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By Penrose R.

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1. Consider the torus T 2 = R2 /Z2 . H 0 (T 2 ) = R, since closed zero-forms are constant functions, and there are as many of them as there are connected components of the manifold. , x and x + 1 represent the same point). One can show that any other closed one-forms are either exact or differ from adx + bdy (a, b constants) by an exact form, so H 1 (T 2 ) = R2 . Likewise, dx ∧ dy generates H 2 . There are other representatives of H 1 (T 2 ). For example, consider δ(x)dx, where δ(x) is a delta function.

Here we give four examples of toric varieties, along with the diagrams (fans) that encode their combinatorial data (see Fig. 1). However, we will not give a general account of going from the diagram to the construction of the variety. The reader can find a much more thorough treatment in Ch. 7. A) The three vectors vi in the toric fan (A) are not linearly independent. They satisfy the relation 1 · v1 + 1 · v2 + 1 · v3 = 0. The coefficients (1, 1, 1) in this relation encode the scaling action under λ ∈ C∗ : zi → λ1 zi .

Here there are two relations: v1 + v2 = 0 and v3 + v4 = 0. There are therefore two C∗ actions encoded by the vectors (1, 1, 0, 0) and (0, 0, 1, 1). Namely, (λ1 , λ2 ) ∈ (C∗ )2 maps (z1 , z2 , z3 , z4 ) → (λ11 z1 , λ11 z2 , λ12 z3 , λ12 z4 ). The set U is the union of two sets: U = {z1 = z2 = 0} ∪ {z3 = z4 = 0}. Then (C4 \ U )/(C∗ )2 = P1 × P1 . ) The southwest vector here is v2 = (−1, −n), all others the same as in (C), which is the special case n = 0. The construction of the toric variety proceeds much as in (C), except the first relation is now 1·v1 +1·v2 +n·v3 = 0, so the first C∗ acts by (1, 1, n, 0).

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