Collectanea Mathematica (electronic version): http://www.imub.ub.es/collect Collect. Math. 59, 2 (2008), 215–237

c 2008 Universitat de Barcelona

Hodge spaces of real toric varieties

Valerie Hower Department of Mathematics, University of Georgia, Athens, GA 30602 E-mail: [email protected] Received September 17, 2007. Revised December 12, 2007.

Abstract We define the Z2 Hodge spaces Hpq (Σ) of a rational fan Σ. If Σ is the normal fan of a reflexive polytope ∆ then we use polyhedral duality to compute the Z2 Hodge Spaces of Σ. In particular, if the cones of dimension at most e in the face fan Σ∗ of ∆ are smooth then we compute Hpq (Σ) for p < e − 1. If Σ∗ is a smooth fan then we completely determine the spaces Hpq (Σ) and we show XΣ is maximal, meaning that the sum of the Z2 Betti numbers of XΣ (R) is equal to the sum of the Z2 Betti numbers of XΣ (C).

1. Introduction In this paper, we define and study the Z2 Hodge spaces of a rational fan Σ. The Z2 Hodge spaces Hpq (Σ) are indexed by pairs of integers p, q with 0 ≤ q ≤ p ≤ dimΣ and are defined to be the homology groups Hp (∧q E ), where ∧q E is the qth exterior power of the cosheaf E on Σ. Geometrically, for σ ∈ Σ the stalk Eσ of the cosheaf E is (Z2 )r , a compact subset of the real orbit Oσ (R) ∼ = (R∗ )r of the real toric variety XΣ (R). The terminology Z2 Hodge spaces is inspired by work of Brion [7] who considered similar spaces associated to a fan Σ. When Σ is a smooth fan, we have Hpq (Σ) = 0 for p 6= q. However, for p > q the spaces Hpq (Σ) are not generally well understood. While these spaces are interesting from a purely combinatorial viewpoint, they are also important in understanding the topology of toric varieties. The Z2 Hodge spaces of Σ are related to the topology of both the real and complex points of the toric variety XΣ in that 1 2 , where (E r , dr ) and (E r , dr ) are two spectral sequences with Hpq (Σ) = E p,q = Ep,q 1

E p,q

+3 Hp (XΣ (R); Z2 )

and

2 Ep,q

+3 Hp+q (XΣ (C); Z2 ) .

Keywords: Real toric variety, Hodge space, reflexive polytope, Smith-Thom inequality. MSC2000: Primary 14M25; Secondary 55T99, 52B12.

215

216

Hower

When the sum of the Z2 Betti numbers for XΣ (R) is equal to the sum of the Z2 Betti numbers for XΣ (C), we say XΣ is maximal. As discussed in Section 3.2, if we have ∞ 1 E = E then E ∞ = E 2 and XΣ is maximal. In [5], Bihan, Franz, McCrory, and van Hamel conjectured that all toric varieties are maximal, and in [13] we found a counterexample to this conjecture. In this paper, we mainly restrict our attention to fans which are the normal fans of reflexive polytopes. The set of reflexive polytopes has a duality operation ∗ which is very beneficial for our purposes. If ∆ is a reflexive polytope then we write Σ for the normal fan of ∆ and Σ∗ for the normal fan of the dual (or polar) polytope ∆∗ . It can occur that XΣ is highly singular while XΣ∗ is nonsingular. For example, if ∆ is the d dimensional cross polytope then ∆∗ is the d dimensional cube. In this case, XΣ∗ = P1 × P1 × · · · × P1 and the singular locus of XΣ has codimension 2. The main question we seek to answer is the following. What happens to the Z2 Hodge spaces of Σ (and the topology of XΣ ) when Σ∗ is a smooth fan? We answer this question by using the polyhedral duality mentioned above and using the language of cosheaves on a fan. Section 4 is devoted to establishing the following theorem. Theorem 1.0.1 Let ∆ be a reflexive polytope and Σ∗ the face fan of ∆. If the cones in Σ∗ of dimension e are Z2 regular then (1) Hpq (Σ) = 0 for q < p < e − 1 (2) Hqq (Σ) = Z2 for q < e − 1 where Σ is the normal fan of ∆. In Section 5 we assume Σ∗ consists of Z2 regular cones. We show that the spectral r r 1 sequence (E , d ) for XΣ collapses at E , and hence XΣ is maximal. Some notation: N ∼ = Zd and M = Hom(N, Z) are dual lattices with dual pairing denoted h·, ·i. All tensor products are taken over Z. We write Σ ⊂ N when Σ ⊂ N ⊗ R is the normal fan of a d dimensional lattice polytope ∆ ⊂ M ⊗ R (written ∆ ⊂ M ). Except where noted, all rational fans are assumed to be polytopal. Many statements hold for the more general complete fans, but the class of polytopal fans is enough for our purposes. For σ ∈ Σ we will abuse notation by writing σ ∩ N for the lattice of points in the linear span of σ. The cones in Σ (resp. faces of σ) of dimension k are denoted Σ(k) (resp. σ(k)). All homology and cohomology groups will be with Z2 coefficients, unless otherwise stated. I would like to thank Matthias Franz and Tom Braden for helpful discussions at the June 2006 Toric Topology conference in Osaka, my advisor Clint McCrory for our ongoing discussions, and Bernd Sturmfels for introducing me to the beauty of real toric varieties.

217

Hodge spaces of real toric varieties 2. The real points of a toric variety

2.1

The natural cell structure on XΣ (R)

A real toric variety is a normal real variety that contains a real algebraic torus (R∗ )d as a dense subset. When XΣ = XΣ (C) is a projective toric variety embedded in CPr , the real variety XΣ (R) is the intersection of XΣ with the real projective space RPr . A discussion of real toric varieties can be found in [20]. Suppose XΣ is projective with moment map µ = µC (see [10, §4.2]). Since µC is invariant under complex conjugation, we have a map µR for the real toric variety XΣ (R). We write the complex torus as 1 T = (R>0 )d × (N ⊗R)/N and the real torus as T (R) = (R>0 )d × 2 N/N . The maps µC and µR are shown below (R>0 )d

×

(N ⊗R)/N

⊂ XΣ (C)

∪ (R>0 )d ∼ =

∪

1 N 2

×

⊂ XΣ (R)

/N

µC

µR



 |

⊂

int ∆

.

∆

Suppose p ∈ int f , f < ∆, and σ ∈ Σ the cone dual to f . Then, we have the following µ−1 C {p}

=

∪ µ−1 R {p}

(N ⊗R)/σ

∼ =

N/σ∩N

1 N 2 1 σ∩ 2 N

/N /σ∩N

In the sequel, we will use N/2N rather than

(1)

∪

∪ =

(S 1 )dim f

∼ = (S 0 )dim f

1 N 2

.

