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Real permutohedral varieties and their generalizations

  • 윤영한 (아주대학교 일반대학원)

초록/요약

Real toric varieties arise as the real loci of toric varieties. However, their topological and combinatorial structures remain far less understood than those of toric varieties themselves. This thesis aims to deepen our understanding of the topological and combinatorial structure of real toric varieties, focusing on real permutohedral varieties and their generalizations. We first study real Coxeter toric varieties and compute their Betti numbers for all types, thereby resolving the previously open cases E7 and E8. We also describe the cohomology rings of the type A and type B real permutohedral varieties in terms of alternating permutations and B-snakes. Next, we investigate nestohedra. For chordal nestohedra, we show that the Betti numbers of the associated real toric varieties are computed by alternating permutations, analogous to how the Betti numbers of the corresponding toric varieties are determined by permutations. For graph associahedra, we establish a decomposition formula for the Betti numbers of the associated real toric varieties, and this result yields examples whose Betti sequences are unimodal but not log-concave. Finally, we turn to Bier spheres. We determine the homotopy types of particular families of their induced subcomplexes and apply these results to explicitly describe the cohomology of the associated real toric varieties.

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목차

1 Introduction 1
2 Preliminaries 5
2.1 Simplicial complexes 5
2.2 Toric manifolds 8
2.3 Real toric manifolds 10
3 Real Coxeter toric varieties 13
3.1 Coxeter complexes 13
3.2 Betti numbers of real Coxeter toric varieties 15
3.3 Cohomology rings of real permutohedral varieties 26
3.4 Cohomology rings of real type B permutohedral varieties 46
4 On the nestohedra 67
4.1 Nestohedra 67
4.2 Graph associahedra 70
4.3 Chordal nestohedra 84
4.4 Hochschild Polytopes 106
4.5 Venustus building sets 113
4.6 Unimodality of the Betti sequence 118
5 On the Bier spheres 126
5.1 Bier spheres 126
5.2 Real toric manifolds associated to Bier spheres 127
References 139
Appendix 150

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