Definition
A Kahler manifold \((M,J, g)\) is a complex manifold \((M, J)\) \(J^2 = -\id\) with a Riemannian metric \(g\) such that \(g(J\cdot -, J\cdot -) = g\) and \(\nabla^gJ = 0\). Call \(\omega = g(J\cdot - , -)\) the Kahler form and note that \([\omega] \in H^2(M, \mathbb R)\).