Abstract
Let \(W(z_1, \ldots , z_n): ({\mathbb {C}}^*)^n \rightarrow {\mathbb {C}}\) be a Laurent polynomial in n variables, and let \({\mathcal {H}}\) be a generic smooth fiber of W. Ruddat et al. (Geom Topol 18:1343–1395, 2014) give a combinatorial recipe for a skeleton for \({\mathcal {H}}\). In this paper, we show that for a suitable exact symplectic structure on \({\mathcal {H}}\), the RSTZ-skeleton can be realized as the Liouville Lagrangian skeleton.
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Notes
The case where Q is not full-dimension can be reduced to this one, by defining \(N'_{\mathbb {R}}={{\,\mathrm{span}\,}}_{\mathbb {R}}(Q) \subset N_{\mathbb {R}}\), and \(M_{\mathbb {R}}\twoheadrightarrow M'_{\mathbb {R}}\). The skeleton for \((M_{\mathbb {R}}, N_{\mathbb {R}}, Q)\) would be that of \((M'_{\mathbb {R}}, N'_{\mathbb {R}}, Q)\) times \(T^d\) where \(d=\dim M_{\mathbb {R}}- \dim M'_{\mathbb {R}}\).
We thank Gammage and Shende for this clarification.
We say two Kähler metrics are comparable if the underlying Riemannian metric, denoted as \(g_1, g_2\) , satisfy
$$\begin{aligned} C^{-1} g_1< g_2 < C g_1 \end{aligned}$$for some positive constant C.
We may smooth \(\varphi \) at a small neighborhood around \(0 \in M_{\mathbb {R}}\), but this is irrelevant since we will use \(\varphi \) only as \(\varphi (\beta u)\) for \(\beta \gg 1\) and u in a neighborhood of \(\partial P\).
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Acknowledgements
I would like to thank my advisor Eric Zaslow for introducing the idea of Lagrangian skeleton and the problem of finding Lagrangian embedding. The idea of using Legendre transformation to identify \(N_{\mathbb {R}}\) and \(M_{\mathbb {R}}\) was inspired by a talk of Helge Ruddat. I thank Vivek Shende and Ben Gammage for the clarification of their work. I also thank Nicolò Sibilla, Gabe Kerr, David Nadler and Ilia Zharkov for many helpful discussions.
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Zhou, P. Lagrangian skeleta of hypersurfaces in \(({\mathbb {C}}^*)^n\). Sel. Math. New Ser. 26, 26 (2020). https://doi.org/10.1007/s00029-020-00555-9
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DOI: https://doi.org/10.1007/s00029-020-00555-9