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Deformation of singular Lagrangian subvarieties. (English) Zbl 1051.14006
The authors develop some ideas of a deformation theory of singular lagrangian subvarieties. Lagrangian submanifolds with singularities have been studied by various authors: Arnold, Givental and others [A. B. Givental’, J. Sov. Math. 52, No. 4, 3246–3278 (1990); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Novejshie Dostizh. 33, 55–112 (1988; Zbl 0900.58013)].
The authors study the behaviour of Lagrangian singularities under deformations by describing the spaces of infinitesimal deformations and obstructions of a Lagrangian subvariety. They focus their attention mainly in the sheaf of Lagrangian deformations of $$L$$.
In section 1, a complex $$({\mathcal C}^*, \delta)$$ is defined in terms of $$L$$ and it is shown in proposition 1 that $$({\mathcal C}, \delta, \wedge)$$ is a differential graded algebra. In proposition 2, the authors establish an isomorphism of $${\mathcal C}_L$$ and $$\Omega_L$$ on the regular locus, which can be extended to an isomorphism of $${\Omega_L^p}^{**}$$ with $${\mathcal C}_L^p$$. In section 3, theorem 1 they calculate the zeroth and first cohomology groups of the complex $${\mathcal C}_L^{*}$$. In theorem 2 of the same section, the authors define an obstruction theory of $$L$$ in terms of the second cohomology group of the complex $${\mathcal C}_L^{\bullet}$$. In section 4 they prove in theorem 3 that for Lagrangian spaces $$L$$ satisfying “Condition P” given in definition 2, the cohomology of $${\mathcal C}_L^{\bullet}$$ is finite dimensional. In section 5, the authors compute the sheaf of Lagrangian deformations of $$L$$ very explicitly for three families of examples: a determinantal variety called the swallowtail, for a special example of a conormal space associated to a plane curve and for special examples of completely integrable Hamiltonian systems.

##### MSC:
 14B12 Local deformation theory, Artin approximation, etc. 32S40 Monodromy; relations with differential equations and $$D$$-modules (complex-analytic aspects) 14B05 Singularities in algebraic geometry 32S30 Deformations of complex singularities; vanishing cycles
Macaulay2
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