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EXPECTED TOPOLOGY OF RANDOM REAL ALGEBRAIC SUBMANIFOLDS
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 14 / Issue 4 / October 2015
- Published online by Cambridge University Press:
- 12 May 2014, pp. 673-702
- Print publication:
- October 2015
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- Article
- Export citation
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Let
$X$ be a smooth complex projective manifold of dimension 
$n$ equipped with an ample line bundle 
$L$ and a rank 
$k$ holomorphic vector bundle 
$E$. We assume that 
$1\leqslant k\leqslant n$, that 
$X$, 
$E$ and 
$L$ are defined over the reals and denote by 
$\mathbb{R}X$ the real locus of 
$X$. Then, we estimate from above and below the expected Betti numbers of the vanishing loci in 
$\mathbb{R}X$ of holomorphic real sections of 
$E\otimes L^{d}$, where 
$d$ is a large enough integer. Moreover, given any closed connected codimension 
$k$ submanifold 
${\it\Sigma}$ of 
$\mathbb{R}^{n}$ with trivial normal bundle, we prove that a real section of 
$E\otimes L^{d}$ has a positive probability, independent of 
$d$, of containing around 
$\sqrt{d}^{n}$ connected components diffeomorphic to 
${\it\Sigma}$ in its vanishing locus.
