T HE RATIONAL HOMOLOGY OF SMOOTH REAL TORIC VARIETIES Alex Suciu Northeastern University visiting the University of Sydney Geometry Topology and Analysis Seminar The University of Sydney November 26, 2012 A LEX S UCIU (N ORTHEASTERN ) R EAL TORIC VARIETIES GTA S EMINAR 1 / 31
T ORIC MANIFOLDS T ORIC MANIFOLDS Let P be an n -dimensional convex polytope; facets F 1 , . . . , F m . Assume P is simple (each vertex is the intersection of n facets). Then P determines a dual simplicial complex, K = K ❇ P , of dimension n ✁ 1: Vertex set [ m ] = t 1 , . . . , m ✉ . Add a simplex σ = ( i 1 , . . . , i k ) whenever F i 1 , . . . , F i k intersect. F IGURE : A prism P and its dual simplicial complex K A LEX S UCIU (N ORTHEASTERN ) R EAL TORIC VARIETIES GTA S EMINAR 2 / 31
T ORIC MANIFOLDS Let χ be an n -by- m matrix with coefficients in G = Z or Z 2 . χ is characteristic for P if, for each vertex v = F i 1 ❳ ☎ ☎ ☎ ❳ F i n , the n -by- n minor given by the columns i 1 , . . . , i n of χ is unimodular. Let T = S 1 if G = Z , and T = S 0 = t✟ 1 ✉ if G = Z 2 . Given q P P , let F ( q ) = F j 1 ❳ ☎ ☎ ☎ ❳ F j k be the maximal face so that q P F ( q ) ✆ . The map χ yields a k -dimensional subtorus T F ( q ) = T F j 1 ❳ ☎ ☎ ☎ ❳ T F jk ⑨ T n . Here, if F is a face, and χ F : G Ñ G n is the corresponding column χ F : T n Ñ T ) ✕ T n ✁ 1 . vector, then T F = ker ( ① A LEX S UCIU (N ORTHEASTERN ) R EAL TORIC VARIETIES GTA S EMINAR 3 / 31
T ORIC MANIFOLDS To the pair ( P , χ ) , M. Davis and T. Januszkiewicz associate the space X = T n ✂ P / ✒ where ( t , p ) ✒ ( u , q ) if p = q and t ☎ u ✁ 1 P T F ( q ) . The projection map X Ñ P has fibers T n over points in the interior of P , T n ✁ 1 = T F over points on a face F , etc. For G = Z , the space X is called a toric manifold , denoted M P ( χ ) . It is a closed, orientable manifold of dimension 2 n . For G = Z 2 , the space X is called a real toric manifold (or, small cover ), denoted N P ( χ ) . It is a closed, not necessarily orientable manifold of dimension n . A LEX S UCIU (N ORTHEASTERN ) R EAL TORIC VARIETIES GTA S EMINAR 4 / 31
T ORIC MANIFOLDS E XAMPLE (T ORIC MANIFOLDS OVER THE n - SIMPLEX ) � 1 ☎☎☎ 0 1 � . Let P = ∆ n be the n -simplex, and χ the n ✂ ( n + 1 ) matrix ... . . . 0 ☎☎☎ 1 1 Then M P ( χ ) = CP n N P ( χ ) = RP n . and P T ✂ P T ✂ P / ✒ CP 1 RP 1 A LEX S UCIU (N ORTHEASTERN ) R EAL TORIC VARIETIES GTA S EMINAR 5 / 31
T ORIC MANIFOLDS E XAMPLE (T ORIC MANIFOLDS OVER THE SQUARE ) � 1 � 1 � 1 � � � 0 0 0 ✌ ✌ ✌ ✌ ✌ ✌ � 0 � 0 � 0 � 0 � 0 � ✁ 2 � � � � � � 1 1 1 1 1 1 ✌ � ✌ ✌ � ✌ ✌ � 1 � ✌ � 1 � 1 ✁ 1 0 1 CP 1 ✂ CP 1 CP 2 # CP 2 CP 2 # CP 2 � 1 � 1 � � 0 0 ✌ ✌ ✌ ✌ � 0 � 0 � 0 � 0 � � � � 1 1 1 1 ✌ � ✌ ✌ � ✌ � 1 � 1 0 1 S 1 ✂ S 1 RP 2 # RP 2 A LEX S UCIU (N ORTHEASTERN ) R EAL TORIC VARIETIES GTA S EMINAR 6 / 31
