Kaehler structures on toric varieties
Let \((X,\omega)\) be a compact connected \(2n\)-dimensional manifold, and let \(\tau: T^ n\to\text{Diff}(X, \omega)\) be an effective Hamiltonian action of the standard \(n\)-torus. Let \(\varphi: X\to \mathbb{R}^ n\) be its moment map. The image, \(\Delta\), of \(\varphi\) is a convex polytope, called the moment polytope. The main purpose of this paper is to study which geometry of \(X\) is determined by \(\Delta\). \(\Delta\) can be described by a set of inequalities of the form \(\langle x, u_ i\rangle\geq \lambda_ i\) \((i=1,\dots, d)\) the \(u_ i\)'s being primitive elements of the lattice, \(\mathbb{Z}^ n\), and \(d\) the number of \((n-1)\)-dimensional faces of \(\Delta\). Let \(l_ i(x)= \langle x, u_ i\rangle- \lambda_ i\) and let \(\Delta^ 0\) be the interior of \(\Delta\). If \(X_ r\) is the fixed point set of an intrinsic involution \(\gamma: X\to X\) which reverses the complex structure and maps \(\omega\) to \(-\omega\) then the restriction of \(\varphi\) to \(X_ r\) is a ramified cover \(\psi: X_ r\to \Delta\) which breaks into \(2^ n\) connected components \(X_ r^ \varepsilon\), \(\varepsilon= (\pm1, \dots, \pm1)\) over \(\Delta^ 0\). The author shows that the metric on \(X_ r^ \varepsilon\) is the pullback to \(\psi\) of the metric \({1\over 2} \sum_{i=1}^ d {{(dl_ i)^ 2} \over {l_ i}}\) on \(\Delta^ 0\). For projective embeddings \(\iota\) of \(X\) is shown \(T^ n\)-equivariantly symplectomorphicity of \(\omega\) and \(\iota^* \omega_{FS}\). If \(c_ i\) is the cohomology class in \(H^ 2 (X,\mathbb{Z})\) dual to this manifold then the author finds \({1\over {2\pi}} [\omega ]=- \sum_{i=1}^ d \lambda_ i c_ i\). Some applications of this formula are obtained: a new proof of the `combinatorial Riemann-Roch' formula of Khovanskij and Pukhlikov; a generalization of a well-known result on the topology of toric surfaces related to the intersection number \(\#(X_ i\cap X_ j)\), where \(X_ i\) is the codimension-2 submanifold of \(X\) defined by \(l_ i\circ \varphi=0\).
scientific article; zbMATH DE number 700984