We introduce a combinatorial system called toric wedge induction, which proceeds inductively by applying wedge operations to toric spaces. Using this method, we extend the classification of complete non-singular toric varieties from the cases with Picard number 2 and 3, due to Kleinschmidt (1988) and Batyrev (1991), respectively, to the case of Picard number 4.
We modify toric wedge induction by strengthening the basis step and weakening the inductive step. With this modification, we prove that any real toric space over a PL sphere with a small number of vertices admits a lift to a toric space over the same PL sphere.
According to the fundamental theorem of toric geometry, a fan completely determines the geometric properties of the associated toric variety. For algebraic geometers, this makes toric varieties natural testing grounds for developing and exploring new ideas. Unlike complex toric varieties, the geometric properties of real toric varieties remain largely unknown, which is often the case in real algebraic geometry. Therefore, exploring the topological aspects of real toric varieties, as potential testing grounds in real algebraic geometry, is important for uncovering connections with their geometric structure.
In this talk, we introduce the notion of real toric varieties and their topological invariants, and investigate some notable properties through examples. In particular, we discuss how real toric varieties associated with graphs lead to rich combinatorial structures and exhibit notable topological behavior.
In this talk, we discuss the multiplicative structure of the cohomology rings of real toric varieties. Although S. Choi and H. Park presented a formula for the cup product in their cohomology in 2020, the cohomology ring does not form a commutative algebra, which makes the product structure significantly more intricate than in the complex case. We illustrate this structure through concrete examples, focusing in particular on real permutohedral varieties.
This is an introductory talk on fiber bundles with the aim of a better understanding of toric manifolds. Being a space that locally looks like a product, a fiber bundle has a nice description especially in the toric case. Basic properties of fiber bundles and explicit examples will be given. I will also review some classification results in various context and ask related questions.
Bott manifolds are smooth projective toric varieties providing interesting avenues among topology, geometry, representation theory, and combinatorics. They are used to understand the geometric structure of Bott–Samelson varieties, which provide desingularizations of Schubert varieties. However, not all Bott manifolds originate from Bott–Samelson varieties. Those that do are specifically referred to as Bott manifolds of Bott–Samelson type. In this talk, we will first consider the basic notions of flag varieties, Schubert varieties, and Bott–Samelson varieties. After that, we will explore a relationship between Bott manifolds of Bott–Samelson type and assemblies of ordered partitions. This talk is based on an ongoing project with Junho Jeong and Jang Soo Kim.
A partial flag variety is a smooth projective homogeneous variety G/P admitting an action of a maximal torus T of G, where G is a semisimple algebraic group over ℂ and P is a parabolic subgroup. Schubert varieties form an interesting family of T-invariant subvarieties of the partial flag varieties. In this talk, we are going to consider toric Schubert varieties in G/P with respect to the action of a quotient of the torus T. This talk is based on an ongoing project with Eunjeong Lee.
A (full) flag variety is defined as the homogeneous space G/B, which is a smooth projective variety, where G is a semisimple algebraic group over ℂ and B is a Borel subgroup. Although a flag variety is not necessarily toric, it admits toric degenerations. Recently, Fujita and Oya provided a family of toric degenerations arising from cluster structures. When the number of seeds in a given cluster structure is finite, we say that the cluster structure is of finite type. In this talk, we consider a family of toric degenerations of flag varieties of finite type arising from cluster structures. In particular, we analyze toric degenerations of the flag variety Flag(ℂ5) of cluster type D6. This talk is based on an ongoing project with Eunjeong Lee and Shin Hong.