Transformation Groups Seminar

Speaker: Mathieu Vallée (Université Libre, Brussels)

"Classification of toric manifold of small Picard number"

Time: 14:30

Room: MC 108

Toric varieties form a specific class of algebraic varieties equipped with a well-behaved action of an algebraic torus. They provide a useful setting for testing conjectures, as they admit a particularly explicit and combinatorial description. The fundamental theorem of toric geometry states that toric varieties correspond to fans, that is, sets of strongly convex polyhedral cones in $\mathbb{R}^n$ that are closed under taking faces and whose relative interiors are pairwise disjoint. Properties of the fan translate directly into geometric properties of the associated toric variety. In particular, a toric variety is complete if and only if the cones of the fan cover the whole space $\mathbb{R}^n$, and it is non-singular if and only if each cone is generated by part of a basis of the integer lattice $\mathbb{Z}^n$. We focus here on characterizing complete non-singular toric varieties, also called toric manifolds. The Picard number of a toric manifold is the rank of its Picard group; this equals the number of the 1-dimensional cones minus the dimension of its associated fan. There are two major directions of research toward this characterization: studying toric manifolds of fixed (small) dimension, or studying those with fixed (small) Picard number. In dimension 2, toric manifolds are completely understood: they are obtained by a sequence of toric blow-ups startying either from the complex projective plane or from a Hirzebruch surface. In any dimension $n$, the unique toric manifold of Picard number 1 is the complex projective space $\mathbb{C}P^n$, whose fan corresponds to the normal fan of a unimodular $n$-simplex. Kleinschmidt (1988) and Batyrev (1991) classified toric manifolds of Picard number 2 and 3, respectively. In this talk, I will present a sequence of joint works with Suyoung Choi and Hyeontae Jang leading to the classification of toric manifolds of Picard number 4 in terms of fans, relying mainly on a combinatorial construction known as the (simplicial) wedge operation; which was used for instance by F. Santos in his construction for disproving the Hirsch conjecture. I will also discuss recent advances toward the case of Picard number 5.

Transformation Groups Seminar

Speaker: Xin Fu (SIMIS, Shanghai)

"Szczarba's twisted shuffle map and equivariant path homology of directed graphs"

Time: 14:30

Room: MC 108

Inspired by the GLMY path homology theory of directed graphs and its generalisations to quivers and marked categories, we associate a path chain complex to a marked simplicial set and define its path homology. In this talk, I will introduce a Borel-type construction for marked simplicial sets equipped with simplicial group actions and twisting functions. This construction is given by a marked version of the twisted Cartesian product using the box product.

A classical theorem of Szczarba states that the twisted shuffle map induces a quasi-isomorphism between the chain complex of a twisted Cartesian product and an associated twisted tensor product. In the marked setting, we prove that this map restricts to an isomorphism of path chain complexes. As an application, digraphs with group actions admit a natural Borel construction as a special case of our framework. This leads to a notion of equivariant path homology, which can be computed via an explicit twisted tensor product. This is joint work with Shing-Tung Yau.

PhD Thesis Defence

Speaker: Tao Gong (Western)

"Toric varieties and Weyl groups"

Time: 12:30

Room: MC 204

We study toric varieties associated with $W$-permutohedra and their quotients by parabolic subgroups. For a Weyl group $W$ and a $W$-permutohedron $P$, the quotient $P/W_K$ by a parabolic subgroup $W_K$ can be identified with a polytope inside $P$, giving rise to toric varieties $X_P$ and $X_{P/W_K}$.

We construct an explicit algebra isomorphism between the rational cohomology ring of $X_{P/W_K}$ and the invariant of the rational cohomology ring of $X_P$$$H^*(X_{P/W_K};\mathbb{Q})\cong H^*(X_P;\mathbb{Q})^{W_K},$$ and generalize this result to non-degenerate $W$-symmetric polytopes, to intermediate lattices, and to finite Coxeter groups via a polytopal algebra model.

From a topological perspective, we prove that $X_P/W_K$ is homotopy equivalent to $X_{P/W_K}$, and extend this result to non-degenerate $W$-symmetric polytopes. In joint work with Crowley and Simpson, we further establish a variety isomorphism between $X_P/W_K$ and $X_{P/W_K}$, and generalize this isomorphism to $W$-symmetric polytopes.

These results give affirmative answers to questions of Horiguchi--Masuda--Shareshian--Song concerning equivalences between $X_P/W_K$ and $X_{P/W_K}$.

Finally, we study the real locus $X_P^{\mathbb{R}}$ of the toric variety $X_P$. We show that the quotient $X_P^{\mathbb{R}}/W$ is contractible, and describe the homotopy type of $X_P^{\R}/W_K$ in low-dimensional cases.