/N and N (σ) :=

N/2N σ∩N/σ∩2N

instead of

1 N 2 1 σ∩ 2 N

/N for convenience. The group σ∩N/σ∩2N will be denoted Nσ . The real moment /σ∩N

map µR induces a polyhedral cell structure on XΣ (R). Explicitly, we realize XΣ (R) as the quotient space ∆ × N/2N XΣ (R) ∼ = ∼ (x, t) ∼ (x0 , t0 ) ⇐⇒ x = x0

(2) and t − t0 ∈ Nγ

where γ is dual to g < ∆ and x ∈ intg. Thus, the cells in (2) are of the form (g, t) with t ∈ N (γ). A discussion of this cell structure can be found in [12, §11.5 B]. Next, we discuss the first step in an algorithm for identifying the faces of the 2d copies of ∆. This algorithm basically follows from [12, Theorem 5.4 in Chapter 11]. The entire algorithm is presented in [14] and involves at step j gluing faces of codimension j. When we identify two faces, we will glue the closed faces. The reason for this is as

218

Hower

follows. Let (g, t) and (g, t0 ) be two open faces identified in the relation ∼ of (2). Then, t − t0 ∈ Nγ where γ is dual to g. Suppose f < g in ∆ and β is the cone dual to f . We have γ < β and Nγ ⊂ Nβ yielding t − t0 ∈ Nβ . Thus, (f, t) and (f, t0 ) are identified in the relation ∼. Step 1: Let f1 , f2 , · · · , fp be the facets of ∆ and ri the ray in Σ dual to fi . If Ri is the image of ri in N/2N then (fi , t) is identified with (fi , t0 ) if and only if t − t0 = Ri . Lemma 2.1.1 After Step 1 we are left with 2d−s connected components, where s = rank span{R1 , R2 , · · · , Rp }




with Ri the image of ri in N/2N and {ri , r2 , · · · , rp } the rays of Σ. To prove Lemma 2.1.1 we fix t ∈ N/2N . The d-cell (∆, t) will be glued to the d-cells (∆, t+R1 ), (∆, t+R2 ), · · · , (∆, t+Rp ) along the facets f1 , f2 , · · · , fp respectively. Next, note that the cell (∆, t + Ri + Rj ) is in the same component as (∆, t) since the cell (∆, t + Ri ) is glued to (∆, (t + Ri ) + Rj ) along the facet fj . Similarly, we can see P P that (∆, t + i∈I Ri ) is in the same connected component as (∆, t) where i∈I Ri ∈ span{R1 , R2 , · · · , Rp }. As the rank of the Z2 vector space is s, there are exactly 2s d d-cells in this connected component yielding 2 /2s = 2d−s connected components in total after Step 1.  Note that in Steps 2 and higher, only faces of codimension greater than 1 are identified. Hence, Lemma 2.1.1 gives rankHd (XΣ (R)) = 2d−s .

2.2

T-homeomorphic versus algebraically isomorphic

Two real toric varieties may be homeomorphic even though they are not isomorphic as algebraic varieties. Since our main interest lies in the topological type of real toric varieties, we introduce the notion of T-homeomorphic real toric varieties. We will first develop the notion of T-homeomorphism for the real variety XΣ (R) in the projective case assuming ∆ is full dimensional. We then discuss this notion for the real part of an orbit closure corresponding to a face f of ∆. Suppose Σ1 ⊂ N1 and Σ2 ⊂ N2 are two fans with N1 ∼ = N2 ∼ = Zd . Let’s assume A is a d × d integer valued matrix with detA an odd integer. We consider A as a map between the d dimensional lattices N1 and N2 . Then we have A(2N1 ) ⊂ 2N2 and hence A determines a map A0 : N1/2N1 −→ N2/2N2 . Since the determinant of A is odd, the preimage A−1 {2N2 } is contained in 2N1 . Thus A0 is an injective map between Z2 vector spaces of the same rank yielding A0 is an isomorphism of vector spaces. Definition 2.2.1 Suppose the d × d matrix A determines a rational isomorphism ∼ =

/ XΣ (R) provided of fans. Then A induces a T-homeomorphism Ae : XΣ1 (R) 2 detA = 1(mod 2). We say XΣ1 (R) and XΣ2 (R) are T-homeomorphic.

219

Hodge spaces of real toric varieties

Note that for σ1 ∈ Σ1 the restriction A0 Nσ is an isomorphism between Nσ1 and

1

Nσ2 = A0 (Nσ1 ) where σ2 ∈ Σ2 . Note that the dual matrix A∗ gives a combinatorial equivalence between the polytopes ∆2 and ∆1 which define XΣ2 and XΣ1 , respectively. Hence, the T-homeomorphism Ae is a homeomorphism which respects the natural cell structure on XΣ1 (R)

∆1 ×N1/2N1 ∼1

e A ∼ =

/ ∆2 ×N2/2N2 .

∼2

Suppose f1 < ∆1 with σ1 ∈ Σ1 dual to f . Using equation (1) in Section 2.1, we see V (σ1 )(R) inherits a cell structure from the cell structure (2) on XΣ1 (R). The notation V (σ1 ) refers to the closure of the torus orbit associated to the cone σ1 . Specifically, we have f × N (σ1 ) V (σ1 )(R) ∼ = ∼f f 0 0 (x, t) ∼ (x , t ) ⇐⇒ x = x0

(3) and t − t0 ∈ Nγ/Nσ

0 where γ is dual to g < f and x ∈ intg. Next, the isomorphism A Nσ : Nσ1



∼ =

1

mentioned above determines an isomorphism between the quotients N (σ1 )

∼ =

/ Nσ 2 / N (σ2 ).

We obtain a homeomorphism induced by the T-homeomorphism Ae

f1 × N (σ1 ) ∼1f1

∼ =

/ f2 × N (σ2 ) ,

∼f22

where f2 < ∆2 is the face for which A∗ (f2 ) = f1 . When this occurs we say V (σ1 )(R) and V (σ2 )(R) are T-homeomorphic.

3. The Z2 Hodge spaces and the spectral sequence E

3.1

r

The cosheaf E and the Z2 Hodge spaces Hpq (Σ)

In this section, we define the notion of a cosheaf on a fan Σ and cosheaf homology. In later sections, we will also use sheaves on Σ and sheaf cohomology. The definition and properties of sheaves on fans are similar to those of cosheaves and are developed in [7, §1.1]. Sheaves on fans are also studied in [6, 1]. The main difference between the sheaves we consider and those in [7] is that our sheaves are sheaves of Z2 vector spaces. A cosheaf F of Z2 vector spaces on a fan Σ is a collection of vector spaces (Fσ )σ∈Σ over Z2 together with face extension maps ρσ,τ : Fσ −→ Fτ for σ < τ satisfying the following two conditions. • If σ < τ < β then ρτ,β ◦ ρσ,τ = ρσ,β • ρσ,σ is the identity map Note that throughout our work, we will use script letters for the cosheaf (or sheaf) and Roman letters for their stalks.

