Eldan’s Stochastic Localization and Tubular Neighborhoods of Complex-Analytic Sets

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Abstract

Let f: Cn→ Ck be a holomorphic function and set Z= f- 1(0). Assume that Z is non-empty. We prove that for any r> 0 , γn(Z+r)≥γn(E+r),where Z+ r is the Euclidean r-neighborhood of Z; γn is the standard Gaussian measure in Cn, and E⊆ Cn is an (n- k) -dimensional, affine, complex subspace whose distance from the origin is the same as the distance of Z from the origin.

Original languageEnglish
Pages (from-to)2008-2027
Number of pages20
JournalJournal of Geometric Analysis
Volume28
Issue number3
DOIs
StatePublished - 1 Jul 2018

Keywords

  • Complex varieties
  • Gaussian measure
  • Stochastic processes
  • Tubular neighborhoods

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