T ORIC MANIFOLDS A construction from toric geometry: Suppose X is a smooth, projective toric variety. Let P be the rational, simple polytope associated to the fan ( P is the Delzant polytope, X Ñ P the moment map.) Let χ be the matrix defined by the rays of the fan. Then X ( C ) = M P ( χ ) and X ( R ) = N P ( χ mod 2 Z ) . But not all toric manifolds arise this way: M = CP 2 ✼ CP 2 is a toric manifold over the square, but it does not admit any (almost) complex structure. Thus, M ✢ X ( C ) . Let P be the dodecahedron. Then characteristic matrices χ exist for P (Garrison and Scott), and N P ( χ ) is a hyperbolic 3-manifold (Andreev). Thus, N P ( χ ) ✢ X ( R ) (by C. Delaunay). A LEX S UCIU (N ORTHEASTERN ) R EAL TORIC VARIETIES GTA S EMINAR 7 / 31
T ORIC MANIFOLDS Davis and Januszkiewicz showed that: M P ( χ ) admits a perfect Morse function with only critical points of even index. Moreover, rank H 2 i ( M P ( χ ) , Z ) = h i ( P ) , where ( h 0 ( P ) , . . . , h n ( P )) is the h -vector of P , which depends only on the number of i -faces of P (0 ↕ i ↕ n ). N P ( χ ) admits a perfect Morse function over Z 2 . Moreover, dim Z 2 H i ( N P ( χ ) , Z 2 ) = h i ( P ) . They also gave presentations for the cohomology rings H ✝ ( M P ( χ ) , Z ) and H ✝ ( N P ( χ ) , Z 2 ) , similar to the ones given by Danilov and Jurkiewicz for toric varieties. A LEX S UCIU (N ORTHEASTERN ) R EAL TORIC VARIETIES GTA S EMINAR 8 / 31
T ORIC MANIFOLDS In work with A. Trevisan, we compute H ✝ ( N P ( χ ) , Q ) , both additively and multiplicatively. The (rational) Betti numbers of N P ( χ ) no longer depend just on the h -vector of P , but also on the characteristic matrix χ . E XAMPLE Recall there are precisely two small covers over the square P : � 1 0 1 0 The torus T 2 = N P ( χ ) , with χ = � . 0 1 0 1 � 1 0 1 0 The Klein bottle K ℓ = N P ( χ ✶ ) , with χ ✶ = � . 0 1 1 1 Then b 1 ( T 2 ) = 2, yet b 1 ( K ℓ ) = 1. Idea: use finite covers involving (up to homotopy) certain generalized moment-angle complexes, and cohomology with coefficients in rank 1 local systems. A LEX S UCIU (N ORTHEASTERN ) R EAL TORIC VARIETIES GTA S EMINAR 9 / 31
G ENERALIZED MOMENT - ANGLE COMPLEXES M OMENT - ANGLE COMPLEXES Let ( X , A ) be a pair of topological spaces Let K be a simplicial complex on vertex set [ m ] . Corresponding (generalized) moment-angle complex (or, polyhedral product): ↕ ( X , A ) σ ⑨ X ✂ m Z K ( X , A ) = σ P K where ( X , A ) σ = t x P X ✂ m ⑤ x i P A if i ❘ σ ✉ . Interpolates between A ✂ m and X ✂ m . Homotopy invariance: ( X , A ) ✔ ( X ✶ , A ✶ ) ù ñ Z K ( X , A ) ✔ Z K ( X ✶ , A ✶ ) Converts simplicial joins to direct products: Z K ✝ L ( X , A ) ✕ Z K ( X , A ) ✂ Z L ( X , A ) A LEX S UCIU (N ORTHEASTERN ) R EAL TORIC VARIETIES GTA S EMINAR 10 / 31
G ENERALIZED MOMENT - ANGLE COMPLEXES Usual moment-angle complex: Z K ( D 2 , S 1 ) . π 1 = π 2 = t 1 ✉ . Real moment-angle complex: Z K ( D 1 , S 0 ) . π 1 = W ✶ K , the derived subgroup of W K , the right-angled Coxeter group associated to K ( 1 ) . E XAMPLE Let K = two points. Then: Z K ( D 2 , S 1 ) = D 2 ✂ S 1 ❨ S 1 ✂ D 2 = S 3 Z K ( D 1 , S 0 ) = D 1 ✂ S 0 ❨ S 0 ✂ D 1 = S 1 D 1 × S 0 S 0 × D 1 D 1 Z K ( D 1 , S 0 ) S 0 S 0 × S 0 A LEX S UCIU (N ORTHEASTERN ) R EAL TORIC VARIETIES GTA S EMINAR 11 / 31
