Kähler metrics on singular toric varieties
Let \(G\) be a compact torus with Lie algebra \(\mathfrak g\) and integral lattice \(\mathbb Z_G\subset \mathfrak g\). Let \(u_1,\dots,u_N\in\mathbb Z_G\) be a set of primitive vectors generating the \(\mathbb R\)-vector space \(\mathfrak g\), and for \(\lambda_1,\dots\lambda_N\in \mathbb R\) consider the polyhedral set \(P=P_{u,\lambda}:=\{\eta\in\mathfrak g^*\,| \,\langle\eta,u_j\rangle -\lambda_j\geq 0, 1\leq j\leq N\}\) when \(P\) has nonempty interior and the collection of inequalities defining \(P\) is minimal. The authors modify a construction of \textit{T. Delzant} [Bull. Soc. Math. Fr. 116, No. 3, 315--339 (1988; Zbl 0676.58029)] who established a one-to-one correspondence between the class of \(2n\)-dimensional compact toric symplectic manifolds and a specific class of convex polytopes in \(\mathbb R^n\), now known as Delzant polytopes. The Delzant polytopes are the images of the momentum maps of those manifolds. In the present paper the authors associate to \(P\) a symplectic stratified space \(M\) with an effective Hamiltonian action of \(G\) and momentum map \(\phi:M\rightarrow\mathfrak g^*\) with \(\phi(M)=P\). The space \(M\) is a symplectic quotient of \(\mathbb C^N\) by a compact subgroup of the standard torus \(T^N\) and therefore a Kählerian complex space. Moreover \(M\) is a toric space, since the \(G\)-action on \(M\) extends to an analytic action of the complexification \(G^{\mathbb C}\) of \(G\) with dense open orbit. For proofs of these facts see the results of \textit{P. Heinzner, A. Huckleberry} and \textit{F. Loose} in [Geom. Funct. Anal. 4, No. 3, 288--297 (1994; Zbl 0816.53018), J. Reine Angew. Math. 455, 123--140 (1994; Zbl 0803.53042), Invent. Math. 126, No. 1, 65--84 (1996; Zbl 0855.58025)]. The \(\phi\)-preimages of the open faces of \(P\) are also quotients of this type and submanifolds of \(M\). The authors give explicit formulas for Kähler potentials on these manifolds, hereby generalising a result of V. Guillemin for compact \(M\): the function \(\sum_{j=1}^N \lambda_j\log(u_j-\lambda_j)+u_j)\circ\phi\) is a Kähler potential on the \(\phi\)-preimage of the interior of \(P\), where \(u_j\) is considered as a function on \(\mathfrak g^*\). For related results see also \textit{D. M. Calderbank, L. David} and \textit{P. Gauduchon} [J. Symplectic Geom. 1, No. 4, 767--784 (2003; Zbl 1155.53336)] and \textit{D. Burns} and \textit{V. Guillemin} [Commun. Anal. Geom. 12, No. 1--2, 281--303 (2004; Zbl 1073.53113)].
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