un 1 99 8 1 SPECIAL LANGRANGIAN GEOMETRY AND SLIGHTLY DEFORMED ALGEBRAIC GEOMETRY ( SPLAG AND SDAG )

The special geometry of calibrated cycles, closely related to mirror symmetry among Calabi–Yau 3-folds, is itself a real form of a new subject, which we call slightly deformed algebraic geometry. On the other hand, both of these geometries are parallel to classical gauge theories and their complexifications. This article explains this parallelism, so that the appearance of invariants of new type in complexified gauge theory (see [D-T] and [T]) can be accompanied by analogous invariants in the theory of special Lagrangian cycles, for which the development is at present much more modest than in gauge theory. We discuss related geometric constructions, arising from mirror symmetry and symplectic geometry. §1. spLag cycles We begin by recalling the geometric construction for a pair L ⊂ S, where S is a smooth symplectic manifold of dimension 2n with a given tame almost complex structure I, and L ⊂ S a smooth, oriented Lagrangian submanifold (of maximal dimension dimL = n = 1 2 dimS); this is now quite popular in the set-up of Calabi– Yau threefolds. The structure on S is an almost Kähler structure, and we say for short that S is an aK manifold. Write ω for the symplectic form and I for the almost complex structure on S, so that the tangent space TSp at a point p is C with the constant symplectic form 〈 , 〉 = ωp and the constant Euclidean metric gp, giving the Hermitian triple (ωp, Ip, gp). We now define the Lagrangian Grassmannian Λ↑p = Λ↑(TSp) to be the Grassmannian of maximal oriented Lagrangian subspaces in TSp. Taking this space over every point of S gives the oriented Lagrangian Grassmannisation of TS π: Λ↑(S) → S with π(p) = Λ↑p. (1.1) Our tame almost complex structure on S gives each fibre the standard form Λ↑p = U(n)/ SO(n) (1.2) This space admits a canonical map det: Λ↑p→ U(1) = S p sending u ∈ U(n) to det u ∈ U(1) = S. (1.3) Typeset by AMS-TEX