### Abstract

A method of "algebraic estimates" is developed, and used to study the stability properties of integrals of the form f_{B}|f(z)|^{-δ}dV, under small deformations of the function f. The estimates are described in terms of a stratification of the space of functions {R(z) = |P(z)|^{ε}/|Q(z)|^{δ}} by algebraic varieties, on each of which the size of the integral of R(z) is given by an explicit algebraic expression. The method gives an independent proof of a result on stability of Tian in 2 dimensions, as well as a partial extension of this result to 3 dimensions. In arbitrary dimensions, combined with a key lemma of Siu, it establishes the continuity of the mapping c → f_{B} |f(z, c)|^{-δ}dV1 ⋯ dV_{n} when f(z, c) is a holomorphic function of (z, c). In particular the leading pole is semicontinuous in f, strengthening also an earlier result of Lichtin.

Original language | English (US) |
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Pages (from-to) | 277-329 |

Number of pages | 53 |

Journal | Annals of Mathematics |

Volume | 152 |

Issue number | 1 |

DOIs | |

State | Published - Jul 2000 |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty