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Medientyp:
E-Artikel
Titel:
Combinatorics and topology of proper toric maps
Beteiligte:
de Cataldo, Mark Andrea;
Migliorini, Luca;
Mustaţă, Mircea
Erschienen:
Walter de Gruyter GmbH, 2018
Erschienen in:
Journal für die reine und angewandte Mathematik (Crelles Journal), 2018 (2018) 744, Seite 133-163
Sprache:
Englisch
DOI:
10.1515/crelle-2015-0104
ISSN:
1435-5345;
0075-4102
Entstehung:
Anmerkungen:
Beschreibung:
Abstract We study the topology of toric maps. We show that if f : X → Y {f\colon X\to Y} is a proper toric morphism, with X simplicial, thenthe cohomology of every fiber of f is pure and of Hodge–Tate type. When the map is a fibration, we give an explicit formula for the Betti numbersof the fibers in terms of a relative version of the f-vector,extending the usual formula for the Betti numbers of a simplicial complete toric variety. We then describe the Decomposition Theoremfor a toric fibration, giving in particular a nonnegative combinatorial invariant attached to each cone in the fan of Y, which is positiveprecisely when the corresponding closed subset of Y appears as a support in the Decomposition Theorem. The description of this invariant involvesthe stalks of the intersection cohomology complexes on X and Y, but in the case when both X and Y are simplicial, there is a simple formulain terms of the relative f-vector.