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Unitary toric manifolds, multi-fans and equivariant index. (English) Zbl 0940.57037

There is one-to-one correspondence between toric varieties (objects in algebraic geometry) and fans (objects in combinatorics). A toric manifold is a compact nonsingular toric variety. In the article under review, the author describes a correspondence between toric manifolds and regular fans from a topological point of view. More precisely, he introduces a geometrical object called a unitary toric manifold and (using equivariant cohomology) associate to it a combinatorial object called a multi-fan. A unitary toric manifold \(M\) is a compact unitary manifold with an action of a compact torus \(T\) having nonempty isolated fixed points, where \(2\dim_{\mathbb{R}}T= \dim_{\mathbb{R}}M\). The multi-fan is a collection of cones (together with two functions on maximal cones) in the homology \(H_2(BT;\mathbb{R})\) of the classifying space \(BT\). The cones may overlap and the degree of the overlap is essentially the Todd genus of the unitary toric manifold. A moment map relates a unitary toric manifold equipped with an equivariant complex line bundle to a twisted polytope, and the equivariant Riemann-Roch index for the equivariant line bundle can be described in terms of the moment map. The author applies this result to establish a generalization of Pick’s formula describing the number of lattice points on the domain bounded by an integral simple plane polygon \(P\) in terms of the area of the bounded domain and the number of lattice points on \(P\). The formula is generalized to any integral plane polygon which may have self-intersections.

MSC:

57S15 Compact Lie groups of differentiable transformations
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
57S25 Groups acting on specific manifolds
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