Alex Suciu Northeastern University visiting the University of - - PowerPoint PPT Presentation

THE 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

ALEX SUCIU (NORTHEASTERN) REAL TORIC VARIETIES GTA SEMINAR 1 / 31

TORIC MANIFOLDS

TORIC MANIFOLDS

Let P be an n-dimensional convex polytope; facets F1, . . . , Fm. 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] = t1, . . . , m✉. Add a simplex σ = (i1, . . . , ik) whenever Fi1, . . . , Fik intersect. FIGURE: A prism P and its dual simplicial complex K

ALEX SUCIU (NORTHEASTERN) REAL TORIC VARIETIES GTA SEMINAR 2 / 31

TORIC MANIFOLDS

Let χ be an n-by-m matrix with coefficients in G = Z or Z2. χ is characteristic for P if, for each vertex v = Fi1 ❳ ☎ ☎ ☎ ❳ Fin, the n-by-n minor given by the columns i1, . . . , in of χ is unimodular. Let T = S1 if G = Z, and T = S0 = t✟1✉ if G = Z2. Given q P P, let F(q) = Fj1 ❳ ☎ ☎ ☎ ❳ Fjk be the maximal face so that q P F(q)✆. The map χ yields a k-dimensional subtorus TF(q) = TFj1 ❳ ☎ ☎ ☎ ❳ TFjk ⑨ Tn. Here, if F is a face, and χF : G Ñ Gn is the corresponding column vector, then TF = ker(① χF : Tn Ñ T) ✕ Tn✁1.

ALEX SUCIU (NORTHEASTERN) REAL TORIC VARIETIES GTA SEMINAR 3 / 31

TORIC 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 TF(q). The projection map X Ñ P has fibers

Tn over points in the interior of P, Tn✁1 = TF over points on a face F, etc.

For G = Z, the space X is called a toric manifold, denoted MP(χ). It is a closed, orientable manifold of dimension 2n. For G = Z2, the space X is called a real toric manifold (or, small cover), denoted NP(χ). It is a closed, not necessarily orientable manifold of dimension n.

ALEX SUCIU (NORTHEASTERN) REAL TORIC VARIETIES GTA SEMINAR 4 / 31

TORIC MANIFOLDS

EXAMPLE (TORIC MANIFOLDS OVER THE n-SIMPLEX) Let P = ∆n be the n-simplex, and χ the n ✂ (n + 1) matrix 1 ☎☎☎ 0 1 ... . . .

0 ☎☎☎ 1 1

• .

Then MP(χ) = CPn and NP(χ) = RPn. P T ✂ P T ✂ P/ ✒ CP1 RP1

ALEX SUCIU (NORTHEASTERN) REAL TORIC VARIETIES GTA SEMINAR 5 / 31

TORIC MANIFOLDS

EXAMPLE (TORIC MANIFOLDS OVER THE SQUARE) ✌

1

• 1
• ✌

1

• ✌

1 ✌

✌

1

• 1
• ✌

1

• ✌

1 1 ✌

✌

1

• 1
• ✌

✁2 1

• ✌ 1

✁1 ✌

CP1 ✂ CP1 CP2# CP2 CP2# CP2 ✌

1

• 1
• ✌

1

• ✌

1 ✌

✌

1

• 1
• ✌

1

• ✌

1 1 ✌

S1 ✂ S1 RP2#RP2

ALEX SUCIU (NORTHEASTERN) REAL TORIC VARIETIES GTA SEMINAR 6 / 31

TORIC 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) = MP(χ) and X(R) = NP(χ mod 2Z). But not all toric manifolds arise this way: M = CP2✼CP2 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 NP(χ) is a hyperbolic 3-manifold (Andreev). Thus, NP(χ) ✢ X(R) (by C. Delaunay).

