An explicit calculation of Weil-Petersson forms

Let M be the moduli space of holomorphic r -bundles E with c 1 ( E ) = 0 and c 2 ( E ) = 1 on the blowup of P 2 = C 2 ∪ ℓ ∞ at 0 ∈ C 2 that are framed along the line ℓ ∞ at infinity. With respect to the standard Kähler metric on P ˜ 2 ﹨ ℓ ∞ , an explicit formula for a potential for the Weil-Petersson Kähler form for M is computed in this elementary case. A 1-parameter family of such potentials corresponding to a family of Kähler metrics degenerating to the pull-back of the flat metric on P 2 ﹨ ℓ ∞ is also constructed, and this is shown to converge to the standard form for such bundles on P 2 .