Donaldson-Friedman construction and deformations of a triple of compact complex spaces (Q1976573)

The article deals with twistor spaces of connected sums \(n\mathbb P_2(\mathbb C)\). The main object is to show that special blowing-ups of \(\mathbb P_1\times \mathbb P_1\) appear as real fundamental divisors of those twistor spaces. A twistor space of an oriented four-dimensional Riemannian manifold \(M\) is a three-dimensional complex manifold \(Z\) together with a fixed point free anti-holomorphic involution \(\sigma:Z\to Z\) (the real structure) and a \(C^\infty\)-map \(\pi: Z\to M\) (the twistor fibration) which gives \(Z\) the structure of an \(S^2\)-bundle over \(M\) with the following properties: \(\pi\circ\sigma = \pi\), every fiber \(L_p:=\pi^{-1}(p)\) is a complex submanifold of \(Z\) and biholomorphic to \(\mathbb P_1\) (a twistor line), and the holomorphic normal bundle of \(L_p\) in \(Z\) is isomorphic to \({\mathcal O}(1)\oplus{\mathcal O}(1)\). A subset \(S \subset Z\) is called real, if \(\sigma(S)=S\). There exists a distinguished holomorphic line bundle on \(Z\) (the fundamental line bundle) which is denoted by \(K^{-\frac 12}\) since its square is the anticanonical line bundle \(K^{-1}\) of \(Z\). The elements of its fundamental system are called fundamental divisors. The twistor spaces of \(M\) are in one-to-one correspondence to the self-dual metrics on \(M\) [\textit{M. F. Atiyah, N. J. Hitchin} and \textit{I. M. Singer}, Proc. R. Soc. Lond., Ser. A 362, 425-461 (1978; Zbl 0389.53011)]. It is known [\textit{H. Pedersen} and \textit{Y. S. Poon}, Proc. Am. Math. Soc. 121, No. 3, 859-864 (1994; Zbl 0808.32028)] that a real irreducible fundamental divisor on a twistor space \(Z\) of \(M=n\mathbb P_{2}(\mathbb C)\) can be blown down to \(\mathbb P_{1}\times \mathbb P_{1}\) preserving the real structure, which is given on \(\mathbb P_{1}\times\mathbb P_{1}\) by \(\tau: \mathbb P_1\times \mathbb P_1\to \mathbb P_1\times \mathbb P_1\), \(((z_{0}:z_1),(w_0:w_1))\mapsto ((\overline{z}_1:\overline{z}_0),(\overline{w}_0:\overline{w}_1))\). The author solves the converse problem for special cases: Given a non-singular rational (resp. elliptic) curve \(C_0\subset \mathbb P_1\times\mathbb P_1\) of bidegree \((2,1)\) (resp. \((2,2))\) with \(\tau(C_0)=C_0\) there exists a set of \(n\) points \(p_i\) (may be infinitely near) such that the blowing-up \(S\) of \(\mathbb P_1\times \mathbb P_1\) in the \(2n\) points \(p_1,\tau(p_1),\dots,p_n,\tau(p_n)\) is a real fundamental divisor of a suitable twistor space of \(n\mathbb P_2(\mathbb C)\). The proof uses the methods for constructing self-dual structures on \(n\mathbb P_2(\mathbb C)\) by \textit{S. K. Donaldson} and \textit{R. Friedman} [Nonlinearity 2, 197-239 (1989; Zbl 0671.53029)].