T n -action on the Grassmannians G n , 2 via hyperplane arrangements - PowerPoint PPT Presentation

T n -action on the Grassmannians G n , 2 via hyperplane arrangements Svjetlana Terzi´ c University of Montenegro based on joint results with Victor M. Buchstaber Workshop on Torus Actions in Topology Fields Institute for Research in Mathematics May 11, 2020. T n -action on the Grassmannians G n , 2 via hyperplane arrangements Svjetlana Terzi´ c (University of Montenegro based on joint results with Victor M. Buchstaber Workshop on Torus Actions in Topology Fields 1 / 29

Complex Grassmann manifolds G n , k = G n , k ( C ) G n , k – k -dimensional complex subspaces in C n , The coordinate-wise T n - action on C n induces T n - action on G n , k . This action is not effective — T n − 1 = T n / ∆ acts effectively. d = k ( n − k ) − ( n − 1 ) - complexity of T n − 1 -action; d ≥ 2 for n ≥ k + 3, k ≥ 2. T n -action extends to ( C ∗ ) n -action on G n , k Problem: Describe the combinatorial structure and algebraic topology of the orbit space G n , k / T n ∼ = G n , n − k / T n . T n -action on the Grassmannians G n , 2 via hyperplane arrangements Svjetlana Terzi´ c (University of Montenegro based on joint results with Victor M. Buchstaber Workshop on Torus Actions in Topology Fields 2 / 29

V. M. Buchstaber and S. Terzi´ c, Topology and geometry of the canonical action of T 4 on the complex Grassmannian G 4 , 2 and the complex projective space CP 5 , Moscow Math. Jour. Vol. 16, Issue 2, (2016), 237–273. V. M. Buchstaber and S. Terzi´ c, Toric Topology of the Complex Grassmann Manifolds , Moscow Math. 19 , no. 3, (2019) 397-463. V. M. Buchstaber and S. Terzi´ c, The foundations of ( 2 n , k ) -manifolds , Sb. Math. 210, No. 4, 508-549 (2019). I. M. Gelfand and V. V. Serganova, Combinatoric geometry and torus strata on compact homogeneous spaces , Russ. Math. Survey 42 , no.2(254), (1987), 108–134. (in Russian) I. M. Gelfand, R. M. Goresky, R. D. MacPherson and V. V. Serganova, Combinatorial Geometries, Convex Polyhedra, and Schubert Cells , Adv. in Math. 63 , (1987), 301–316. M. M. Kapranov, Chow quotients of Grassmannians I , Adv. in Soviet Math., 16, part 2, Amer. Math. Soc. (1993), 29–110. T n -action on the Grassmannians G n , 2 via hyperplane arrangements Svjetlana Terzi´ c (University of Montenegro based on joint results with Victor M. Buchstaber Workshop on Torus Actions in Topology Fields 3 / 29

We describe here the orbit space G n , 2 / T n in terms of : 1. ”soft” chamber decomposition L ( A n , 2 ) for ∆ n , 2 , A = Π ∪ { x i = 0 , 1 ≤ i ≤ n } ∪ { x i = 1 , 1 ≤ i ≤ n } - hyperplane arrangement in R n ; Π = { x i 1 + . . . + x i l = 1 , 1 ≤ i 1 < . . . < i l ≤ n , 2 ≤ l ≤ [ n 2 ] } ; L ( A ) - face lattice for A ; ◦ L ( A n , 2 ) = L ( A ) ∩ ∆ n , 2 ; 2. spaces of parameters F C for C ∈ L ( A n , 2 ) - parametrize ( C ∗ ) n - orbits in µ − 1 n , 2 ( C ) ⊂ G n , 2 ; 3. universal space of parameters F . T n -action on the Grassmannians G n , 2 via hyperplane arrangements Svjetlana Terzi´ c (University of Montenegro based on joint results with Victor M. Buchstaber Workshop on Torus Actions in Topology Fields 4 / 29

