Algebraic Geometry by Raynaud M. (Ed), Shioda T. (Ed)

By Raynaud M. (Ed), Shioda T. (Ed)

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C ((m + 1) factors) on Cm+1 and depends on the choice of a base point of U. If the latter is chosen to be [1] = (1 : 1 : . . : 1), τ = ∑m+1 j=1 τ j [e j ] is determined by [z] = (eτ1 ([z]) : . . : eτm+a ([z]) ) mod Tm (16) a; equivalently, τ ([z]) = m+1 ∑ log |z j | [e j ], (17) j=1 for any [z] = (z1 : . . : zm+1 ) in U; in view of (11), we infer τ= 1 m+1 ∑ log 2 j=1 j − log 0 [e j ] (18) 0 as a tm a -valued function on ∆a ; we thus have τ, ξ = 1 2 m+1 ∑ (log j − log 0 )( j − λ j ). (19) j=1 By (14) and (13), the symplectic potential and the induced metric on ∆0a have then the following expressions: G= 1 2 m+1 ∑ j log j− 0 log 0 , (20) j=1 2 In general, the K¨ahler potential F, the symplectic potential G, the induced metric g∆a , and the dual momentum map τ , all regarded as defined on ∆0a via the momentum map µa , are related to each other by (14) and by: dF = d τ , ξ , cf.

13. X. Dai. Adiabatic limits, nonmultiplicativity of signature, and Leray spectral sequence. J. Am. Math. , 4:265–321, 1991. 14. M. Goresky and R. MacPherson. Intersection homology theory. Topology, 19:135–162, 1980. 15. M. Goresky and R. MacPherson. Intersection homology II. Invent. , 71:77–129, 1983. 16. T. Hausal, E. Hunsicker, and R. Mazzeo. The hodge cohomology of gravitational instantons. Duke Math. J. 122, no. 3:485–548, 2004. 17. N. Hitchin. L2 -cohomology of hyperk¨ahler quotients. Comm.

Without loss of generality, we can then assume that M = P(1 ⊕ L), where 1 stands for the trivial line bundle P1 × C, and L denotes a holomorphic line bundle over P1 of nonpositive degree −k, for some nonnegative integer k. If k = 0, L is the trivial line bundle and M is then the product M = P1 × P1 . If k > 0, L = Λk , where Λ = O(−1) denotes the tautological line bundle over P1 : for any y in P1 , the fiber Λy is y itself, viewed as a complex line3 in C2 . The resulting ruled surface P(1 ⊕ L) is then the k-th Hirzebruch surface, here denoted by Fk or simply M.

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