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 Ncrith
of critical points of a random holomorphic section. We are interested in the metric dependence of
Ncrith, especially metrics h which minimize Ncrith.
We concentrate on the asymptotic minimization problem for the sequence of tensor powers (LN, hN) → M
of the line bundle and their critical point densities KcritN,h(z). We prove
that KcritN,h(z) has a complete asymptotic expansion in N whose coefficients
are curvature invariants of h. The first two terms in the expansion of NcritN,h
are topological invariants of (L,M). The third term is a topological invariant plus a constant β2(m)
(depending only on the dimension m of M) times the Calabi functional ∫Mρ2dVolh,
where ρ is the scalar curvature of the Kähler metric ωh := (i/2)Θh.
We give an integral formula for β2(m) and show, by a computer assisted calculation, that
β2(m) > 0 for m ≤ 5, hence that NcritN,h is asymptotically minimized
by the Calabi extremal metric (when one exists). We conjecture that β2(m) > 0
in all dimensions, i.e., the Calabi extremal metric is always the asymptotic minimizer.