In this paper, we derive new adjunction inequalities for embedded surfaces with non-negative self-intersection number in four-manifolds. These formulas are proved by using relations between Seiberg-Witten invariants which are induced from embedded surfaces. To prove these relations, we develop the relevant parts of a Floer theory for four-manifolds which bound circle-bundles over Riemann surfaces.

We show that under suitable conditions a branched cover satisfies the same upper curvature bounds as its base space. First we do this when the base space is a metric space satisfying Alexandrov's curvature condition CAT(κ) and the branch locus is complete and convex. Then we treat branched covers of a Riemannian manifold over suitable mutually orthogonal submanifolds. In neither setting do we require that the branching be locally finite. We apply our results to hyperplane complements in several Hermitian symmetric spaces of nonpositive sectional curvature in order to prove that two moduli spaces arising in algebraic geometry are aspherical. These are the moduli spaces of the smooth cubic surfaces in ℂP³ and of the smooth complex Enriques surfaces.

We identify a distinguished hyperbolic partial differential equation of importance in the study of principal foliations. We apply Riemann's method to obtain an obstruction to the occurrence of elliptic sectors in principal foliations on surfaces and as a consequence we obtain the first global result on the Carathéodory conjecture.

Motivated by the picture of mirror symmetry suggested by Strominger, Yau and Zaslow, we make a conjecture concerning the Gromov-Hausdorff limits of Calabi-Yau n-folds (with Ricci-flat Kähler metric) as one approaches a large complex structure limit point in moduli; a similar conjecture was made independently by Kontsevich, Soibelman and Todorov. Roughly stated, the conjecture says that, if the metrics are normalized to have constant diameter, then this limit is the base of the conjectural special lagrangian torus fibrations associated with the large complex structure limit, namely an n-sphere, and that the metric on this Sⁿ is induced from a standard (singular) Riemannian metric on the base, the singularities of the metric corresponding to the limit discriminant locus of the fibrations. This conjecture is trivially true for elliptic curves; in this paper we prove it in the case of K3 surfaces. Using the standard description of mirror symmetry for K3 surfaces and the hyperkähler rotation trick, we reduce the problem to that of studying Kähler degenerations of elliptic K3 surfaces, with the Kähler class approaching the wall of the Kähler cone corresponding to the fibration and the volume normalized to be one. Here we are able to write down a remarkably accurate approximation to the Ricci-flat metric: if the elliptic fibres are of area ∊ > 0, then the error is O(e^−C/∊) for some constant C > 0. This metric is obtained by gluing together a semi-flat metric on the smooth part of the fibration with suitable Ooguri-Vafa metrics near the singular fibres. For small ∊, this is a sufficiently good approximation that the above conjecture is then an easy consequence.

We study the special algebraic properties of alternating 3-forms in 6 dimensions and introduce a diffeomorphism-invariant functional on the space of differential 3-forms on a closed 6-manifold M. Restricting the functional to a de Rham cohomology class in H³(M, R), we find that a critical point which is generic in a suitable sense defines a complex threefold with trivial canonical bundle. This approach gives a direct method of showing that an open set in H³(M, R) is a local moduli space for this structure and introduces in a natural way the special pseudo-Kähler structure on it.