220

Hower

We define Hp (F ) to be the pth homology group of the complex (C∗ (F ), ∂∗ ), where M Cp (F ) := Fσ σ∈Σ(d−p)

and ∂p : Cp (F ) −→ Cp−1 (F ) is the direct sum of the maps X

M

ρσ,τ : Fσ −→

σ∈τ (d−p)

Fτ .

(4)

σ∈τ (d−p)

Note that if we were working with cosheaves of k vector spaces with chark 6= 2 then we would need to introduce signs in (4) to guarantee ∂ 2 = 0. Next, we define the cosheaf N on Σ by Nσ := σ∩N/σ∩2N

for σ ∈ Σ

with the extension map ρσ,τ given by inclusion ⊂

ρσ,τ : Nσ

/ Nτ

for σ < τ .

If dim σ = q then Nσ is a rank q vector space over Z2 . We encountered the cosheaf N in Section 2.1 when identifying the 2d copies of ∆. The cosheaf E is defined so that Eσ := N (σ) =

N/2N

Nσ

.

If σ < τ then the extension map $σ,τ : N (σ) −→ N (τ ) is induced from the identity on N/2N which takes Nσ to Nτ . Thus, the cosheaf E is the cokernel of the inclusion N ,→ N/2N , where N/2N is the constant cosheaf assigning N/2N to each cone in Σ. Definition 3.1.1 The Z2 Hodge spaces of Σ, denoted Hpq (Σ), are defined to be the homology groups Hpq (Σ) := Hp (∧q E ), where ∧q E is the qth exterior power of the cosheaf E . Remark 3.1.2 Since ∧q Eσ is zero for q > codim σ, we have Cp (∧q E ) = 0 for p < q. Hence the Z2 Hodge spaces Hpq (Σ) are indexed by integers p, q with 0 ≤ q ≤ p ≤ d.

3.2

The spectral sequence E

r

We recall that the cells in XΣ (R) are of the form (f, t) with f < ∆ and t ∈ N (σ), σ ∈ Σ is dual to f . It follows that Cj (XΣ (R)) =

M

H0 (N (σ)),

(5)

σ∈Σ(d−j)

where H0 (N (σ)) is the group algebra of the finite group N (σ). Note that elements in H0 (N (σ)) are formal combinations of elements in N (σ) with coefficients in Z2 . The

Hodge spaces of real toric varieties

221

boundary map ∂ for the cellular chain complex C∗ (XΣ (R)) is induced from the face extension maps $σ,τ : N (σ) −→ N (τ ) from Section 3.1, and ∂j is the direct sum of the maps X M ($σ,τ )∗ : H0 (N (σ)) −→ H0 (N (τ )). σ∈τ (d−j)

σ∈τ (d−j)

In [5] Bihan et al. want to understand the relationship between the topology of XΣ (R) and that of XΣ (C). The authors show that the chain complex C∗ (XΣ (R)) r r can be filtered in such a way that the associated spectral sequence (E , d ) converges 1 to H∗ (X(R)) and is known to collapse at E when XΣ is complete and has isolated singularities or when the dimension of XΣ is at most 3 [5]. Our notation is slightly different than in [5] and hence we briefly review their construction. Using (5) we may specify a filtration on C∗ (XΣ (R)) by giving a filtration of H0 (N (σ)), the Z2 group algebra of N (σ). We use the augmentation homomorphism σ σ : H0 (N (σ)) −→ Z2 X

ng g 7−→

X

ng .

g∈N (σ)

We define Iσ , an ideal in H0 (N (σ)), via Iσ := ker σ . This gives a filtration of H0 (N (σ)) 0 = Iσj+1 ⊂ Iσj ⊂ · · · ⊂ Iσ2 ⊂ Iσ ⊂ Iσ0 = H0 (N (σ)) where j = rank N (σ) = codim σ. We reindex by setting Jσp = Iσd−p so that 0 = Jσd−j−1 ⊂ Jσd−j ⊂ · · · ⊂ Jσd−2 ⊂ Jσd−1 ⊂ Jσd = H0 (N (σ))

(6)

is an increasing filtration of H0 (N (σ)). The filtrations of H0 (N (σ)) of the form
(6) for σ ∈ Σ determine an increasing filtration F of C∗ (XΣ (R)) and ∂j Fq Cj (XΣ (R)) ⊂ e r , der ) be the associated spectral sequence as in [16]. Using Fq Cj−1 (XΣ (R)). We let (E [5, Proposition 7.1] we arrive at the following equality e1 ∼ E p,q = Hp+q, d−p (Σ),

where Hp+q, d−p (Σ) is the Z2 Hodge space of Σ defined in Section 3.1. Next, since e 1 = 0 for q < d − 2p, we obtain a natural homomorphism E p,q e1 E d−(p+q),2(p+q)−d −→ Hp+q (XΣ (R)).

(7)

At this point, we introduce reindexing which we will use in the sequel. We rotate e 1 term clockwise by 90◦ and then shear so that the diagonal lines p + q = c the E r

r

become vertical lines p = c. We write (E , d ) for the spectral sequence obtained after reindexing. r r 1 The total grading of (E , d ) is p, and furthermore we have the identity E p,q = 1

e 1 and E Hpq (Σ). Figure 1 shows the terms E p,q p,q for a 4 dimensional fan Σ. r r r r r For the spectral sequence (E , d ) the boundary map d satisfies dp,q : E p,q −→ r E p−1,q+r . We show the first five boundary maps in Figure 2.

222

Hower

H44 (Σ) H43 (Σ) H33 (Σ) q p

H42 (Σ) H32 (Σ) H22 (Σ)

H41 (Σ) H31 (Σ) H21 (Σ) H11 (Σ)

H40 (Σ)

q

H30 (Σ) H20 (Σ) H10 (Σ) H00 (Σ)

H00 (Σ)

H11 (Σ) H10 (Σ)

H22 (Σ) H21 (Σ) H20 (Σ)

H33 (Σ) H32 (Σ) H31 (Σ) H30 (Σ)

H44 (Σ) H43 (Σ) H42 (Σ) H41 (Σ) H40 (Σ)

p

e 1 (left) and E 1 (right) Figure 1: The terms E p,q p,q

0

1

2

3

Figure 2: The differentials d , d , d , d , and d

4

Next, we point out that after reindexing the natural homomorphism (7) becomes a map which we denote f∗R and will use in future sections where 1

fqR : Hqq (Σ) = E q,q −→ Hq (XΣ (R)).

(8)

From [5, Proposition 7.1] we have 1 2 E p,q ∼ = Ep,q

for 0 ≤ p, q ≤ d

(9)

where (E r , dr ) is the Z2 Leray spectral sequence for the moment map µC for XΣ (C). We refer the reader to [16, §XI.7] or [17, §5] for construction of the Leray spectral sequence and to Section 2.1 for properties of µC . We now have a way to compare the topology of the real and complex points of a toric variety. The Smith-Thom inequality [8, §1.1] states the sum of the Z2 Betti numbers for XΣ (R) is less than or equal to the sum of the Z2 Betti numbers for XΣ (C). We say XΣ is maximal if equality is obtained. Using the the Smith-Thom inequality and (9) we have the following diagram P

1

rank(E p,q )

=

P

≤

P

∨|

P

bi (X(R))

2 ) rank(Ep,q ∨|

bj (X(C)) .

