yang–Mills Metrics on Kähler Surfaces

Abstract

In this paper, we consider the limit behavior of a sequence of deformed Hermitian–Yang–Mills metrics

F_m

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$F_m$$\end{document}

on

L^{\otimes m}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^{\otimes m}$$\end{document}

where L is an ample line bundle over a Kähler surface

(X,\omega )

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(X, \omega )$$\end{document}

. If the cohomology class

c_1(L)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$c_1(L)$$\end{document}

admits a solution of the J-equation, then we prove that

F_m

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$F_m$$\end{document}

will converge to it. Furthermore, we also consider a boundary case. In this case, we prove that

F_m

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$F_m$$\end{document}

will converge to a singular Kähler metric away from a finite number of curves with negative self-intersection on the surface.

Keywords

Deformed Hermitian–Yang–Mills metric / J-equation / Monge–Ampére equation / Kähler surface

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Funding

National Key R&D Program of China(2022YFA1005400)

NFSC(12031017)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature