Abstract
Let \((L, h)\rightarrow (X, \omega )\) denote a polarized toric Kähler manifold. Fix a toric submanifold \(Y\) and denote by \(\hat{\rho }_{tk}:X\rightarrow \mathbb {R}\) the partial density function corresponding to the partial Bergman kernel projecting smooth sections of \(L^k\) onto holomorphic sections of \(L^k\) that vanish to order at least \(tk\) along \(Y\), for fixed \(t>0\) such that \(tk\in \mathbb {N}\). We prove the existence of a distributional expansion of \(\hat{\rho }_{tk}\) as \(k\rightarrow \infty \), including the identification of the coefficient of \(k^{n-1}\) as a distribution on \(X\). This expansion is used to give a direct proof that if \(\omega \) has constant scalar curvature, then \((X, L)\) must be slope semi-stable with respect to \(Y\) (cf. Ross and Thomas in J Differ Geom 72(3): 429–466, 2006). Similar results are also obtained for more general partial density functions. These results have analogous applications to the study of toric K-stability of toric varieties.
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Notes
To be more precise, we should say that \([\omega _P]\) is integral iff \(\mu \) (which is only determined up to an additive constant) can be chosen to make \(P\) integral.
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Acknowledgments
We thank Julius Ross, Richard Thomas and Steve Zelditch for useful conversations. The second author was supported by a Leverhulme Research Fellowship while this work was being completed.
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Pokorny, F.T., Singer, M. Toric partial density functions and stability of toric varieties. Math. Ann. 358, 879–923 (2014). https://doi.org/10.1007/s00208-013-0978-2
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DOI: https://doi.org/10.1007/s00208-013-0978-2