On the classification of real toric manifolds

On the classification of real toric manifolds

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Suyoung Choi, Ajou University

Online Talk 

Zoom link: https://princeton.zoom.us/j/92116764865

Passcode: 114700

Toric manifolds of Picard numbers  smaller than 4 have been entirely classified by Kleinschmidt(1988) and Batyrev(1991). Recently, their results have been recovered by Choi-Park (2016).  Simplicial complexes which cannot be described as the wedge of a lower dimensional simplicial complex are called seeds. Choi-Park showed that, for a fixed Picard number $p$, there are at most finite seeds which support toric manifolds. In addition, they provided an algorithm to construct all toric manifolds of Picard number $p$ from such seeds. By using this, one can reprove the results of Kleinschmidt and Batyrev, and there will be a chance to solve the case of Picard number $4$. However, it was not successful yet, since it is quite challenging to list up all seeds (for the given Picard number) which support toric manifolds although we know that they are of finite number.

Note that if a simplicial complex supports a toric manifold, then it also supports a real toric manifold. Furthermore, the number of real toric manifolds over a given simplicial complex is always finite. Therefore, the classification of  smooth real toric manifolds is an essential pre-step of the classification of toric manifolds.

In this talk, we provide an improved algorithm for finding real toric manifolds over wedged simplicial complexes. It is indeed much faster than other known algorithms including Garrison-Scott's algorithm. In addition, we discuss the possible listing of seeds which support (real) toric manifolds of Picard number 4.

This talk is based on ongoing projects with Mathieu Vallée.