ALEX SUCIU (NORTHEASTERN) REAL TORIC VARIETIES GTA SEMINAR 7 / 31

TORIC MANIFOLDS

Davis and Januszkiewicz showed that: MP(χ) admits a perfect Morse function with only critical points of even index. Moreover, rank H2i(MP(χ), Z) = hi(P), where (h0(P), . . . , hn(P)) is the h-vector of P, which depends only

• n the number of i-faces of P (0 ↕ i ↕ n).

NP(χ) admits a perfect Morse function over Z2. Moreover, dimZ2 Hi(NP(χ), Z2) = hi(P). They also gave presentations for the cohomology rings H✝(MP(χ), Z) and H✝(NP(χ), Z2), similar to the ones given by Danilov and Jurkiewicz for toric varieties.

ALEX SUCIU (NORTHEASTERN) REAL TORIC VARIETIES GTA SEMINAR 8 / 31

TORIC MANIFOLDS

In work with A. Trevisan, we compute H✝(NP(χ), Q), both additively and multiplicatively. The (rational) Betti numbers of NP(χ) no longer depend just on the h-vector of P, but also on the characteristic matrix χ. EXAMPLE Recall there are precisely two small covers over the square P: The torus T 2 = NP(χ), with χ = 1 0 1 0

0 1 0 1

• .

The Klein bottle Kℓ = NP(χ✶), with χ✶ = 1 0 1 0

0 1 1 1

• .

Then b1(T 2) = 2, yet b1(Kℓ) = 1. Idea: use finite covers involving (up to homotopy) certain generalized moment-angle complexes, and cohomology with coefficients in rank 1 local systems.

ALEX SUCIU (NORTHEASTERN) REAL TORIC VARIETIES GTA SEMINAR 9 / 31

GENERALIZED MOMENT-ANGLE COMPLEXES

MOMENT-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): ZK (X, A) = ↕

σPK

(X, A)σ ⑨ X ✂m where (X, A)σ = tx P X ✂m ⑤ xi P A if i ❘ σ✉. Interpolates between A✂m and X ✂m. Homotopy invariance: (X, A) ✔ (X ✶, A✶) ù ñ ZK (X, A) ✔ ZK (X ✶, A✶) Converts simplicial joins to direct products: ZK✝L(X, A) ✕ ZK (X, A) ✂ ZL(X, A)

ALEX SUCIU (NORTHEASTERN) REAL TORIC VARIETIES GTA SEMINAR 10 / 31

GENERALIZED MOMENT-ANGLE COMPLEXES

Usual moment-angle complex: ZK (D2, S1).

π1 = π2 = t1✉.

Real moment-angle complex: ZK (D1, S0).

π1 = W ✶

K , the derived subgroup of WK , the right-angled Coxeter

group associated to K (1).

EXAMPLE Let K = two points. Then: ZK (D2, S1) = D2 ✂ S1 ❨ S1 ✂ D2 = S3 ZK (D1, S0) = D1 ✂ S0 ❨ S0 ✂ D1 = S1

D1 S0 D1 × S0 S0 × D1 ZK(D1, S0) S0 × S0

ALEX SUCIU (NORTHEASTERN) REAL TORIC VARIETIES GTA SEMINAR 11 / 31

GENERALIZED MOMENT-ANGLE COMPLEXES

EXAMPLE Let K be a circuit on 4 vertices. Then: ZK (D2, S1) = S3 ✂ S3 ZK (D1, S0) = S1 ✂ S1 EXAMPLE More generally, let K be an n-gon. Then: ZK (D2, S1) = #

n✁3 r=1 r ☎

n ✁ 2 r + 1

• Sr+2 ✂ Sn✁r

ZK (D1, S0) = an orientable surface of genus 1 + 2n✁3(n ✁ 4) The second equality was proved by H.S.M. Coxeter in 1937.