(10)

Hodge spaces of real toric varieties r

223

1

If the spectral sequence E collapses at E then the left vertical inequality in (10) is equality. This forces both the right vertical inequality and the lower horizontal inequality to both be equalities. In this case we obtain XΣ is maximal.

3.3

The diagonal entries Hqq (Σ)

In this section we work with the Z2 torus invariant Chow groups of XΣ . The integral torus invariant Chow groups are discussed in [10, §5.1] or [11]. When considering T T (X ), where the Z2 torus invariant Chow groups, we define ATq (XΣ ) := Zq (XΣ )/Rq+1 Σ T Zq (XΣ ) is the Z2 vector space generated by cycles [V (τ )] for τ ∈ Σ of codimension q. The relations are generated by torus invariant divisors of the following form. For σ ∈ Σ of codimension q + 1, each u ∈ σ ⊥ ∩ M gives a rational function χu on V (σ). T (X ) is generated by divχu except the coefficients are taken mod 2. The subspace Rq+1 Σ That is, the coefficient of [V (τ )] in the relation divχu is hu, nσ,τ i (mod 2)

(11)

where nσ,τ is a lattice generator of τ ∩N/σ∩N . Equivalently, we may take u ∈ M (σ) := σ ⊥ ∩M/σ ⊥ ∩2M and n N σ,τ the nonzero element of τ/Nσ to obtain the coefficient (11). By tensoring the integer complexes with Z2 we have Hqq (Σ) ∼ = ATq (XΣ ), for 0 ≤ q ≤ d. A discussion of this canonical isomorphism may also be found in [14]. Lemma 3.3.1 Assume for each τ ∈ Σ(d − k − 1) we have V (τ )(R) is T-homeomorphic to RPk+1 . If q ≤ k then ATq (XΣ ) is generated by the orbit closure of any q dimensional torus orbit in XΣ . Proof. We first look at the relations coming from a single cone σ of codimension q + 1. Since σ is a face of a cone in Σ(d−k −1), V (σ)(R) is T-homeomorphic to RPq+1 . Thus, M (σ) ∼ = Hom(N (σ), Z2 ) is generated by elements a1 , a2 , · · · , aq+1 and σ is contained in the q + 2 cones ω1 , ω2 · · · , ωq+2 of codimension q. Let a ˘i ∈ N (σ) be dual to ai ∈ M (σ) for 1 ≤ i ≤ q. Then, since V (σ)(R) is T-homeomorphic to RPq+1 , we have nσ,ωi , the nonzero element of Nωi/Nσ ⊂ N (σ), is of the form (

nσ,ωi =

a ˘i

if

a ˘1 + a ˘2 + · · · + a ˘q+1 if

1≤i≤q+1 i=q+2 .

Hence, the map M (σ) −→ ZqT (XΣ )

(12)

is given by basis elements as follows a1 7−→ [V (ω1 )] + [V (ωq+2 )] a2 7−→ [V (ω2 )] + [V (ωq+2 )] ··· aq+1 7−→ [V (ωq+1 )] + [V (ωq+2 )] . Extending linearly to all of M (σ), we see that in the image of (12) we obtain the sum of any even number of [V (ωi )]’s. For any r with 1 ≤ r ≤ q + 2, the cycle [V (ωr )] is

224

Hower

not in the image of (12). Hence for any r, the cycle [V (ωr )] is a representative of the generator of coker(M (σ) −→

q+2 M

[V (ωi )]). Next, we consider the map

i=1

M

M

M (σ) −→

σ∈Σ(d−q−1)

[V (ω)],

(13)

ω∈Σ(d−q)

where the map on each M (σ) is described above. If ω and ω 0 are codimension q cones in Σ then [V (ω)] + [V (ω 0 )] is in Claim 3.3.2 the image of (13). To see the claim, note that if ω∩ω 0 is a codimension q+1 cone then [V (ω)]+[V (ω 0 )] is in the image of (13), as discussed above. Else, find a sequence of codimension q and codimension q + 1 cones ω2

ω1

ω = ω0

σ0,1

σ1,2

···

ωn = ω 0

ωn−1

···

σn−1,n

where σi−1,i < ωi−1 and σi−1,i < ωi for 1 ≤ i ≤ n. There exists ui−1,i ∈ M (σi−1,i ) with div(χui−1,1 ) = [V (ωi−1 )] + [V (ωi )]. Adding, we obtain n X

ui−1,i ∈

i=1

and

n X

M

M (σ)

σ∈Σ(d−q−1)

ui−1,i 7−→ [V (ω)] + [V (ω 0 )]

i=1

which proves the claim. Thus, the sum of any even number of [V (ω)]’s is in the image of (13) and the cycle [V (ω 0 )] is not. Hence, ATq (XΣ ) is generated by [V (ω 0 )] where ω 0 is any cone of codimension q.  Next, we recall the natural map fqR : ATq (XΣ ) ∼ = Hqq (Σ) −→ Hq (XΣ (R)) given in (8) which arises from the edge homomorphisms of the spectral sequence e r , der ). Thinking cellularly, the real points of an algebraic cycle [V (β)] in Z T (XΣ ) is (E q represented by the sum of all 2q cells in the orbit Oβ (R). Hence, if g is a q dimensional face of ∆ with β the cone dual to g then fqR ([V (β)]) is the homology class of Cβ where P Cβ := t∈N (β) (g, t) is the cellular chain in Cq (XΣ ) obtained by adding all 2q copies of g in XΣ (R). We recall that a cone σ ⊂ N is Z2 regular provided the image in N/2N of the rays of σ forms a basis for the Z2 vector space Nσ . Bihan et al. [5] show that if Σ consists of Z2 regular cones then the map fqR is an isomorphism for 0 ≤ q ≤ d. The counterexample in [13] shows that in general fqR is neither injective nor surjective.

