J ul 2 00 4 CRITICAL POINTS AND SUPERSYMMETRIC VACUA , II : ASYMPTOTICS AND EXTREMAL METRICS

Motivated by the vacuum selection problem of string/M theory, we study a new geometric invariant of a positive Hermitian line bundle (L, h) → M over a compact Kähler manifold: the expected distribution Kcrit h (z) of critical points d log |s(z)|h = 0 of a Gaussian random holomorphic section s ∈ H(M,L) with respect to h. It is a measure on M whose total mass is the average number N crit h of critical points of a random holomorphic section. We are interested in the metric dependence of N crit h , especially metrics h which minimize N crit h . We concentrate on the asymptotic minimization problem for the sequence of tensor powers (L , h ) → M of the line bundle and their critical point densities Kcrit h (z). We prove that Kcrit h (z) has a complete asymptotic expansion in N whose coefficients are curvature invariants of h. The first two terms in the expansion of N crit h are topological invariants of (L,M). The third term is a topological invariant plus a constant β 2 (depending only on the dimension m of M) times the Calabi functional ∫ M ρ2dVolh, where ρ is the scalar curvature of the curvature form of h. We give an integral formula for β 2 and show, by a computer assisted calculation, that β 2 > 0 for m ≤ 4, hence that N crit h is asymptotically minimized by the Calabi extremal metric (when one exists). We conjecture that β 2 > 0 in all dimensions, i.e. the Calabi extremal metric is always the asymptotic minimizer.