On the topology of some torus invariant
subsets in the Grassmannians
K. Ogiwara(Osaka City University)

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Toric varieties in Gk (Cn )(I.M. Gel’fand, and R.D. MacPherson)
Geometry in Grassmannians and a Generalization of theDilogarithm, 1982

(I.M. Gel’fand, R.M. Goresky, R.D. MacPherson, and V.V.
Serganova)
Combinatorial geometries, convex polyhedra, and Schubert cells,
1987.

(Victor M. Buchstaber and Svjetlana Terzić)Toric topology of the complex grassmann manifolds, 2018.

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Toric varieties in Gk (Cn )Gk (Cn ): set of all k-dimensional vector
subspaces in Cn(C∗ )n : algebraic torus, where C∗ = C \ {0}(C∗ )n ↷ Gk (Cn )For an orbit O ⊂ Gk (Cn ), the closure O is atoric variety.

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Toric varieties in Gk (Cn )∃µ (moment map) : Gk (Cn ) → Rn s.t.
• µ(O) is a (convex) polytope in Rn ,
{ﬁxed points of O} ←→ {vertices of µ(O)};• O is smooth ⇔ µ(O) is a simple polytope.
Question. Which O’s are smooth?
→ Find simple µ(O)′ s.

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Toric varieties in Gk (Cn )Actually, µ(O) is a matroid polytope.
(combinatorial)Theorem (Noji-O).
A simple matroid polytope is a product of simplices.
(geometric)Theorem (Noji-O).
If O is smooth, then O is a product of complex
projective spaces.
(µ(O) = ∆d1 × · · · × ∆dr ⇒ O = CP d1 × · · · × CP dr )

1Remark on our results

We found a paper which prove the following:
Theorem.
Every simple 0/1-polytope is the (cartesian) product ofsome 0/1-simplices. (Volker Kaibel and Martin Wolﬀ, Simple
0/1-polytopes, Europ. J. Combinatorics (2000) 21, 139–144. )Matroid polytopes are 0/1-polytopes and this easily
implies our combinatorial Theorem in the previous slide.
However, at least for matroid polytopes, I and Nojiproved a stronger statement in our paper to prove ourcombinatorial Theorem in the previous slide.

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Toric varieties in Gk (Cn )∃µ (moment map) : Gk (Cn ) → Rn s.t.
• µ(O) is a (convex) polytope in Rn ,
{ﬁxed points of O} ←→ {vertices of µ(O)};• O is smooth ⇔ µ(O) is a simple polytope.

One viewpoint.
The more edges emanate from a vertex of µ(O), the“more singular ” O is, at the corresponding ﬁxed point.
Question.
How many edges do emanate from a vertex of µ(O)?

1Toric varieties in Gk (Cn )We may assume that O is non-degenerate i.e.
dim O = n − 1,
becausedim O = n − r⇒
O = O1 × · · · × Or
and each Oi can be regarded as non-degeneratenitorus orbit closure in Gki (C ).

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Toric varieties in Gk (Cn )When k = 0 or 1,Gk (Cn ) = {a point}.
When k = 1 or n − 1,Gk (Cn ) = complex projective space.
And then (non-degenerate O)= Gk (Cn ).
(∵ O is unique open dense orbit in Gk (Cn ),since Gk (Cn ) is the complex projective space)Thus, we consider the cases when 2 ≤ k ≤ n − 2.

2µ(O) for non-degenerate O ⊂ G2 (C4 )O ⊂ G2 (C )4µ(∃O) = Octahedronµ(∃O) = Pyramid3A iniquality for the number of edges from a vertex

For a polytope P , letD(P ) = max dv − dim P.
v∈V (P )V (P ): set of all vertices of polytope anddv : the number of edges which emanate from v
Then

P is simple ⇔ D(P ) = 0.

“Theorem” (O).
When 2 ≤ k ≤ n − 2, for non-degenerate O√
33(n − 3) − 1 ≤ D(µ(O)).
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3A iniquality for the number of edges from a vertex

When 5 ≤ n,√√

0 < 3 34 (5 − 3) − 1 ≤ 3 34 (n − 3) − 1 ≤ D(µ(O)).

This means that O is not smooth when 5 ≤ n,which reprove the previous theorem.
Remark.
More generally, the above assertion holds for matroid polytopesof connected matroids on [n] of rank k.

4Remark on our viewpoint

Other viewpoint can be taken. Indeed, we can think that themore facets intersect at a vertex of µ(O), the “more singular” Ois, at corresponding ﬁxed point.
For generic O ⊂ Gk (Cn ),µ(O) = ∆n,k is called a hypersimlex.
The ∆n,k has n facets at each vertex
while it has k(n − k) edges at each vertices.
Since for a general (n − 1)-dimensional polytope, the minimum
number of its edgs and facets at a vertex of it are both n − 1, thetwo viewpoints give diﬀerent impressions of the singularity of O.