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Foliations of multiprojective spaces and a conjecture of Bernstein and Lunts. (English) Zbl 1225.32033

The author uses foliations of multiprojective spaces defined by Hamiltonian functions on the underlying affine space to prove the three-dimensional case of a conjecture of Bernstein and Lunts, according to which the symbol of a generic fist-order differential operator gives rise to a hypersurface of the cotangent bundle which does not contain involutive conical subvarieties apart from the zero section and fibres of the bundle.

MSC:

32S65 Singularities of holomorphic vector fields and foliations
32C38 Sheaves of differential operators and their modules, \(D\)-modules
37F75 Dynamical aspects of holomorphic foliations and vector fields
16S32 Rings of differential operators (associative algebraic aspects)
37J30 Obstructions to integrability for finite-dimensional Hamiltonian and Lagrangian systems (nonintegrability criteria)
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