ALEX SUCIU (NORTHEASTERN) REAL TORIC VARIETIES GTA SEMINAR 12 / 31

GENERALIZED MOMENT-ANGLE COMPLEXES

If (M, ❇M) is a compact manifold of dim d, and K is a PL-triangulation of Sm on n vertices, then ZK (M, ❇M) is a compact manifold of dim (d ✁ 1)n + m + 1. (Bosio–Meersseman) If K is a polytopal triangulation of Sm, then

ZK (D2, S1) if n + m + 1 is even, or ZK (D2, S1) ✂ S1 if n + m + 1 is odd

is a complex manifold. This construction generalizes the classical constructions of complex structures on S2p✁1 ✂ S1 (Hopf) and S2p✁1 ✂ S2q✁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).

ALEX SUCIU (NORTHEASTERN) REAL TORIC VARIETIES GTA SEMINAR 13 / 31

GENERALIZED MOMENT-ANGLE COMPLEXES DAVIS–JANUSZKIEWICZ SPACES

(X, ✝) pointed space ZK (X) = ZK (X, ✝) Davis–Januszkiewicz space: ZK (CP✽).

π1 = t1✉. H✝(ZK (CP✽), Z) = S/IK , where S = Z[x1, . . . , xm], deg xi = 2.

Real Davis–Januszkiewicz space: ZK (RP✽).

π1 = WK : right-angled Coxeter group associated to K (1) = (V, E). WK = ①v P V ⑤ v2 = 1, vw = wv if tv, w✉ P E②. H✝(ZK (RP✽), Z2) = R/IK , where R = Z2[x1, . . . , xm], deg xi = 1.

Toric complex: ZK (S1).

π1 = GK : right-angled Artin group associated to K (1). GK = ①v P V ⑤ vw = wv if tv, w✉ P E②. H✝(ZK (S1), Z) = E/JK , where E = ➍[e1, . . . , em], deg ei = 1.

ALEX SUCIU (NORTHEASTERN) REAL TORIC VARIETIES GTA SEMINAR 14 / 31

FINITE ABELIAN COVERS

FINITE ABELIAN COVERS

Let X be a connected, finite-type CW-complex, π = π1(X, x0). Let p : Y Ñ X a (connected) regular cover, with group of deck transformations Γ. We then have a short exact sequence 1

π1(Y, y0)

p✼

π1(X, x0)

ν

Γ 1 .

Conversely, every epimorphism ν: π ։ Γ defines a regular cover X ν Ñ X (unique up to equivalence), with π1(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: X ab

• X ν

p

• X

Ð Ñ π1(X)

ν

• ab π1(X)ab

χ

• Γ

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FINITE ABELIAN COVERS

Let Cq(X ν; k) be the group of cellular q-chains on X ν, with coefficients in a field k. We then have natural isomorphisms Cq(X ν; k) ✕ Cq(X; kΓ) ✕ Cq(r X) ❜kπ kΓ. Now suppose Γ is finite abelian, k = k, and char k = 0. Then, all k-irreps of Γ are 1-dimensional, and so Cq(X ν; k) ✕ à

ρPHom(Γ,k✂)

Cq(X; kρ✆ν), where kρ✆ν denotes the field k, viewed as a kπ-module via the character ρ ✆ ν: π Ñ k✂. Thus, Hq(X ν; k) ✕ ➚

ρPHom(Γ,k✂) Hq(X; kρ✆ν).

ALEX SUCIU (NORTHEASTERN) REAL TORIC VARIETIES GTA SEMINAR 16 / 31

THE BETTI NUMBERS OF REAL TORIC MANIFOLDS

THE BETTI NUMBERS OF REAL TORIC MANIFOLDS

Let P be an n-dimensional, simple polytope with m facets, and let K = K❇P be the simplicial complex dual to ❇P. Let χ: Zm

2 Ñ Zn 2 be a characteristic matrix for P.

Then ker(χ) ✕ Zm✁n

2

acts freely on ZK (D1, S0), with quotient the real toric manifold NP(χ). NP(χ) comes equipped with an action of Zm

2 / ker(χ) ✕ Zn 2; the

• rbit space is P.

