The blowup formula for Donaldson invariants
In this beautiful paper the authors present a blowup formula for the Donaldson invariants of a (simply-connected) closed 4-manifold \(X\), with \(b^+\) odd and greater than one, and those of its blowup \(\widehat X=X \# \overline {CP^2}\). The formula is independent of \(X\) and states that there are polynomials \(B_k(x)\) such that \(\widehat D(e^kz) =D(B_k(x)z)\) for all \(z\in A(X) =\text{Sym}_* (H_0(X) \oplus H_2(X))\). Here \(e \in H_2 (X)\) denotes the homology class of the exceptional divisor, \(x\in H_0(X)\) is the generator [1] corresponding to the orientation, \(D=D_X\) and \(\widehat D=D_{\widehat X}\) are the SU(2) Donaldson invariants corresponding to \(X\) and \(\widehat X\) (they take the (possible) nonzero values in the same degrees mod 4 as \(b^+_X =b^+_{\widehat X})\). Given the formal power series \[ B(x,t)= \sum_{k= 0}^\infty B_k(x) {t^k\over k!}, \] the authors prove that \(B(x,t)= \exp(-t^2x/6)\) \(\sigma_3 (x,t)\), where, as a function of \(t\), \(\sigma_3\) is a particular quasi-periodic Weierstrass sigma-function associated to the \(\wp\)-function which satisfies the differential equation \((y')^2= 4y^3- g_2y- g_3\), where \(g_2= 4(x^2/3-1)\) and \(g_3= (8x^3-36x)/27\). A similar result is also obtained for the Donaldson invariants associated to SO(3) bundles over the above manifolds.
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