Moment map � n ucker embedding G n , k → C P N − 1 , N = � The Pl¨ , is given by k � � L → P ( L ) = P I ( A L ) , I ⊂ { 1 , . . . n } , | I | = k , P I ( A L ) - Pl¨ ucker coordinates of L in a fixed basis. The moment map µ n , k : G n , k → R n is defined by 1 | P ( L ) | 2 = � � | P I ( A L ) | 2 Λ I , | P I ( A L ) | 2 , µ n , k ( L ) = | P ( L ) | 2 where Λ I ∈ R n has 1 at k places and it has 0 at the other ( n − k ) places, the sum goes over the subsets I ⊂ { 1 , . . . , n } , | I | = k . Im µ n , k = convexhull (Λ I ) = ∆ n , k – hypersimplex. ∆ n , k is in the hyperplane x 1 + · · · + x n = k in R n , dim ∆ n , k = n − 1. µ n , k : G n , k / T n → ∆ n , k . µ n , k is T n -invariant, it unduces the map ˆ T n -action on the Grassmannians G n , 2 via hyperplane arrangements Svjetlana Terzi´ c (University of Montenegro based on joint results with Victor M. Buchstaber Workshop on Torus Actions in Topology Fields 5 / 29

T n -action, moment map and Aut G n , k Lemma Let H < AutG n , k consists of the elements which commutes with the canonical T n -action on G n , k . Then H = T n − 1 ⋊ S n for n � = 2 k; H = Z 2 × ( T n − 1 ⋊ S n ) for n = 2 k. Let f ∈ Aut G n , k and assume there exists (combinatorial) isomorphism ¯ f : ∆ n , k → ∆ n , k such that the diagram commutes: f − − − − → G n , k G n , k   � µ n , k � µ n , k (1)   ¯ f ∆ n , k − − − − → ∆ n , k . T n -action on the Grassmannians G n , 2 via hyperplane arrangements Svjetlana Terzi´ c (University of Montenegro based on joint results with Victor M. Buchstaber Workshop on Torus Actions in Topology Fields 6 / 29

Proposition Let H < Aut G n , k consists of those elements which satisfy (1). Then H = T n − 1 ⋊ S n for n � = 2 k ; H = Z 2 × ( T n − 1 ⋊ S n ) for n = 2 k . ¯ t = id ∆ n , k for t ∈ T n − 1 ; ¯ s ( x 1 , . . . , x n ) = ( x s ( 1 ) , . . . , x s ( n ) ) for s ∈ S n ; ¯ c n , k ( x 1 , . . . , x n ) = ( 1 − x 1 , . . . , 1 − x n ) for c n , k ∈ Z 2 , n = 2 k - duality automorphism. Corollary µ − 1 µ − 1 n , k ( s ( x )) for x ∈ ∆ n , k and s ∈ S n ˆ n , k ( x ) is homeomorphic to ˆ µ − 1 µ − 1 ˆ n , k ( x ) is homeomorphic to ˆ n , k ( 1 − x ) for x ∈ ∆ n , k , when n = 2 k. T n -action on the Grassmannians G n , 2 via hyperplane arrangements Svjetlana Terzi´ c (University of Montenegro based on joint results with Victor M. Buchstaber Workshop on Torus Actions in Topology Fields 7 / 29

Strata on G n , k Let M I = { L ∈ G n , k | P I ( L ) � = 0 } , I ⊂ { 1 , . . . , n } , | I | = k . M I is an open and dense set in G n , k and G n , k = � M I . M I contains exactly one T n - fixed point x I . Set Y I = G n , k \ M I . Let σ ⊂ { I , I ⊂ { 1 , . . . , n } , | I | = k } and define the stratum W σ by W σ = ( ∩ I ∈ σ M I ) ∩ ( ∩ I / ∈ σ Y I ) if this intersection is nonempty . The main stratum is W = ∩ I ∈{ ( n k ) } M I - an open and dense set in G n , k . ′ , W σ ∩ W σ ′ = ∅ for σ � = σ W σ is ( C ∗ ) n - invariant, G n , k = ∪ σ W σ W σ are no open, no closed and their geometry is not nice. T n -action on the Grassmannians G n , 2 via hyperplane arrangements Svjetlana Terzi´ c (University of Montenegro based on joint results with Victor M. Buchstaber Workshop on Torus Actions in Topology Fields 8 / 29

