Total curvatures of Kaehler manifolds in complex projective spaces (Q1077742)

Let \(f: M\to {\mathbb{C}}P^ n\) be a holomorphic isometric immersion of a compact Kähler manifold into \({\mathbb{C}}P^ n\). Then it is proved, that the Lipschitz-Killing curvature of f can be computed as the average number of critical points of ''height functions'' \(h\circ f: M\to {\mathbb{C}}P^ 1\). Here height functions are defined as orthogonal projections onto complex lines \({\mathbb{C}}P^ 1\subset {\mathbb{C}}P^ n\). It should be remarked that all such immersions are algebraic and hence the mentioned number of critical points is constant a.e. on the Grassmannian of lines \({\mathbb{C}}P^ 1\subset {\mathbb{C}}P^ n\).