Toric topology of torus actions of the positive complexity

Toric topology of torus actions of the positive complexity

Victor Buchstaber, Steklov Mathematical Institute, RAS and Svjetlana Terzić, University of Montenegro

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Passcode: 114700

The problems related to the standard  action of the compact torus $T^{n}$ on a complex Grassmann manifold $G_{n,2}$, $n\geq 3$  are widely known in algebraic topology, algebraic geometry and mathematical physics.  This action has the complexity $n-3$ for a given $n$. The talks are devoted to toric topology of the family $\{(G_{n,2}, T^n)\}$ whose members are connected by the natural equivariant embeddings.  

 In the seminal papers of Gel'fand, Serganova, Goresky, MacPherson, it was studied the action of the algebraic torus $(\mathbb{C} ^{\ast})^{n}$ on $G_{n,2}$ using  the canonical moment map $\mu : G_{n,2} \to \Delta_{n,2}$, where $\Delta_{n,2}$ is the hypersimplex.   Their results  were formulated  in  terms of  the strata $\{W_{\sigma}\}$ for  the $(\mathbb{C} ^{\ast})^{n}$  -action  on $G_{n,2}$ and the decomposition of $\Delta _{n,2}$ into the chambers.

 In the first talk (October 22) it will be given the description of the  orbit space $G_{n,2}/T^n$ in  the new notions: an universal space of parameters $\mathcal{F}_{n}$; virtual spaces    of parameters $\widetilde{F}_{\sigma}\subset \mathcal{F}_{n}$ of  the strata $W_{\sigma}$; the correspondence which  to  the set of the strata defining a  chamber  assigns the decomposition of the space $\mathcal{F}_{n}$  into the corresponding virtual spaces of parameters.

 In modern algebraic geometry it is known the notion of the wonderful compactification  based on the arrangement of smooth  subvarieties in a smooth algebraic variety. In the second talk (November 5)   we describe   our smooth manifolds $\mathcal{F}_{n}$   in terms of the wonderful compactification and  show that the   family ${\mathcal{F}_{n}\} $ can be identified with the family   $\{\overline{M(0,n)}\}$, where $\overline{M(0,n)}$ is   the Deligne-Mumford compactification   of  the moduli space $M(0,n)$,  which plays an important role in  known  problems of modern  mathematical physics. In toric geometry and toric topology are obtained many results in terms of subspace arrangements. The proofs of the results presented in the first talk essentially use our description of the chamber decomposition of $\Delta _{n,2}$ based on the special  hyperplane arrangement.

This description will be also presented  in the second  talk.