The totally geodesic coisotropic submanifolds in Kähler manifolds (Q1597482)

A real submanifold \(\Sigma\) in a symplectic manifold \((M^{2n}, \omega)\) is said to be a coisotropic submanifold if \(T_p\Sigma^\omega \subset T_p\Sigma\) for all \(p\in\Sigma\), where \(T_p\Sigma^\omega= \{X\in T_pM\mid \omega (X,Y)=0\), \(\forall Y\subset T_p\Sigma\}\) is the symplectic complement of \(T_p \Sigma\). Let \((M^{2n}, g,J)\) be a complete connected Kähler manifold of nonnegative holomorphic curvature. Assume that \(W\) and \(V\) are closed, totally geodesic immersed coisotropic submanifolds in \(M\) and assume that either \(W\) or \(V\) is compact. It is shown that if the holomorphic curvature of \(M\) is positive on either \(W\) or \(V\), then \(W\) and \(V\) must intersect. Assume that here \(V\) is compact and that the holomorphic curvature \(H(X)=\langle R(X,JX)X,JX \rangle\) of \(M\) satisfies \(\liminf_{t\to \infty}\int^t_0 H(\gamma'(t)) dt>0\), where \(\gamma(t): [0,\infty)\to M\) is any normalized geodesic issuing orthogonally from \(V\). It is shown that then \(M\) must be compact. Moreover, assume that the holomorphic curvature of \(M\) is positive on \(V\). Then the induced homomorphism of the fundamental groups \(\pi_1(V) \to\pi_1(M)\) is surjective. As an application, a fixed-point theorem is given. If \((M^{2n},g,J)\) has positive holomorphic curvature and \(f\) is a symplectomorphism of \(M\), whose graph is a totally geodesic submanifold of \((M\times M,g\times g)\), then \(f\) has a fixed point.