Todd genera of complex torus manifolds

Ishida, Hiroyuki; Masuda, Mikiya

doi:10.2140/agt.2012.12.1777

Cited by 4 publications

(5 citation statements)

References 9 publications

(7 reference statements)

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“…Then we have a map f λ : | wedge v (K)| → S n as we defined in Subsection 3.4. By Lemma 4.2 of [19], the restriction…”

Section: Toric Objects Over the Complex K(j)mentioning

confidence: 98%

“…Let (K, λ) be a characteristic map of dimension n and I ∈ K a face of K. One defines the cone over I be the positive hull pos{λ(i) | i ∈ I} and denote it by ∠λ I . From now on, we assume (K, λ) is complete and follow an argument of Section 4 of [19]. First, we consider the (geometric) simplicial complex |K| which is an (n − 1)-dimensional sphere.…”

Section: Topological Toric Manifolds and Characteristic Mapsmentioning

confidence: 99%

“…Then we have a map f λ : | wedge v (K)| → S n as we defined in Subsection 3.4. By Lemma 4.2 of [19], the restriction f λ | St{v i } (i = 1, 2) is an embedding of an n-disc into S n , where St{v i } denotes the open star of v i in wedge v (K). Observe that St{v 1 } ∩ St{v 2 } contains a maximal simplex {v 1 , v 2 } ∪ τ , where τ is a maximal simplex of Lk K {v}.…”

Section: Toric Objects Over the Complex K(j)mentioning

confidence: 99%

“…We can check whether a characteristic map is fan-giving. The following is a version of Lemma 4.1 of [19].…”

Section: Topological Toric Manifolds and Characteristic Mapsmentioning

confidence: 99%

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Wedge operations and torus symmetries

Choi¹,

Park²

2016

Tohoku Math. J. (2)

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A fundamental result of toric geometry is that there is a bijection between toric varieties and fans. More generally, it is known that some class of manifolds having well-behaved torus actions, called topological toric manifolds M 2n , can be classified in terms of combinatorial data containing simplicial complexes with m vertices. We remark that topological toric manifolds are a generalization of smooth toric varieties. The number m − n is known as the Picard number when M 2n is a compact smooth toric variety.In this paper, we investigate the relationship between the topological toric manifolds over a simplicial complex K and those over the complex obtained by simplicial wedge operations from K. As applications, we do the following.(1) We classify smooth toric varieties of Picard number 3. This is a reproving of a result of Batyrev.(2) We give a new and complete proof of projectivity of smooth toric varieties of Picard number 3 originally proved by Kleinschmidt and Sturmfels.(3) We find a criterion for a toric variety over the join of boundaries of simplices to be projective. When the toric variety is smooth, it is known as a generalized Bott manifold which is always projective. (4) We classify and enumerate real topological toric manifolds when m−n = 3. In particular, when P is a polytope whose Gale diagram is a pentagon with assigned numbers (a1, a3, a5, a2, a4), then every real topological toric manifold over P is a real toric variety, and the number #DJ of them up to Davis-Januszkiewicz equivalence is #DJ = 2 a 1 +a 4 −1 + 2 a 2 +a 5 −1 + 2 a 3 +a 1 −1 + 2 a 4 +a 2 −1 + 2 a 5 +a 3 −1 − 5.When P is a polytope whose Gale diagram is a heptagon with arbitrary assigned numbers, no real topological toric manifold over P is a real toric variety, and we have #DJ = 2. (5) When m − n ≤ 3, any real topological toric manifold is realizable as fixed points of the conjugation of a topological toric manifold.

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“…Then we have a map f λ : | wedge v (K)| → S n as we defined in Subsection 3.4. By Lemma 4.2 of [19], the restriction…”

Section: Toric Objects Over the Complex K(j)mentioning

confidence: 98%

Section: Topological Toric Manifolds and Characteristic Mapsmentioning

confidence: 99%

Section: Toric Objects Over the Complex K(j)mentioning

confidence: 99%

“…We can check whether a characteristic map is fan-giving. The following is a version of Lemma 4.1 of [19].…”

Section: Topological Toric Manifolds and Characteristic Mapsmentioning

confidence: 99%

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Wedge operations and torus symmetries

Choi¹,

Park²

2016

Tohoku Math. J. (2)

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“…Such an almost complex structure cannot be integrable. Indeed, according to the result of Ishida-Karshon [171] (Theorem 6.6.8, see also [172]), a quasitoric manifold with an invariant complex structure is biholomorphic to a compact toric variety.…”

Section: Under This Correspondence An Edge E Coming Out Of a Vertexmentioning

confidence: 99%

Toric Topology

Buchstaber¹,

Panov²

2012

Preprint

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Toric topology emerged in the end of the 1990s on the borders of equivariant topology, algebraic and symplectic geometry, combinatorics and commutative algebra. It has quickly grown up into a very active area with many interdisciplinary links and applications, and continues to attract experts from different fields.The key players in toric topology are moment-angle manifolds, a family of manifolds with torus actions defined in combinatorial terms. Their construction links to combinatorial geometry and algebraic geometry of toric varieties via the related notion of a quasitoric manifold. Discovery of remarkable geometric structures on moment-angle manifolds led to seminal connections with the classical and modern areas of symplectic, Lagrangian and non-Kähler complex geometry. A related categorical construction of moment-angle complexes and their generalisations, polyhedral products, provides a universal framework for many fundamental constructions of homotopical topology. The study of polyhedral products is now evolving into a separate area of homotopy theory, with strong links to other areas of toric topology. A new perspective on torus action has also contributed to the development of classical areas of algebraic topology, such as complex cobordism.The book contains lots of open problems and is addressed to experts interested in new ideas linking all the subjects involved, as well as to graduate students and young researchers ready to enter into a beautiful new area. IntroductionChapter guide Acknowledgments Chapter 1. Geometry and combinatorics of polytopes 1.1. Convex polytopes 1.2. Gale duality and Gale diagrams 1.3. Face vectors and Dehn-Sommerville relations 1.4. Characterising the face vectors of polytopes Polytopes: additional topics 1.5. Nestohedra and graph-associahedra 1.6. Flagtopes and truncated cubes 1.7. Differential algebra of combinatorial polytopes 1.8. Families of polytopes and differential equations Chapter 2. Combinatorial structures 2.1. Polyhedral fans 2.2. Simplicial complexes 2.3. Barycentric subdivision and flag complexes 2.4. Alexander duality 2.5. Classes of triangulated spheres 2.6. Triangulated manifolds 2.7. Stellar subdivisions and bistellar moves 2.8. Simplicial posets and simplicial cell complexes 2.9. Cubical complexes Chapter 3. Combinatorial algebra of face rings 3.1. Face rings of simplicial complexes 3.2. Tor-algebras and Betti numbers 3.3. Cohen-Macaulay complexes 3.4. Gorenstein complexes and Dehn-Sommerville relations 3.5. Face rings of simplicial posets Face rings: additional topics 3.6. Cohen-Macaulay simplicial posets 3.7. Gorenstein simplicial posets 3.8. Generalised Dehn-Sommerville relations Chapter 4. Moment-angle complexes 4.1. Basic definitions iii iv CONTENTS 4.2. Polyhedral products 4.3. Homotopical properties 4.4. Cell decomposition 4.5. Cohomology ring 4.6. Bigraded Betti numbers 4.7. Coordinate subspace arrangements Moment-angle complexes: additional topics 4.8. Free and almost free torus actions on moment-angle complexes 4.9. Massey products in the cohomolo...

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