## 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 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 β_{2}(m) (depend- ing 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/2Θ_{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 ex- tremal 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.

Original language | English (US) |
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Pages (from-to) | 381-427 |

Number of pages | 47 |

Journal | Journal of Differential Geometry |

Volume | 72 |

Issue number | 3 |

DOIs | |

State | Published - 2006 |

## ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory
- Geometry and Topology