Universal Lagrangian bundles

Abstract

Abstract. The obstruction to construct a Lagrangian bundle over a fixed integral affine manifold was constructed by Dazord and Delzant in [DD87] and shown to be given by 'twisted' cup products in [Sep11]. This paper uses the topology of universal Lagrangian bundles, which classify Lagrangian bundles topologically (cf. [Sep10b]), to reinterpret this obstruction as the vanishing of a differential on the second page of a Leray-Serre spectral sequence. Using this interpretation, it is shown that the obstruction of Dazord and Delzant depends on an important cohomological invariant of the integral affine structure on the base space, called the radiance obstruction, which was introduced by Goldman and Hirsch in [GH84]. Some examples, related to non-degenerate singularities of completely integrable Hamiltonian systems, are discussed.

Real Lagrangians in Calabi–Yau threefolds

“…Given a Lagrangian fibration , there is a natural affine structure on , for which the discriminant locus agrees with the discriminant locus of [19, 20, 55]. Moreover, this affine structure is integral if and only if the symplectic form on represents an integral cohomology class by [52, Remark 5.10] — see also [32, Remark 1.2]. Conversely, fixing an affine manifold with singularities, describing the obstructions to constructing a torus fibration over it is a technically challenging problem, and is discussed further in Appendix B.…”

Poisson manifolds of strong compact type over 2-tori

“…Here by a strongly integral affine structure we mean an integral affine structure with integral translational part [Sep13,Remark 5.10].…”