225

Hodge spaces of real toric varieties

3.4

The right-most column Hdq (Σ)

Let {r1 , r2 , · · · , rk } be the rays of Σ and Ri the image of ri in N/2N ∼ = (Z2 )d . Proposition 3.4.1 Suppose the rank of the Z2 vector space V := span(R1 , R2 , · · · , Rk ) is s ≤ d. Then, ! d−s r rankHdq (Σ) = and dd,q = 0 for r ≥ 1. q−s Proof. We choose a basis for V consisting of a subset of the Ri , and reorder if needed so that {R1 , R2 , · · · , Rs } is a basis for V . Note that the elements of the form Ri ∧ ∗ ∧ ∗ ∧ · · · ∧ ∗ generate ker(∧q N/2N −→ ∧q N (ri )), where the ∗-entries are any elements of N/2N . Hence, qN

ker(∧ /2N −→

s M

∧q N (ri ))

i=1

is generated by elements of the following form R1 ∧ R2 ∧ · · · ∧ Rs ∧ ∗ ∧ ∗ ∧ · · · ∧ ∗ . n This is a subspace of ∧q N/2N of rank d−s q−s , where k = 0 if k < 0. Hence, the kernel of the boundary map M 0 dd,q : ∧q N/2N −→ ∧q N (r)







r∈Σ(1)

has rank at most

d−s
q−s

. This shows bd (XΣ (R)) ≤

! d X d−s q=s

q−s

= 2d−s

0

(14)

and equality holds if and only if for each q, kerdd,q has rank d−s q−s and all higher differr entials dd,q , r ≥ 1 with source the rightmost column are zero. From Lemma 2.1.1 we

have bd (XΣ (R)) = 2d−s yielding rankHdq (Σ) = r dd,q , r ≥ 1 are zero.






d−s q−s



and the higher boundaries 

226

Hower

4. Z2 Hodge spaces for reflexive polytopes

4.1

A correspondence between sheaves and cosheaves

A reflexive polytope is a lattice polytope ∆ with 0 ∈ int∆ and such that the polar polytope ∆∗ is also a lattice polytope. A discussion of reflexive polytopes can be found in [2, §4.1]. Throughout this section we will use the following notation ∆⊂M ∆∗ ⊂ N Σ⊂N

a reflexive polytope the polar polytope of ∆ the normal fan of ∆ (= the face fan of ∆∗ ) the normal fan of ∆∗ (= the face fan of ∆) .

Σ∗ ⊂ M

If τ ∈ Σ and dim τ = j > 0 then τ = poshullf ∗ where f ∗ < ∆∗ is a face of dimension j − 1. We define τ ∗ to be the cone in Σ∗ of dimension d − j + 1 which is dual to f ∗ . The correspondence τ ∈ Σ ←→ τ ∗ ∈ Σ∗ gives a one-to-one inclusion reversing correspondence between the cones in Σ of positive dimension and the positive dimensional cones in Σ∗ , where dimτ ∗ = d − dimτ + 1. Let H be a sheaf on the fan Σ∗ . We use the correspondence above to create don Σ by defining for τ ∈ Σ a cosheaf H c := H τ

  Hτ ∗

if dim τ > 0

 0

if dim τ = 0

(15)

with extension map for σ < τ ρbσ,τ :=

  ρτ ∗ ,σ∗

if dim σ > 0

 0

if dim σ = 0

(16)

where ρτ ∗ ,σ∗ is a restriction map for the sheaf H . Using (15) we have equality of don Σ, as depicted below. chain groups for the sheaf H on Σ∗ and the cosheaf H Cd (H )

b

0

/

Cd−1 (H )

/

b

C 0 (H )

/ Cd−2 (Hb ) /

C 1 (H )

/ ···

/ ···

/

/

/

C2 (H )

b

C d−3 (H )

/

C1 (H )

/ C0 (Hb )

C d−2 (H )

/ C d−1 (H )

b

From (16) the horizontal sheaf and cosheaf boundary maps are equal. Hence we may equate sheaf cohomology groups for H on Σ∗ with cosheaf homology don Σ groups for H d) ∼ Hp (H = H d−p−1 (H ) ,

p > 0.

Hodge spaces of real toric varieties

227

One more construction which we will use in the sequel is that of the cosheaf A ◦ . If A is a cosheaf on Σ then A ◦ is defined by   Aσ

A◦σ :=  0

if dim σ > 0 if dim σ = 0

with the extension map for σ < τ defined by ρ◦σ,τ :=

  ρσ,τ

if dim σ > 0

 0

if dim σ = 0 .

Note that by definition Cp (A ) = Cp (A ◦ ) for p ≤ d − 1, and hence we have Hp (A ) = Hp (A ◦ ) for p ≤ d − 2. 4.2

c, Gb, and C on Σ The cosheaves F

The sheaf F on Σ∗ is defined as follows. For σ ∗ ∈ Σ∗ the stalk is Fσ∗ :=

(σ ∗ )⊥ ∩ N (σ ∗ )⊥ ∩ 2N

and the face restriction map ρτ ∗ ,σ∗ : Fσ∗

⊂

/F ∗ τ

is given by inclusion for τ ∗ < σ ∗ in Σ∗ . The sheaf G on Σ∗ is then defined to be the cokernel of the inclusion F ,→ N/2N so that 0 −→ F −→ N/2N −→ G −→ 0

(17)

is a short exact sequence of sheaves on Σ∗ . Claim 4.2.1 For τ ∈ Σ, τ ∗ ∈ Σ∗ as in the previous section with dim τ = l > 0, we have the following containment of Z2 vector spaces τ ∩N (τ ∗ )⊥ ∩ N ⊂ . ∗ ⊥ (τ ) ∩ 2N τ ∩ 2N

(18)

Note that the Z2 vector space on the left has rank l − 1 while the one on the right has rank l. To prove the claim, let f < ∆ and f ∗ < ∆∗ be such that τ = poshullf ∗ = poshull{q1 , q2 , · · · , qs }

τ ∗ = poshullf = poshull{p1 , p2 , · · · , pk }.

228

Hower

Then we have hpi , qj i = −1 ∀ i, j. This gives (τ ∗ )⊥ = span{qi − qj | 1 ≤ i, j ≤ s} and the claim follows. Next, we note that the inclusion (18) is compatible with the face extension maps for cones in Σ. Thus, we obtain an injective homomorc ,→ N . We define the cosheaf C to be the cokernel of this phism of cosheaves F homomorphism so that c −→ N −→ C −→ 0 0 −→ F

(19)

is an exact sequence of cosheaves on Σ. Note that if dim σ = 0 then the stalks Fbσ , Nσ , and Cσ are all zero. Moreover, for each σ ∈ Σ of positive dimension the stalk Cσ is a rank one Z2 vector space. Lemma 4.2.2 For σ < τ in Σ with dim σ > 0 the map Cσ −→ Cτ is an isomorphism. Proof. We prove the lemma by contradiction. Assume we have τ1 < τ2 in Σ and Cτ1 −→ Cτ2 is the zero map. We have the follow diagram (τ1∗ )⊥ ∩N (τ1∗ )⊥ ∩2N

/

0

/

τ1 ∩N τ1 ∩2N

/

(τ2∗ )⊥ ∩N (τ2∗ )⊥ ∩2N

/0

φ



0

/ Cτ 1

/



τ2 ∩N τ2 ∩2N

 / Cτ 2

/0

where the rows are short exact sequences and vertical maps are cosheaf extension maps. We have assumed φ = 0 and hence τ1 ∩ N (τ ∗ )⊥ ∩ N ⊂ ∗2 ⊥ . τ1 ∩ 2N (τ2 ) ∩ 2N

(20)

