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The canonical spectrum of projective toric manifolds

School of Mathematical Sciences, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, P. R. China

https://doi.org/10.1142/S0129167X18500489Cited by:0 (Source: Crossref)

Abstract

Let $X$ $X$ be a complex projective toric manifold. We associated to $X$ $X$ , a positive and closed $(1, 1)$ $(1, 1)$ -current called the canonical toric Kähler current of $X$ $X$ denoted by $ω_{X, can}$ $ω_{X, can}$ , and a new invariant called the canonical spectrum of $X$ $X$ . This spectrum is obtained as the set of the eigenvalues of a singular Laplacian defined by $ω_{X, can}$ $ω_{X, can}$ and which is described uniquely by the combinatorial structure of $X$ $X$ . The construction of this Laplacian and the study of its spectral properties are the consequence of a generalized spectral theory of Laplacians on compact Kähler manifolds that we develop in this paper.

Keywords:

AMSC: 35J05, 32W20, 58C40, 14M25

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