We study the topology of toric maps. We show that if f W X ! Y is a proper toric morphism, with X simplicial, then the 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 numbers of 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 The- orem for a toric fibration, giving in particular a nonnegative combinatorial invariant attached to each cone in the fan of Y , which is positive precisely when the corresponding closed sub- set of Y appears as a support in the Decomposition Theorem. The description of this invariant involves the 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 formula in terms of the relative f -vector.
Combinatorics and topology of proper toric maps / de Cataldo, Mark Andrea; Migliorini, Luca; Mustaţă, Mircea. - In: JOURNAL FÜR DIE REINE UND ANGEWANDTE MATHEMATIK. - ISSN 0075-4102. - STAMPA. - 744 (2018):(2018), pp. 133-163. [10.1515/crelle-2015-0104]
Combinatorics and topology of proper toric maps
Migliorini, Luca;
2018
Abstract
We study the topology of toric maps. We show that if f W X ! Y is a proper toric morphism, with X simplicial, then the 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 numbers of 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 The- orem for a toric fibration, giving in particular a nonnegative combinatorial invariant attached to each cone in the fan of Y , which is positive precisely when the corresponding closed sub- set of Y appears as a support in the Decomposition Theorem. The description of this invariant involves the 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 formula in terms of the relative f -vector.File | Dimensione | Formato | |
---|---|---|---|
[14355345 - Journal für die reine und angewandte Mathematik] Combinatorics and topology of proper toric maps.pdf
accesso aperto
Tipo:
Versione (PDF) editoriale
Licenza:
Licenza per accesso libero gratuito
Dimensione
343.43 kB
Formato
Adobe PDF
|
343.43 kB | Adobe PDF | Visualizza/Apri |
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.