G ENERALIZED MOMENT - ANGLE COMPLEXES E XAMPLE Let K be a circuit on 4 vertices. Then: Z K ( D 2 , S 1 ) = S 3 ✂ S 3 Z K ( D 1 , S 0 ) = S 1 ✂ S 1 E XAMPLE More generally, let K be an n -gon. Then: � n ✁ 2 � n ✁ 3 S r + 2 ✂ S n ✁ r Z K ( D 2 , S 1 ) = # r = 1 r ☎ r + 1 Z K ( D 1 , S 0 ) = an orientable surface of genus 1 + 2 n ✁ 3 ( n ✁ 4 ) The second equality was proved by H.S.M. Coxeter in 1937. A LEX S UCIU (N ORTHEASTERN ) R EAL TORIC VARIETIES GTA S EMINAR 12 / 31
G ENERALIZED MOMENT - ANGLE COMPLEXES If ( M , ❇ M ) is a compact manifold of dim d , and K is a PL-triangulation of S m on n vertices, then Z K ( M , ❇ M ) is a compact manifold of dim ( d ✁ 1 ) n + m + 1. (Bosio–Meersseman) If K is a polytopal triangulation of S m , then Z K ( D 2 , S 1 ) if n + m + 1 is even, or Z K ( D 2 , S 1 ) ✂ S 1 if n + m + 1 is odd is a complex manifold. This construction generalizes the classical constructions of complex structures on S 2 p ✁ 1 ✂ S 1 (Hopf) and S 2 p ✁ 1 ✂ S 2 q ✁ 1 (Calabi–Eckmann). In general, the resulting complex manifolds are not symplectic, thus, not Kähler. In fact, they may even be non-formal (Denham–Suciu). A LEX S UCIU (N ORTHEASTERN ) R EAL TORIC VARIETIES GTA S EMINAR 13 / 31
G ENERALIZED MOMENT - ANGLE COMPLEXES D AVIS –J ANUSZKIEWICZ SPACES ( X , ✝ ) pointed space � Z K ( X ) = Z K ( X , ✝ ) Davis–Januszkiewicz space: Z K ( CP ✽ ) . π 1 = t 1 ✉ . H ✝ ( Z K ( CP ✽ ) , Z ) = S / I K , where S = Z [ x 1 , . . . , x m ] , deg x i = 2. Real Davis–Januszkiewicz space: Z K ( RP ✽ ) . π 1 = W K : right-angled Coxeter group associated to K ( 1 ) = ( V , E ) . W K = ① v P V ⑤ v 2 = 1 , vw = wv if t v , w ✉ P E ② . H ✝ ( Z K ( RP ✽ ) , Z 2 ) = R / I K , where R = Z 2 [ x 1 , . . . , x m ] , deg x i = 1. Toric complex: Z K ( S 1 ) . π 1 = G K : right-angled Artin group associated to K ( 1 ) . G K = ① v P V ⑤ vw = wv if t v , w ✉ P E ② . H ✝ ( Z K ( S 1 ) , Z ) = E / J K , where E = ➍ [ e 1 , . . . , e m ] , deg e i = 1. A LEX S UCIU (N ORTHEASTERN ) R EAL TORIC VARIETIES GTA S EMINAR 14 / 31
� � � � � F INITE ABELIAN COVERS F INITE ABELIAN COVERS Let X be a connected, finite-type CW-complex, π = π 1 ( X , x 0 ) . Let p : Y Ñ X a (connected) regular cover, with group of deck transformations Γ . We then have a short exact sequence p ✼ ν � 1 . � π 1 ( Y , y 0 ) � π 1 ( X , x 0 ) � Γ 1 Conversely, every epimorphism ν : π ։ Γ defines a regular cover X ν Ñ X (unique up to equivalence), with π 1 ( X ν ) = ker ( ν ) . If Γ is abelian, then ν = χ ✆ ab factors through the abelianization, while X ν = X χ is covered by the universal abelian cover of X : ab � π 1 ( X ) ab X ab X ν π 1 ( X ) Ð Ñ χ p ν X Γ A LEX S UCIU (N ORTHEASTERN ) R EAL TORIC VARIETIES GTA S EMINAR 15 / 31
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