Suppose τ2 = poshull{q1 , · · · , ql }, τ1 = poshull{q1 , · · · , ql−r }, and τ2∗ = poshull{p1 · · · , pk } . Due to the inclusion (20) we have hq1 , p1 i = 0(mod 2). This contradicts the fact the ∆ is reflexive which guarantees hq1 , p1 i = −1.  Remark 4.2.3 If we were considering that analogous sheaves of Z modules or of Q vector spaces then the map Cσ −→ Cτ from Lemma 4.2.2 would be injective without the assumption of reflexivity. Next, note that c −→ Nd 0 −→ F /2N −→ Gb −→ 0

(21)

is a short exact sequence of cosheaves of Σ where Nd /2N = N/2N ◦ as cosheaves on Σ. We can combine (21) with (19) into the following commutative diagram

229

Hodge spaces of real toric varieties

0

0

0



0

0

K

 / c F

 / Nd /2N

 / b G







Φ

/ N/2N ◦

/N 

/ E◦



C

/0

/0



0

0



0 where the rows and columns are exact and K := kerΦ. Next, we use the Snake Lemma on the commutative diagram to obtain K ∼ = C as cosheaves yielding 0 −→ C −→ Gb −→ E ◦ −→ 0

(22)

is a short exact sequence of cosheaves on Σ. As (∧q E )◦ = ∧q E ◦ we have Hp (∧q E ) = Hp (∧q E ◦ ) for p ≤ d − 2. 4.3

Vanishing of the homology groups Hp (∧q Gb )

We remind the reader that a cone σ ∈ Σ is Z2 regular provided the image in N/2N of the rays of σ forms a basis for Nσ . Lemma 4.3.1 Assume the cones in Σ∗ of dimension at most e are Z2 regular. Then Hp (∧q Gb ) = 0 for 1 ≤ p < e − 1. Proof. First, we will assume that e = d. Following [7, §1.2], as the cones in Σ∗ are Z2 regular the sheaf G on Σ∗ can be written as follows. G =

G (ri )

M ri ∈Σ∗ (1)

is a direct sum of the sheaves G (ri ) with G(ri )τ ∗ =

  Z2

if ri ∈ τ ∗ (1)

 0

else.

230

Hower

Moreover, we have ∧q G =

G (r1 , r2 , · · · , rk )

M r1 ,r2 ,··· ,rk distinct

is the direct sum of sheaves G (r1 , r2 , · · · , rk ) with G(r1 , r2 , · · · , rk )

τ∗

=

  Z2

if r1 , r2 , · · · , rk ∈ τ ∗ (1)

 0

else.

The proof of the proposition in [7, Section 1.2] yields H p (G (r1 , r2 , · · · , rk )) = 0 for p > 0 because Σ∗ is a complete fan. Hence we have H p (∧q G ) = H p





G (r1 , r2 , · · · , rk )

M r1 ,r2 ,··· ,rk distinct

∼ =

M

H p (G (r1 , r2 , · · · , rk ))

r1 ,r2 ,··· ,rk distinct

= 0 for p > 0. q G we have H (∧q Gb ) = H d−p−1 (∧q G ) = 0 for p such that Hence, as ∧q Gb = ∧d p p > 0 and d − p − 1 > 0, which proves Lemma 4.3.1 when e = d. Next, assume e < d. We still have H p (G (r1 , r2 , · · · , rk )) = 0 for p > 0 S L however the sheaf G cannot be written as ri ∈Σ∗ (1) G (ri ). Let Σ∗≤e := i≤e Σ∗ (i) be the subfan of Σ∗ consisting of the cones of dimension at most e, and let G 0 be the restriction of G to Σ∗≤e . Then as Σ∗≤e consists of Z2 regular cones, we have the following equality M G (ri )0 G0 = ri ∈Σ∗ (1)

where G (ri )0 is the restriction of G (ri ) to Σ∗≤e . Moreover, we have ∧q G 0 =

M

G (r1 , r2 , · · · , rk )0 .

r1 ,r2 ,··· ,rk distinct

Hence we obtain H p (∧q G ) ∼ = H p (∧q G 0 ) M ∼ =

H p (G (r1 , r2 , · · · , rk )0 )

r1 ,r2 ,··· ,rk distinct

∼ =

M

H p (G (r1 , r2 , · · · , rk ))

r1 ,r2 ,··· ,rk distinct

= 0, for p > d−e. In the above equalities we used that H p (A ) = H p (A 0 ) for p > d−e, where A is any sheaf on Σ∗ and A 0 is the restriction to Σ∗≤e . Thus, Hp (∧q Gb ) = 0 for p such that p > 0 and d − p − 1 > d − e. 

Hodge spaces of real toric varieties

4.4

231

Vanishing of the Z2 Hodge spaces Hpq (Σ)

In this section, we use the results from the previous two sections to determine the vanishing of certain Z2 Hodge spaces. See Section 6 for examples. Theorem 4.4.1 Assume the cones in Σ∗ of dimension at most e are Z2 regular. Then Hp (∧q E ) = 0 for q < p < e − 1. Proof. We use the short exact sequence 0 −→ C −→ Gb −→ E ◦ −→ 0 of cosheaves of Σ and the associated Koszul degree q exact sequence 0 −→ S q C −→ ∧1 Gb⊗S q−1 C −→ · · · −→ ∧q−1 Gb⊗S 1 C −→ ∧q Gb −→ ∧q E ◦ −→ 0, (23) where S q−k C is the (q − k)th symmetric power of the cosheaf C . A similar exact sequence of sheaves of Q vector spaces can be found in [7, §1.2] and a treatment of this sequence is in [14, §3.3]. We break (23) into short exact sequences below 0 −→ S q C −→ ∧1 Gb ⊗ S q−1 C −→ W1 −→ 0

(24)

0 −→ W1 −→ ∧2 Gb ⊗ S q−2 C −→ W2 −→ 0

(25)

···

(26)

0 −→ Wq−1 −→ ∧q Gb −→ ∧q E ◦ −→ 0 .

(27)

These induce long exact sequences on homology groups. From Lemma 4.2.2 we see that C q−k S C

∼ = (Z2 )◦ ∼ = (Z2 )◦

where Z2 is the constant cosheaf on Σ. Thus we have Hp (∧k Gb ⊗ S q−k C ) ∼ = Hp (∧k Gb ) ⊗ S q−k C = 0 for 1 ≤ p < e − 1, where we have used Lemma 4.3.1. We begin with the long exact sequence induced from (24) · · · −→ Hp (∧1 Gb ⊗ S q−1 C ) −→ Hp (W1 ) −→ Hp−1 (S q C ) −→ · · · . We have Hp (∧1 Gb ⊗ S q−1 C ) = 0 Hp−1 (S q C ) = 0

for 1 ≤ p < e − 1 for 1 ≤ p − 1 < e − 1

and hence Hp (W1 ) = 0 for 1 < p < e − 1. Next we use the long exact sequence induced from (25) · · · −→ Hp (∧2 Gb ⊗ S q−2 C ) −→ Hp (W2 ) −→ Hp−1 (W1 ) −→ · · · .

