Semiflat

Semiflat, also spelled semi-flat, is a term used in differential and complex geometry to describe a class of metrics on certain complex manifolds that are flat along the fibers of a torus fibration. The construction arises in the study of algebraic completely integrable systems and plays a role in the Strominger–Yau–Zaslow (SYZ) program for mirror symmetry as well as in the geometry of Hitchin moduli spaces.

Concretely, one often considers a holomorphic torus fibration π: M → B where the base B carries a special Kähler metric. A semiflat metric on M is a Riemannian metric that is flat in the torus fiber directions and whose variation along the base is dictated by the special Kähler data. In many examples this metric is compatible with a hyperkähler structure, providing a hyperkähler metric on the smooth locus of M when the fibration comes from an integrable system.

Significance: the semiflat metric serves as a tractable model for the geometry of M in the large complex structure limit and encodes the interaction between flat fiber geometry and the base’s complex structure, in line with predictions from SYZ mirror symmetry. It also acts as a starting point for incorporating corrections near singular fibers, often referred to as instanton corrections.

Limitations: the semiflat metric is generally defined only away from singular fibers and is not complete on M. Near singular fibers, the full metric requires additional corrections to capture the complete hyperkähler geometry.

Related topics include special Kähler geometry, holomorphic torus fibrations, integrable systems, and Hitchin moduli spaces.