Hodge Spaces of Real Toric Varieties by Valerie Hower ( Under the direction of

We introduce the notion of a cosheaf on a fan Σ and define the Z2 Hodge spaces Hpq(Σ), which are the homology groups Hp(∧E ) of the qth exterior power of the cosheaf E on Σ. Geometrically, for σ ∈ Σ the stalk Eσ of the cosheaf E is the compact real torus in the real orbit Oσ(R) of the real toric variety XΣ(R). The Z2 Hodge spaces Hpq(Σ) are indexed by pairs p, q with 0 ≤ q ≤ p ≤ d, where d = dimΣ. 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. 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 k in the face fan Σ of ∆ are smooth then we compute Hpq(Σ) for p ≤ k − 2. Moreover, if Σ is a smooth fan then we completely determine the spaces Hpq(Σ). The Z2 Hodge spaces of Σ are related to the topology of both the real and complex points of the toric variety XΣ in the following way: Hpq(Σ) = E 1 p,q = E 2 p,q, where (E r , d r ) is a spectral sequence with E 1 p,q +3 Hp(XΣ(R),Z2) and (E, d) is a spectral sequence with E p,q +3 Hp+q(XΣ(C),Z2) . When Σ is a smooth fan, we show the spectral sequence (E r , d r ) for XΣ collapses at E 1 and hence XΣ is maximal, meaning that ∑