232

Hower

We have Hp (∧2 Gb ⊗ S q−2 C ) = 0 Hp−1 (W1 ) = 0

for 1 ≤ p < e − 1 for 1 < p − 1 < e − 1

and hence Hp (W2 ) = 0 for 2 < p < e − 1. We continue this process to obtain Hp (∧q E ◦ ) = 0 for q < p < e − 1. Moreover, as e − 1 ≤ d − 1 we have Hp (∧q E ) = Hp (∧q E ◦ ) for q < p < e − 1.  When Σ∗≤e consists of Z2 regular cones, the proposition below gives additional information about the Z2 Hodge spaces of Σ. Proposition 4.4.2 If the cones in Σ∗≤e are Z2 regular then for q < e − 1 we have Hqq (Σ) has rank 1 and is generated by the orbit closure of any q dimensional torus orbit. Let τ ∗ ∈ Σ∗ (e), τ ∗ = poshull f where f = conv{p1 , p2 , · · · , pe }. The proposition follows from Lemma 3.3.1 and the following claim. Claim 4.4.3

The real variety V (τ )(R) in XΣ (R) is T-homeomorphic to RPe−1 .

Proof. Since τ ∗ is Z2 regular, we can find a matrix A of the form 



| | | | |  A := p1 p2 · · · pe t1 · · · td−e   | | | | | with t1 , t2 , · · · , td−e ∈ M , detA = 1(mod 2), and A giving a combinatorial equivalence between a polytope Ψ and ∆. Moreover, if xi is the ith standard basis vector for Rd then we have Axi = pi for 1 ≤ i ≤ e and hence A(conv{x1 , x2 , · · · , xe }) = f . The matrix A∗ determines a rational isomorphism between Σ and Γ, the normal fan of Ψ. As detA∗ = 1(mod 2), we obtain a Thomeomorphism between XΣ (R) and XΓ (R). Using Section 2.2 we have V (τ )(R) is T-homeomorphic to RPe−1 .  5. Collapsing of the spectral sequence E r

r

1

In this section, we show that the spectral sequence E for XΣ collapses at E when the cones in Σ∗ are Z2 regular. Using our work in Section 4.4, we have 1 that the ranks of the entries in the E term are as follows

Hodge spaces of real toric varieties

q

1 1 0 1 0 0 1 0 0 0 p

1 0 0 0 0

1 0 0 0 0 0

∗ ∗ ∗ ∗ ∗ ∗ 0

1 ∗ ∗ ∗ ∗ ∗ ∗ 0

233

(28)

where the ∗ entries are possibly nonzero and occur in the columns p > d − 2. Remark 5.0.4 We can completely determine the ∗ entries. From Section 3.4, if 1 s is the rank of the image of the rays of Σ in N/2N then the rank of E d,q is the  binomial coefficient d−s . It follows from [7, §1.4] that the Euler characteristic q−s q of the qth row is (−1) hq where h = (h0 , h1 , h2 , · · · , hd ) is the h-vector of the 1 polytope ∆. Thus, we may determine the ranks of the vector spaces E d−1,q , 1 giving us knowledge of the ranks of all the entries in E p,q . Examples can be found in Section 6. 1

By looking at the E term for XΣ , we see that the only possible nonzero 1 r higher differentials have target E d−2,d−2 . To show that the spectral sequence E 1 for XΣ collapses at E , we need only show the following lemma. Lemma 5.0.5 The map R fd−2 : Hd−2,d−2 (Σ) −→ Hd−2 (XΣ (R))

from (8) is nonzero. Proof. Let g be a d−2 dimensional face of ∆ with β = poshull {q1 , q2 , · · · , qk } the R cone dual to g.PAs mentioned in Section 3.3, fd−2 sends [V (β)] to the homology class of Cβ := t∈N (β) (g, t). Suppose Cβ = ∂C, C = (g1 , t1 ) + (g2 , t2 ) + · · · + (gr , tr ) ∈ Cd−1 (XΣ ) f where where gi < ∆, σi ∈ Σ is dual to gi , and ti ∈ N (σi ). We include N ⊂ N f ∼ f. Note that ∆∗ × [−1, 1] is reflexive and N = N ⊕ Z and ∆∗ × [−1, 1] ⊂ N the normal fan of ∆∗ × [−1, 1] consists of Z2 regular cones. Let Ξ be the face fan of ∆∗ × [−1, 1]. Then, Ξ is the normal fan of B∆ := (∆∗ × [−1, 1])∗ the bipyramid with base ∆. We define the cone βe ∈ Ξ to be the positive hull of the f. Note that βe is rays {(q1 , 1), (q1 , −1), (q2 , 1), (q2 , −1), · · · , (qk , 1), (qk , −1)} in N

234

Hower

dual to g considered as a face in B∆ . Moreover, we have e/2N N e e = N (β) e∩N e/βe∩2N β e

∼ = ∼ =

N/2N

⊕ hed+1 i ⊕ hed+1 i

β∩N/β∩2N N/2N β∩N/β∩2N

= N (β), where hed+1 i is the Z2 vector space generated by the (d + 1)st standard basis e in H vector. This gives that inclusion of [V (β)] d−2 (XΞ (R)) is also represented fi ∈ Ξ, cellularly by Cβ . Similarly, for each σi ∈ Σ appearing in the chain C, σ fi ) ∼ with rays of the form {(r, 1), (r, −1)| r is a ray of σ} satisfies N (σ = N (σi ). Hence, C can be viewed as a chain in Cd−1 (XΞ (R)). We have e =C ∂C β

(29)

where ∂e is the cellular boundary map for XΞ (R). Equation (29) holds because e generates fi < γ in Ξ implies γ must be of the form γ = σ e with σ ∈ Σ. As [V (β)] σ Hd−2,d−2 (Ξ), the map Hd−2,d−2 (Ξ) −→ Hd−2 (XΞ (R)) must be zero. However, the 1 1 ranks of the entries E for XΞ have the form (28) and E d−2,d−2 is in the 4th 1 column from the right. There cannot be higher boundaries with target E d−2,d−2 which contradicts the fact that Hd−2,d−2 (Ξ) −→ Hd−2 (XΞ (R)) is the zero map.  r

1

This shows that the spectral sequence E for XΣ collapses at E and hence we have the following. Theorem 5.0.6 If Σ∗ consists of Z2 regular cones then XΣ is maximal. The following definition can be found in [3] where Batyrev studies Fano polyhedra, a very special class of reflexive poltyopes. Definition 5.0.7 Let vert∆ = {v1 , v2 , · · · , vn } ⊂ Zd . The d dimensional polytope ∆ is a Fano polyhedron provided (1) 0 ∈ int∆ (2) Each face of ∆ is a simplex (3) If vi1 , vi2 , · · · , vid are the vertices of a (d − 1) dimensional face of ∆ then det[vi1 vi2 · · · vid ] = ±1. We have proved the maximality of toric varieties associated to the Fano polyhedra, which are also called smooth Fano polyhedra by some authors (e.g. [18]).

