Adiabatic limits of anti-self-dual connections on collapsed \(K3\) surfaces (Q2039755)

The SYZ conjecture, formulated by \textit{A. Strominger} et al. [AMS/IP Stud. Adv. Math. 23, 333--347 (2001; Zbl 0998.81091)], has an important role in understanding the geometric realization of mirror symmetry and predicts that the mirror of a Calabi-Yau manifold can be constructed via dual special Lagrangian fibrations. An alternative version of this conjecture has been stated by \textit{M. Gross} and \textit{P. M. H. Wilson} [J. Differ. Geom. 55, No. 3, 475--546 (2000; Zbl 1027.32021); \textit{M. Kontsevich} and \textit{Y. Soibelman}, in: Symplectic geometry and mirror symmetry. Proceedings of the 4th KIAS annual international conference, Seoul, South Korea, August 14--18, 2000. Singapore: World Scientific. 203--263 (2001; Zbl 1072.14046)] making use of the collapsing of Ricci-flat Kähler metrics. In the context of homological mirror symmetry, \textit{K. Fukaya} [Proc. Symp. Pure Math. 73, 205--278 (2005; Zbl 1085.53080)] has conjectured a gauge theory in analogy to the collapsing of Ricci-flat Kähler metrics, which connects the adiabatic limits of anti-self-dual connections on Calabi-Yau manifolds to special Lagrangian cycles on the mirror Calabi-Yau manifolds. The main result of this paper proves a version of Fukaya's conjecture for anti-self-dual connections on \(K3\) surfaces with elliptic fibration. More precisely, let \(M\) be a projective \(K3\) surface with an elliptic fibration \(f \colon M \to N \cong \mathbb{P}^1_{\mathbb{C}}\) admitting a section. Let \(\alpha\) be an ample class on \(M\). Assume that the singularities of the fibers of \(f\) are only of Kodaira type \(I_1\) and type \(II\). Let \(P\) be a principal \(\text{SU}(n)\)-bundle over \(M\) and \((\mathcal{V}, H)\) be the smooth Hermitian vector bundle of rank \(n\) obtained by the twisted product. Let \(\Xi_{t_k}\) be a sequence of connections on \(P\) which are anti-self-dual with respect to a sequence of Ricci flat metrics collapsing the fibers of \(M\). The first result of the paper under review is Theorem 3.1, which states that if the holomorphic vector bundle \(\mathcal{V}\) with the holomorphic structure induced by \(\Xi_{t}\) satisfies certain non-degeneracy assumptions, then there exists a Zariski open subset \(N^o\) inside the locus of smooth fibers in \(N\), such that \(u_k(\Xi_{t_k})\) converges subsequentially to \(\Xi_0\) in the locally \(C^{0, \alpha}\)-sense on \(f^{-1}(N^o)\), for some sequence of unitary gauge transformations \(u_k\) on \(P\). In addition, the restriction of \(\Xi_0\) to any fiber is unitary gauge equivalent to a smooth flat \(\text{SU}(n)\)-connection. Under the same assumptions of Theorem 3.1, the authors show in Theorem 3.2 that \(\Xi_0\) is the Fourier-Mukai transform of a certain flat \(\text{U}(1)\)-connection on the multi-section of \(f\) induced by the sequence of spectral covers. In Section 3.1 is then explained how to deduce Fukaya's Conjecture from the above results. The paper is organized as follows. The preliminary notions, which are useful for understanding these results, are summarized in Section 2. Section 3 is devoted to state the main results and prove Fukaya's conjecture, while Section 4 concerns the proof of Theorem 3.1 assuming some estimates which are proved in the next Sections 5--8. Section 9 contains the proof of Theorem 3.2.