Convergence of the mirror to a rational elliptic surface

Abstract

The construction introduced by Gross, Hacking and Keel in (Several Complex Variables (Springer, New York, NY, 1976))allows one to construct a formal mirror family to a pair (S,D) where S is a smooth rational projective surface and D a certain type of Weil divisor supporting an ample or anti‐ample class. In that paper, they proved two convergence results when the intersection matrix of D is not negative semi‐definite and when the matrix is negative definite. In the original version of that paper, they claimed that if the intersection matrix were negative semi‐definite, then family extends over an analytic neighbourhood of the origin but gave an incorrect proof. In this paper, we correct this error. We reduce the construction of the mirror to such a surface to calculating certain log Gromov–Witten invariants. We then relate these invariants to the invariants of a new space where we can find explicit formulae for the invariants. From this we deduce analytic convergence of the mirror family, at least when the original surface has an I4 fibre.