On the Topology of Random Real Complete Intersections

Michele Ancona*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Given a smooth real projective variety X and m ample line bundles L1, ⋯ Lm on X also defined over R, we study the topology of the real locus of the complete intersections defined by global sections of L1⊗d⊕⋯⊕Lm⊗d. We prove that the Gaussian measure of the space of sections defining real complete intersections with high total Betti number (for example, maximal complete intersections) is exponentially small, as d grows to infinity. This is deduced by proving that, with very high probability, the real locus of a complete intersection defined by a section of L1⊗d⊕⋯⊕Lm⊗d is isotopic to the real locus of a complete intersection of smaller degree.

Original languageEnglish
Article number32
JournalJournal of Geometric Analysis
Issue number1
StatePublished - Jan 2023
Externally publishedYes


  • Bergman kernels
  • Random subvarieties
  • Topology of real algebraic varieties


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