Abstract
Let M be a Fano manifold equipped with a Kähler form \(\omega \in 2\pi c_1(M)\) and K a connected compact Lie group acting on M as holomorphic isometries. In this paper, we show the minimality of a K-invariant Lagrangian submanifold L in M with respect to a globally conformal Kähler metric is equivalent to the minimality of the reduced Lagrangian submanifold \(L_0=L/K\) in a Kähler quotient \(M_0\) with respect to the Hsiang–Lawson metric. Furthermore, we give some examples of Kähler reductions by using a circle action obtained from a cohomogenenity one action on a Kähler–Einstein manifold of positive Ricci curvature. Applying these results, we obtain several examples of minimal Lagrangian submanifolds via reductions.
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Acknowledgements
The author would like to thank Jürgen Berndt and Yoshihiro Ohnita for some suggestions. He also thanks to Takahiro Hashinaga for helpful discussions and Anna Gori for sharing a result of joint work with Lucio Bedulli about the formula of moment map. A part of this work was done while the author was staying at King’s College London and University of Tübingen by the JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers, Mathematical Science of Symmetry, Topology and Moduli, Evolution of International Research Network based on OCAMI. He is grateful for the hospitalities of the college and the university.
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Kajigaya, T. Reductions of minimal Lagrangian submanifolds with symmetries. Math. Z. 289, 1169–1189 (2018). https://doi.org/10.1007/s00209-017-1992-y
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DOI: https://doi.org/10.1007/s00209-017-1992-y