TY - JOUR

T1 - On the Topology of Random Real Complete Intersections

AU - Ancona, Michele

N1 - Publisher Copyright:
© 2022, Mathematica Josephina, Inc.

PY - 2023/1

Y1 - 2023/1

N2 - 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.

AB - 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.

KW - Bergman kernels

KW - Random subvarieties

KW - Topology of real algebraic varieties

UR - http://www.scopus.com/inward/record.url?scp=85142163338&partnerID=8YFLogxK

U2 - 10.1007/s12220-022-01092-x

DO - 10.1007/s12220-022-01092-x

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AN - SCOPUS:85142163338

SN - 1050-6926

VL - 33

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

IS - 1

M1 - 32

ER -