Furthermore, ZK (D1, S0) is homotopy equivalent to the maximal abelian cover of ZK (RP✽), corresponding to the sequence 1

W ✶

K

WK

ab Zm 2

1 .

ALEX SUCIU (NORTHEASTERN) REAL TORIC VARIETIES GTA SEMINAR 17 / 31

THE BETTI NUMBERS OF REAL TORIC MANIFOLDS

Thus, NP(χ) is, up to homotopy, the regular Zn

2-cover of

ZK (RP✽), corresponding to the sequence 1

π1(NP(χ)) WK

χ✆ab

Zn

2

1 .

To sum up, we have a diagram ZK (RP✽)ab ✔ ZK (D1, S0)

/Zm✁n

2

• ZK (RP✽)χ✆ab ✔ NP(χ)

/Zm

2

• /Zn

2

• P

ZK (RP✽) with vertical arrows regular covers, and horizontal arrow the “stratified" (small) cover defining NP(χ).

ALEX SUCIU (NORTHEASTERN) REAL TORIC VARIETIES GTA SEMINAR 18 / 31

THE BETTI NUMBERS OF REAL TORIC MANIFOLDS

LEMMA (S-T) Let λ: Zm

2 Ñ k✂ be a character, and let Kλ be the induced subcomplex

• n vertex set supp(λ) := ti P [m] ⑤ λ(xi) ✘ 1✉. Then

Hq(ZK (RP✽); kλ) ✕ r Hq✁1(Kλ; k) Sketch of proof: The map S1 ã Ñ RP✽ induces a map TK = ZK (S1) ã Ñ ZK (RP✽). The map Kλ ã Ñ K induces a map TKλ ã Ñ TK. Let ¯ λ: Zm ։ Zm

2 λ

Ý Ñ k✂. We then get cellular (chain) retractions Cq(TK; k¯

λ)

• Cq(ZK (RP✽); kλ)
• Cq(TKλ; k¯

λ) ✕

r

Cq✁1(Kλ; k) Hence: dimk Hq(ZK (RP✽); kλ) ➙ dimk r Hq✁1(Kλ; k).

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THE BETTI NUMBERS OF REAL TORIC MANIFOLDS

For the reverse inequality, note that Hq(ZK (RP✽); k) ✕ à

ρPHom(Zm

2 ,k✂)

Hq(ZK (RP✽); kρ). The fibration Z2 Ñ S✽ Ñ RP✽ gives ZK (S✽, Z2) ✔ (ZK (RP✽))ab. A stable splitting theorem of Bahri, Bendersky, Cohen, and Gitler [2010] gives Hq(ZK (S✽, Z2); k) ✕ à

I⑨[m]

r Hq✁1(KI; k). Putting things together finishes the proof.

ALEX SUCIU (NORTHEASTERN) REAL TORIC VARIETIES GTA SEMINAR 20 / 31

THE BETTI NUMBERS OF REAL TORIC MANIFOLDS

For each subset S ❸ [n], Let χS = ➦

iPS χi, where χi is the i-th row of χ.

Let Kχ,S be the induced subcomplex of K on vertex set supp(χS) = tj P [m] ⑤ the j-th entry of χS is non-zero✉. THEOREM (S–T) The Betti numbers of the real toric manifold NP(χ) are given by dim Hq(NP(χ), Q) = ➳

S❸[n]

dim r Hq✁1(Kχ,S, Q). Now, NP(χ) is orientable if and only if bn(NP(χ)) = 1. Hence, we recover a result of Nakayama and Nishimura [2005]. COROLLARY NP(χ) is orientable ð ñ there is a subset S ❸ [n] such that Kχ,S = K.

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THE BETTI NUMBERS OF REAL TORIC MANIFOLDS

EXAMPLE Again, let P be the square, K = K❇P the 4-cycle. Let T 2 = NP(χ), χ = 1 0 1 0

0 1 0 1

• , and Kℓ = NP(χ✶), χ✶ =

1 0 1 0

0 1 1 1

• .

