Some results on equivariant contact geometry for partial flag varieties (Q2816976)

For a connected, simply-connected complex simple group \(G\) of type ADE with its Lie algebra \(\mathfrak{g}\), if \(N\) is the closed subvariety of \(\mathfrak{g}\) consisting of the nilpotent elements, then an adjoint \(G\)-orbit in \(\mathfrak{g}\) is said to be nilpotent if it is contained in \(N\). The set of nilpotent orbits has a unique minimal non-zero element \({\mathcal O}_{\min}\). A natural \(\mathbb C ^{*}\)-action on nilpotent orbits produces a smooth, closed subvariety \(\mathbb P({\mathcal O}_{\min})={\mathcal O}_{\min}/\mathbb C ^{*}\) of \(\mathbb P({\mathfrak g})={\mathfrak g}/\mathbb C ^{*}\). The projective variety \(\mathbb P({\mathcal O}_{\min})\) is \(G\)-equivariantly diffeomorphic to \(G/P\), for some parabolic subgroup \(P\) of \(G\). In [Invent. Math. 142, No. 1, 1--15 (2000; Zbl 0994.53024)], \textit{S.~Kebekus} et al. studied the complex projective manifolds carrying a complex contact structure to support the LeBrun-Salamon conjecture stating that there does not exist a non-symmetric quaternion-Kähler manifold with positive scalar curvature in higher dimensions. They proved that if \(X\) is a \((2n+1)\)-dimensional projective contact manifold with a line bundle \(L\) and the canonical bundle \(K_X\) is not nef, i.e., that there is a curve \(C\subset X\) such that \(K_X.C<0\), then either \(X\) is Fano with the second Betti number equal 1, or there exists a smooth projective manifold \(Y\) such that \((X,L)\) is isomorphic to a pair of projective spaces. NEWLINENEWLINENEWLINENEWLINE In this paper, the authors study equivariant contact structures on complex projective varieties arising as partial flag varieties \(G/P\), where \(G\) is a connected, simply-connected complex simple group of type ADE and \(P\) is a parabolic subgroup, and they consider a special case of the LeBrun-Salamon conjecture for partial flag varieties of these types. They prove that if \(G\) is a connected, simply-connected complex simple group of type ADE, and \(X\) is a partial flag variety of \(G\) with \(b_2(X)=1\) that is endowed with a \(G\)-invariant complex contact structure, then there exists a \(G\)-equivariant isomorphism \(X\cong \mathbb P({\mathcal O}_{\min})\) of contact varieties. Additionally, they present a canonical, global description of the unique \(\mathrm{SO}_{2n}(\mathbb{C})\)-invariant contact structure on the isotropic Grassmannian of 2-planes in \(\mathbb{C}^{2n}\).