## Journal of Differential Geometry

- J. Differential Geom.
- Volume 72, Number 3 (2006), 381-427.

### Critical points and supersymmetric vacua, II: Asymptotics and extremal metrics

Michael R. Douglas, Bernard Shiffman, and Steve Zelditch

#### Abstract

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 ∈ H^{0}(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 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^{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 coefficients
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 β_{2}(m)
(depending only on the dimension m of M) times the Calabi functional ∫_{M}ρ^{2}dVol_{h},
where ρ is the scalar curvature of the Kähler metric ω_{h} := (i/2)Θ_{h}.
We give an integral formula for β_{2}(m) and show, by a computer assisted calculation, that
β_{2}(m) > 0 for m ≤ 5, hence that N^{crit}_{N,h} is asymptotically minimized
by the Calabi extremal metric (when one exists). We conjecture that β_{2}(m) > 0
in all dimensions, i.e., the Calabi extremal metric is always the asymptotic minimizer.

#### Article information

**Source**

J. Differential Geom., Volume 72, Number 3 (2006), 381-427.

**Dates**

First available in Project Euclid: 28 March 2006

**Permanent link to this document**

https://projecteuclid.org/euclid.jdg/1143593745

**Digital Object Identifier**

doi:10.4310/jdg/1143593745

**Mathematical Reviews number (MathSciNet)**

MR2219939

**Zentralblatt MATH identifier**

1236.32013

#### Citation

Douglas, Michael R.; Shiffman, Bernard; Zelditch, Steve. Critical points and supersymmetric vacua, II: Asymptotics and extremal metrics. J. Differential Geom. 72 (2006), no. 3, 381--427. doi:10.4310/jdg/1143593745. https://projecteuclid.org/euclid.jdg/1143593745