S ❍ t1✉ t2✉ t1, 2✉ χS ( 0 0 0 0 ) ( 1 0 1 0 ) ( 0 1 0 1 ) ( 1 1 1 1 ) Kχ,S ❍ tt1✉, t3✉✉ tt2✉, t4✉✉ K χ✶

S

( 0 0 0 0 ) ( 1 0 1 0 ) ( 0 1 1 1 ) ( 1 1 0 1 ) Kχ✶,S ❍ tt1✉, t3✉✉ tt2, 3✉, t3, 4✉✉ tt1, 2✉, t1, 4✉✉ Hence:

b0(T 2) = ˜ b✁1(❍) = 1 b0(Kℓ) = ˜ b✁1(❍) = 1 b1(T 2) = ˜ b0(Kχ,t1✉) + ˜ b0(Kχ,t2✉) = 2 b1(Kℓ) = ˜ b0(Kχ✶,t1✉) + ˜ b0(Kχ✶,t2✉) = 1 b2(T 2) = ˜ b1(Kχ,t1,2✉) = 1 b2(Kℓ) = ˜ b1(Kχ✶,t1,2✉) = 0

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THE HESSENBERG MANIFOLDS

THE HESSENBERG MANIFOLDS

Every Weyl group W determines a smooth, complex projective toric variety TW.

Fan given by the reflecting hyperplanes of W. Polytope PW is the convex hull of a regular orbit W ☎ x0. dimC TW = rank W.

Tn = TSn is the Hessenberg variety, of cx dim n ✁ 1; polytope is the permutahedron Pn (the iterated truncation of the simplex ∆n✁1). Tn equals the De Concini–Procesi wonderful model YG, where G is the maximal building set for the Boolean arrangement in CPn✁1. Thus, Tn can be obtained by iterated blow-ups:

Blow up CPn✁1 at the n coordinate points. Blow up along the proper transforms of the (n

2) coordinate lines.

Blow up along the proper transforms of the (n

3) coordinate planes...

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THE HESSENBERG MANIFOLDS

There is another De Concini–Procesi model, YH, which is isomorphic to the compactified moduli space M0,n+2, and which admits an Sn-equivariant birational morphism onto Tn. The Betti numbers of YH(R) were computed by Etingof, Hernandez, Kamnitzer, and Rains [2010]. The real locus of TW, denoted TW (R), is a smooth, connected, compact real toric variety of dimension equal to the rank of W. Henderson and Lehrer [2009] gave a formula for the equivariant Euler characteristic of TW (R). Building on this work, Henderson computed the Betti numbers of Tn(R). THEOREM (HENDERSON 2010) bi(Tn(R)) = A2i n 2i

• ,

where A2i is the Euler secant number, defined as the coefficient of x2i/(2i)! in the Maclaurin expansion of sec(x),

ALEX SUCIU (NORTHEASTERN) REAL TORIC VARIETIES GTA SEMINAR 24 / 31

THE HESSENBERG MANIFOLDS

We may recover Henderson’s computation, using our general formula. To start with, note that: Pn has 2n ✁ 2 facets: each subset ❍ ✘ Q ⑨ [n] determines a facet F Q with vertices in which all coordinates in positions in Q are smaller than all coordinates in positions not in Q. The corresponding column vectors of the characteristic matrix χ: Z2n✁2

2

Ñ Zn✁1

2

are given by: χi = i-th standard basis vector of Rn✁1 (1 ↕ i ➔ n), χn = ➦

i➔n χi,

χQ = ➦

iPQ χi.

EXAMPLE P3 is a truncated triangle (i.e., a hexagon). Characteristic matrix: χ =

• 1

1 1 1 1 1 1 1

• .

T3(R) is obtained from RP2 by blowing up 3 points.

