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A special Lagrangian type equation for holomorphic line bundles

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Abstract

Let L be a holomorphic line bundle over a compact Kähler manifold X. Motivated by mirror symmetry, we study the deformed Hermitian–Yang–Mills equation on L, which is the line bundle analogue of the special Lagrangian equation in the case that X is Calabi–Yau. We show that this equation is the Euler-Lagrange equation for a positive functional, and that solutions are unique global minimizers. We provide a necessary and sufficient criterion for existence in the case that X is a Kähler surface. For the higher dimensional cases, we introduce a line bundle version of the Lagrangian mean curvature flow, and prove convergence when L is ample and X has non-negative orthogonal bisectional curvature.

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Acknowledgments

We would like to express our gratitude to Murad Alim, Tristan C. Collins, Siu-Cheong Lau, Duong H. Phong, Valentino Tosatti and Mu-Tao Wang for many helpful discussions and comments.

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Authors and Affiliations

  1. Department of Mathematics, University of California at Davis, One Shields Ave, Davis, CA, 95616, USA

    Adam Jacob

  2. Department of Mathematics, Harvard University, 1 Oxford St, Cambridge, MA, 02138, USA

    Shing-Tung Yau

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  1. Adam Jacob
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  2. Shing-Tung Yau
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Correspondence to Adam Jacob.

Additional information

A. Jacob is supported in part by NSF Grant No. DMS-1204155.

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Jacob, A., Yau, ST. A special Lagrangian type equation for holomorphic line bundles. Math. Ann. 369, 869–898 (2017). https://doi.org/10.1007/s00208-016-1467-1

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  • DOI: https://doi.org/10.1007/s00208-016-1467-1

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