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Theory of multi-fans

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arxiv math/0106229 v3 pith:A73ZXTOX submitted 2001-06-27 math.SG math.AT

Akio Hattori , Mikiya Masuda This is my paper · ORCID
classification math.SGmath.AT
keywords torusvarietymanifoldtheorytoricalgebraiccombinatorialmulti-fan
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We introduce the notion of a multi-fan. It is a generalization of that of a fan in the theory of toric variety in algebraic geometry. Roughly speaking a toric variety is an algebraic variety with an action of algebraic torus of the same dimension as that of the variety, and a fan is a combinatorial object associated with the toric variety. Algebro-geometric properties of the toric variety can be described in terms of the associated fan. We develop a combinatorial theory of multi-fans and define ``topological invariants'' of a multi-fan. A smooth manifold with an action of a compact torus of half the dimension of the manifold and with some orientation data is called a torus manifold. We associate a multi-fan with a torus manifold, and apply the combinatorial theory to describe topological invariants of the torus manifold. A similar theory is also given for torus orbifolds. As a related subject a generalization of the Ehrhart polynomial concerning the number of lattice points in a convex polytope is discussed.

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