Strata on G n , k Lemma ◦ µ n , k ( W σ ) = P σ , P σ = convhull (Λ I , I ∈ σ ) Such P σ is called an admissible polytope { W σ } coincide with the strata of Gel’fand-Serganova: W σ = { L ∈ G n , k : µ n , k (( C ∗ ) n · L ) = P σ } , Any face of an admissible polytope is an admissible polytope. ◦ µ n , k ( W ) = ∆ n , k , µ n , k ( fixed point ) = vertex. ∆ n , k and its faces are admissible polytopes. Theorem All points from W σ have the same stabilizer T σ ( ( C ∗ ) σ ). Torus T σ = T n / T σ acts freely on W σ . T n -action on the Grassmannians G n , 2 via hyperplane arrangements Svjetlana Terzi´ c (University of Montenegro based on joint results with Victor M. Buchstaber Workshop on Torus Actions in Topology Fields 9 / 29

Moment map decomposes as µ n , k : W σ → W σ / T σ ˆ µ n , k ◦ → P σ . Theorem ◦ µ n , k : W σ / T σ → ˆ P σ is a locally trivial fiber bundle with a fiber an open algebraic manifold F σ . Thus, ◦ W σ / T σ ∼ P σ × F σ . = F σ – the space of parameter for W σ ; F σ ∼ = W σ / ( C ∗ ) σ . ◦ G n , k / T n = ∪ σ W σ / T σ ∼ To summarize: = ∪ σ ( P σ × F σ ) ◦ G n , k / T n = W / T n − 1 ∼ = ∆ n , k × F . Goal: Describe P σ , F σ and the corresponding compactification F for F T n -action on the Grassmannians G n , 2 via hyperplane arrangements Svjetlana Terzi´ c (University of Montenegro based on joint results with Victor M. Buchstaber Workshop on Torus Actions in Topology Fields 10 / 29

Grassmannians G n , 2 Admissible polytopes ∆ n , 2 ⊂ R n − 1 = { x ∈ R n : x 1 + . . . + x n = 2 } ; dim P σ ≤ n − 1, for any σ . Proposition If dim P σ ≤ n − 3 then P σ ⊂ ∂ ∆ n , 2 . ∂ ∆ n , 2 = ( ∪ n ∆ n − 2 ) ∪ ( ∪ n ∆ n − 1 , 2 ) µ − 1 n , k ( ∂ ∆ n , 2 ) = ( ∪ n C P n − 2 ) ∪ ( ∪ n G n − 1 , 2 ) If dim P σ = n − 2 and P σ ⊂ ∂ ∆ n , 2 : P σ = ∆ n − 2 or P σ ⊆ ∆ n − 1 , 2 is an admissible polytope for G n − 1 , 2 . T n -action on the Grassmannians G n , 2 via hyperplane arrangements Svjetlana Terzi´ c (University of Montenegro based on joint results with Victor M. Buchstaber Workshop on Torus Actions in Topology Fields 11 / 29

Admissible ( n − 2 ) - polytopes ◦ Let dim P σ = n − 2 and P σ ∩ ∆ n , 2 � = ∅ - interior admissible polytope Proposition The interior admissible polytopes of dimension n − 2 coincide with the polytopes obtained by the intersection with ∆ n , 2 of the planes Π : x i 1 + . . . + x i l = 1 , 1 ≤ i 1 < . . . < i l ≤ n , 2 ≤ l ≤ [ n 2 ] . T n -action on the Grassmannians G n , 2 via hyperplane arrangements Svjetlana Terzi´ c (University of Montenegro based on joint results with Victor M. Buchstaber Workshop on Torus Actions in Topology Fields 12 / 29