Hodge spaces of real toric varieties

235

If ∆ is a Fano polyhedron then ∆∗ defines one of the so called smooth toric Fano manifolds. The toric Fano manifolds XΣ∗ (C) are important in algebraic geometry since their anticanonical sheaf is very ample. A classification of the Fano polyhedra is known for dimension at most 4. There are 5 Fano polyhedra of dimension 2. Batyrev classified the 18 Fano polyhedra of dimension 3 in [4] and the 123 Fano polyhedra of dimension 4 in [3].

6. Examples

In this section, we illustrate our results with a few examples. We use torhom [9] to compute the Z2 Hodge spaces Hpq (Σ) and the f -vector of the polytope ∆. Example 6.0.8

The five dimensional cross polytope.

Let ∆ be the five dimensional cross polytope. Then, ∆ is the convex hull of the ten vertices {±e1 , ±e2 , ±e3 , ±e4 , ±e5 } . The polar polytope ∆∗ is the five dimensional cube, which defines the nonsingular toric variety P1 × P1 × P1 × P1 × P1 . The f -vector for ∆ is (10, 40, 80, 80, 32). We compute the ranks of the Z2 Hodge spaces for Σ

q

1 1 0 1 0 0 1 0 0 0 p

31 45 25 5 0

1 4 6 4 1 0 .

We compute the Z2 Betti numbers for the real points XΣ (R) by adding along the columns [1 1 1 1 106 16]. We compute the Z2 Betti numbers for the complex points XΣ (C) by adding along the diagonals [1 0 1 0 1 5 27 49 37 4 1]. Example 6.0.9

A seven dimensional example.

We define ∆ to be the convex hull of the following nine vertices {−e1 , −e2 , −e3 , −e4 , −e5 , −e6 , −e7 , e1 + e2 + e3 + e4 , e5 + e6 + e7 } . The f -vector for ∆ is (9, 36, 84, 125, 120, 70, 20). The polar polytope ∆∗ is the product P3 × P4 , where Pi defines i dimensional projective space. Moreover, ∆∗

236

Hower

defines the nonsingular toric variety P3 × P4 . Below are the ranks of the Z2 Hodge spaces Hpq (Σ)

q

1 1 0 1 0 0 1 0 0 0 p

1 0 0 0 0

1 0 0 0 0 0

15 31 34 21 7 1 0

1 2 1 0 0 0 0 0 .

We compute the Z2 Betti numbers for the real points XΣ (R) by adding along the columns [1 1 1 1 1 1 109 4] and for the complex points XΣ (C) by adding along the diagonals [1 0 1 0 1 0 1 1 8 21 35 31 16 2 1]. Example 6.0.10 A six dimensional example. Let ∆ be the convex hull of the twelve vertices below {−e1 , ±e2 , ±e3 , ±e4 , ±e5 , e6 , e1 − e6 , −e1 − e2 − e3 − e4 − e5 − e6 } . The f -vector for ∆ is (12, 62, 174, 267, 207, 64). The polar polytope ∆∗ has 64 vertices. Using polymake we determine that ∆∗ defines a toric variety with isolated singularities. That is, Σ∗≤5 consists of Z2 regular cones. The results of Section 4.4 give the Z2 Hodge spaces Hpq (Σ) for p < 4, as shown by the torhom computation below 1 58 0 15 113 0 1 38 96 0 q 1 0 34 45 0 1 0 0 10 10 0 1 0 0 0 0 0 0 p . r

We do not have the theory to guarantee collapsing of the spectral sequence E 1 at E . However, using torhom we compute the Z2 Betti numbers for the real points XΣ (R) [1 1 1 1 97 322 1]. r

1

We conclude that the spectral sequence E collapses at E . Again, we use Equation (10) to obtain the collapsing of the spectral sequence E r at E 2 . We can

Hodge spaces of real toric varieties

237

therefore compute the Z2 Betti numbers for XΣ (C) by adding along the diagonals [1 0 1 0 1 10 45 83 111 113 58 0 1].

References 1. G. Barthel, J. Brasselet, K. Fieseler, and L. Kaup, Combinatorial intersection cohomology for fans, Tohoku Math. J. (2) 54 (2002), 1–41. 2. V.V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebraic Geom. (3) (1994), 493–535. 3. V.V. Batyrev, On the classification of toric Fano 4-folds, J. Math. Sci. (New York) 94 (1999), 1021–1050. 4. V.V. Batyrev, Toric Fano threefolds, Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), 704–717. 5. F. Bihan, M. Franz, C. McCrory, and J. van Hamel, Is every toric variety an M-variety?, Manuscripta Math. 120 (2006), 217–232. 6. P. Bressler and V.A. Lunts, Intersection cohomology on nonrational polytopes, Compositio Math. 135 (2003), 245–278. 7. M. Brion, The structure of the polytope algebra, Tohoku Math. J. (2) 49 (1997), 1–32. 8. A. Degtyarev, I. Itenberg, and V. Kharlamov, Real Enriques Surfaces, Lecture Notes in Mathematics, 1746, Springer-Verlag, Berlin, 2000. 9. M. Franz, Maple package torhom, version 1.3.0, September 13, 2004, available at http : //www − fourier.ujf − grenoble.fr/ ∼ franz/maple/torhom.html. 10. W. Fulton, Introduction to Toric Varieties, Annals of Mathematics Studies, 131, Princeton University Press, Princeton, NJ, 1993. 11. W. Fulton and B. Sturmfels, Intersection theory on toric varieties, Topology 36 (1997), 335–353. 12. I.M. Gelfand, M.M. Kapranov, and A.Z. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants, Birkh¨auser, Boston, MA, 1994. 13. V. Hower, A counterexample to the maximality of toric varieties, Proc. Amer. Math. Soc., to appear. 14. V. Hower, Hodge spaces of real toric varieties, Ph.D. dissertation, University of Georgia, 2007. 15. S. Lang, Algebra, 3rd Edition, Addison-Wesley, Reading, MA, 1993. 16. S. Mac Lane, Homology, Springer-Verlag, Berlin-G¨ottingen-Heidelberg, 1963. 17. J. McCleary, A User’s Guide To Spectral Sequences, 2nd Edition, Cambridge Studies in Advanced Mathematics, 58, Cambridge University Press, Cambridge, 2001. 18. B. Nill. Classification of pseudo-symmetric simplicial symmetric polytopes, Algebraic and geometric combinatorics, 269–282, Contemp. Math. 423, Amer Math. Soc., Providence, RI, 2007. 19. T. Oda, Convex Bodies and Algebraic Geometry, Springer-Verlag, Berlin, 1988. 20. F. Sottile, Toric ideals, real toric varieties, and the moment map, Topics in algebraic geometry and geometric modeling, 225–240, Contemp. Math. 334, Amer. Math. Soc., Providence, RI, 2003.