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Cubics, integrable systems, and Calabi-Yau threefolds. In: Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), Israel Math. Conf. Proc. : Supersymmetric Yang–Mills theory and integrable systems. Nucl. Phys. : Lectures on special Kähler geometry and electric-magnetic duality rotations. Nucl. Phys. Proc. Suppl. : Superspace formulation of new non-linear sigma models. J. Nucl. Phys. : Symplectic techniques in physics. : Monopoles, Minimal Surfaces and Algebraic Curves.

19. 7). 20. Locally, we may specify a Kähler geometry by giving a holomorphic function F(z1 , . . , zn ) such that Im ∂ 2F >0 ∂zi ∂zj is positive definite. The function √ −1 1 2 (z ) + · · · + (zn )2 F(z , . . , z ) = 2 leads to the flat metric on Cn . A nontrivial example in one dimension is provided by the holomorphic function τ3 , F(τ ) = 6 defined on the upper half plane n 1 H = {τ : Im τ > 0}. The corresponding Kähler form √ −1 Im(τ )dτ ∧ dτ ω= 2 has Gauss curvature 1/2(Im τ )3 . Note that the coordinate conjugate to τ is w = ∂F/∂τ = τ 2 /2.

A basic fact is that is a complex lagrangian submanifold of T ∗ M, where T ∗ M has the canonical holomorphic symplectic structure. (See [GS, Sect. ) Furthermore, intersects each fiber of T ∗ M in a full lattice. The quotient A = T ∗ M/ is a family of abelian varieties parametrized by M; it is the bundle of Albanese varieties of X → M. Since is complex lagrangian, the canonical holomorphic symplectic form on T ∗ M passes to a holomorphic symplectic form ηˆ on the quotient A. Now a local lagrangian section of π : X U → U over an open set U ⊂ M induces a local isomorphism X U ∼ = A U , and this isomorphism maps ηˆ to η.

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