Toric Legendrian subvarieties (Q2468657)

The author studies the Legendrian subvarieties \(X\) of a projective space \(\mathbb P(\mathbb C ^{2n}),\) where \(\mathbb C ^{2n}\) is endowed with a symplectic form \(\omega \) and for each smooth point of the affine cone \(\hat X\) the tangent space to \(\hat X\subset \mathbb C ^{2n}\) at this point is Lagrangian. The full classification of smooth toric Legendrian subvarieties in \(\mathbb P ^{2n-1}\) is given. It is proved that if \(X\subset \mathbb P ^{2n-1}\) is a smooth toric Legendrian subvariety and \(n\geq 4,\) then it is either a linear subspace or \(n=4\) and then \(X=\mathbb P ^1\times\mathbb P ^1\times\mathbb P ^1\). And for \(n=3\) it is either \(\mathbb P ^1\times Q_1\) or \(\mathbb P ^2\) blown up in three non-collinear points or plane \(\mathbb P ^2\subset \mathbb P ^5.\) It is also proved under some minor assumptions that the group of linear automorphisms preserving a given Legendrian subvariety preserves the contact structure of the ambient projective space.