Existence of anti-self-dual connections with small instanton number on the smooth blow-up

Let $X$ satisfy the hypotheses of the Bogomolov–Miyaoka–Yau conjecture for four-manifolds with a Seiberg–Witten basic class, and let

$$\widetilde X=X\#\overline{\mathbb{C}\mathbb{P}}^2$$

be its smooth blow-up. Let $(E,H)$ be a complex rank-two Hermitian vector bundle over $\widetilde X$ satisfying the fundamental bounds

$$0>p_1(\mathfrak{su}(E))[\widetilde X]\geq-c_2(X).$$

Let $\tilde g$ be a Riemannian metric on $\widetilde X$ that is generic in the sense of Freed and Uhlenbeck. Blow-up anti-self-dual connection conjecture. There exists a smooth, projectively $\tilde g$-anti-self-dual unitary connection $A$ on $E$ such that

$$(F_A^+)_0=0\in\Omega^+(\widetilde X;\mathfrak{su}(E)),$$

where $F_A\in\Omega^2(\mathfrak{u}(E))$. This is presented as a reformulation sufficient, via the expected dimension of the moduli space, to imply the Bogomolov–Miyaoka–Yau inequality; its resolution is not specified in the supplied text.