Wedge Operations and Torus Symmetries II

Choi, Suyoung ; Park, Hanchul

Canadian journal of mathematics, Tome 69 (2017) no. 4, pp. 767-789

Résumé

A fundamental idea in toric topology is that classes of manifolds with well-behaved torus actions (simply, toric spaces) are classified by pairs of simplicial complexes and (non-singular) characteristic maps. In a previous paper, the authors provided a new way to find all characteristic maps on a simplicial complex $K(J)$ obtainable by a sequence of wedgings from $K$ .The main idea was that characteristic maps on $K$ theoretically determine all possible characteristic maps on a wedge of $K$ .We further develop our previous work for classification of toric spaces. For a star-shaped simplicial sphere $K$ of dimension $n-1$ with $m$ vertices, the Picard number Pic $(K)$ of $K$ is $m-n$ . We call $K$ a seed if $K$ cannot be obtained by wedgings. First, we show that for a fixed positive integer $\ell $ , there are at most finitely many seeds of Picard number $\ell $ supporting characteristic maps. As a corollary, the conjecture proposed by V. V. Batyrev inis solved affirmatively.Secondly, we investigate a systematicmethod to find all characteristic maps on $K(J)$ using combinatorial objects called (realizable) puzzles that only depend on a seed $K$ . These two facts lead to a practical way to classify the toric spaces of fixed Picard number.

Détail

DOI : 10.4153/CJM-2016-037-4

Mots-clés : 57S25, 14M25, 52B11, 13F55, 18A10, puzzle, toric variety, simplicial wedge, characteristic map

Bibliographie

[1] [1] Bahri, A., Bendersky, M., Cohen, F. R., and Gitler, S., Operations on polyhedral products and a new topological construction of infinite families oftoric manifolds. Homology Homotopy Appl. 17(2015), no. 2, 137–160. Google Scholar | DOI

[2] [2] Batyrev, V. V., On the classification of smooth projective toric varieties. Tohoku Math. J. (2) 43(1991), no. 4, 569–585. Google Scholar | DOI

[3] [3] Buchstaber, V. M. and Panov, T. E., Torus actions and their applications in topology and combinatorics. University Lecture Series, 24. American Mathematical Society, Providence, RI, 2002. Google Scholar

[4] [4] Choi, S. and Park, H., Wedge operations and torus symmetries. Tohoku Math. J. (2), 68(2016), no. 1,91–138. Google Scholar | DOI

[5] [5] Choi, S., Wedge operations and a new family of projective toric manifolds. To appear in Israel J. Math. arxiv:1 507.08919. Google Scholar

[6] [6] Erokhovets, N., Buchstaber invariant of simple polytopes. Russian Math. Surveys 63(2008), no. 5, 962–964 (Russian), Google Scholar | DOI

[7] [7] Grünbaum, B., Convex polytopes. Second ed. Graduate Texts in Mathematics, 221. Springer-Verlag, New York, 2003. Google Scholar

[8] [8] Kleinschmidt, P.,A classification oftoric varieties with few generators. Aequationes Math. 35(1988), no. 2-3, 254–266. http://dx.doi.Org/10.1007/BF01830946 Google Scholar

[9] [9] Kleinschmidt, P. and Sturmfels, B., Smooth toric varieties with small Picard number are projective. Topology 30(1991), no. 2, 289–299. http://dx.doi.Org/10.1016/0040-9383(91 )90015-V Google Scholar

[10] [10] Mani, P., Spheres with few vertices. J. Combinatorial Theory Ser. A 13(1972), 346–352. http://dx.doi.Org/10.1016/0097-315(72)90068-4 Google Scholar

[11] [11] Oda, T., Convex bodies and algebraic geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete(3), 15. Springer-Verlag, Berlin, 1988. Google Scholar