ALEX SUCIU (NORTHEASTERN) REAL TORIC VARIETIES GTA SEMINAR 25 / 31

THE HESSENBERG MANIFOLDS

EXAMPLE P4 is a truncated octahedron; it has 14 facets (6 squares and 8 hexagons). Characteristic matrix: χ = 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

• ALEX SUCIU (NORTHEASTERN)

REAL TORIC VARIETIES GTA SEMINAR 26 / 31

THE HESSENBERG MANIFOLDS

The dual simplicial complex, Kn = K❇Pn, is the barycentric subdivision of the boundary of the (n ✁ 1)-simplex. Given a subset S ❸ [n ✁ 1], the induced subcomplex on vertex set supp(χS) depends only on r := ⑤S⑤, so denote it by Kn,r. Kn,r is the order complex associated to a rank-selected poset of a certain subposet of the Boolean lattice Bn. Thus, Kn,r is Cohen–Macaulay; in fact, Kn,2r✁1 ✔ Kn,2r ✔ ➎A2r Sr✁1. Hence: bi(Tn(R)) = ➳

S⑨[n✁1]

˜ bi✁1((Kn)χ,S) =

n✁1

➳

r=1

n ✁ 1 r

• ˜

bi✁1(Kn,r) = n ✁ 1 2i ✁ 1

• +

n ✁ 1 2i

• A2i =

n 2i

• A2i.

ALEX SUCIU (NORTHEASTERN) REAL TORIC VARIETIES GTA SEMINAR 27 / 31

THE HESSENBERG MANIFOLDS

Recently, Choi and Park [2012] have extended this computation to a wider class of real toric manifolds. Given a finite simple graph Γ, let B(Γ) be the building set obtained from the connected induced subgraphs of Γ, and let PB(Γ) be the corresponding graph associahedron. Using the above theorem, they compute the Betti numbers of the smooth, real toric variety XΓ(R) defined by PB(Γ), solely in terms

• f some new numerical graph invariants, ai(Γ), that arise out of

this machinery. When Γ = Kn is a complete graph, XKn = Tn, and one recovers the above calculation.

ALEX SUCIU (NORTHEASTERN) REAL TORIC VARIETIES GTA SEMINAR 28 / 31

FORMALITY PROPERTIES

FORMALITY PROPERTIES

A finite-type CW-complex X is formal if its Sullivan minimal model is quasi-isomorphic to (H✝(X, Q), 0)—roughly speaking, H✝(X, Q) determines the rational homotopy type of X. (Notbohm–Ray) If X is formal, then ZK (X) is formal. In particular, the toric complexes ZK (S1) and the generalized Davis–Januszkiewicz spaces ZK (BG) are always formal. (Félix, Tanré) More generally, if both X and A are formal, and the inclusion i : A ã Ñ X induces a surjection i✝ : H✝(X, Q) Ñ H✝(A, Q), then ZK (X, A) is formal.

ALEX SUCIU (NORTHEASTERN) REAL TORIC VARIETIES GTA SEMINAR 29 / 31

FORMALITY PROPERTIES

(Baskakov, Denham–A.S.) Moment angle complexes ZK (D2, S1) are not always formal: they can have non-trivial triple Massey

• products. For instance, take K to be

(Denham–A.S.) There exist polytopes P and dual triangulations K = K❇P for which ZK (D2, S1) is not formal. (A.S.) There are real moment-angle complexes (even manifolds) ZL(D1, S0) which are not formal. (Panov–Ray) Complex toric manifolds MP(χ) are always formal. Question: are the real toric manifolds NP(χ) always formal?

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REFERENCES

REFERENCES

• A. Suciu, A. Trevisan, Real toric varieties and abelian covers of

generalized Davis–Januszkiewicz spaces, preprint 2012.

• A. Suciu, The rational homology of real toric manifolds, to appear

in Oberwolfach Reports. Further references:

• G. Denham, A. Suciu, Moment-angle complexes, monomial ideals,

and Massey products, Pure Appl. Math. Q. 3 (2007), no. 1, 25–60.

• S. Papadima, A. Suciu, Toric complexes and Artin kernels, Adv.
• Math. 220 (2009), no. 2, 441–477.

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