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 of critical points of a Gaussian random holomorphic section s ∈ H⁰(M, L) with respect to the Chern connection ∇_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 inter- ested 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^N, H^N) → M of the line bundle and their critical point densities K^crit_{N, h}(z). We prove that K^crit_{N, h}(z) has a complete asymptotic expansion in N whose co- efficients are curvature invariants of h. The first two terms in the expansion of N^crit_{N, h} are topological invariants of (L, M). The third term is a topological invariant plus a constant β₂(m) (depend- ing only on the dimension m of M) times the Calabi functional ∫_Mρ²dVol_h, where ρ is the scalar curvature of the Kähler metric ω_h :=i/2Θ_h. We give an integral formula for β₂(m) and show, by a computer assisted calculation, that β₂(m) > 0 for m ≤ 5, hence that N^crit_{N, h} is asymptotically minimized by the Calabi ex- tremal metric (when one exists). We conjecture that β₂(m) > 0 in all dimensions, i.e., the Calabi extremal metric is always the asymptotic minimizer.