[Paper Review] Betti numbers of random real hypersurfaces and determinants of random symmetric matrices

Damien Gayet, Jean-Yves Welschinger|arXiv (Cornell University)|Jul 6, 2012

Geometry and complex manifolds18 references50 citations

TL;DR

This paper establishes asymptotic upper bounds for the expected Betti numbers of random real hypersurfaces in smooth real projective manifolds, showing they grow as the square root of the degree. The bounds depend on the Kählerian volume of the real locus and the expected absolute determinant of random real symmetric matrices, with coefficients decaying exponentially away from mid-dimensional Betti numbers as dimension increases.

ABSTRACT

We asymptotically estimate from above the expected Betti numbers of random real hypersurfaces in smooth real projective manifolds. Our upper bounds grow as the square root of the degree of the hypersurfaces as the latter grows to infinity, with a coefficient involving the K\\"ahlerian volume of the real locus of the manifold as well as the expected determinant of random real symmetric matrices of given index. In particular, for large dimensions, these coefficients get exponentially small away from mid-dimensional Betti numbers. In order to get these results, we first establish the equidistribution of the critical points of a given Morse function restricted to the ran- dom real hypersurfaces.

Motivation & Objective

• To improve prior upper bounds on the expected total Betti number of random real hypersurfaces in real projective manifolds.
• To derive individual upper bounds for each Betti number, not just the total sum.
• To establish the equidistribution of critical points of Morse functions restricted to random real hypersurfaces.
• To connect the asymptotic behavior of Betti numbers to the expected absolute determinant of random real symmetric matrices.
• To show that coefficients in the bounds decay exponentially away from mid-dimensional Betti numbers in high dimensions.

Proposed method

• The authors use a Gaussian probability measure on spaces of real holomorphic sections of high tensor powers of an ample line bundle over a real projective manifold.
• They define a fake Betti number as the minimal number of critical points of index i for any Morse function on the real hypersurface.
• They analyze the empirical measure of critical points of index i and prove its weak convergence to a density proportional to the Kähler volume form.
• The key technical tool is the asymptotic equidistribution of critical points of Morse functions restricted to random real hypersurfaces.
• The expected Betti numbers are bounded using the expected absolute determinant of symmetric matrices of given signature, denoted $ e_{\mathbb{R}}(i,n-1-i) $.
• The analysis relies on random matrix theory and integration over symmetric matrix spaces with Gaussian measure.

Experimental results

Research questions

• RQ1 How do the individual expected Betti numbers of random real hypersurfaces grow with the degree of the hypersurface in a real projective manifold?
• RQ2 What is the role of the Kählerian volume of the real locus in determining the asymptotic growth of Betti numbers?
• RQ3 How does the expected absolute determinant of random real symmetric matrices of given signature influence the Betti number bounds?
• RQ4 To what extent do the coefficients in the upper bounds decay away from mid-dimensional Betti numbers as the manifold dimension increases?
• RQ5 Does the empirical distribution of critical points of Morse functions on random real hypersurfaces converge weakly to a smooth measure as the degree tends to infinity?

Key findings

• The expected Betti number $ E(b_i) $ grows at most as $ \frac{1}{\sqrt{\pi}} e_{\mathbb{R}}(i,n-1-i) \mathrm{Vol}_h(\mathbb{R}X) \sqrt{d}^n $ as $ d \to \infty $, with the coefficient $ e_{\mathbb{R}}(i,n-1-i) $ depending on the expected absolute determinant of symmetric matrices of signature $ (i,n-1-i) $.
• For $ n=1 $, the bound becomes an equality: $ E(b_0) \sim \frac{\mathrm{Length}_h(\mathbb{R}X)}{\sqrt{\pi}} \sqrt{d} $, recovering Kostlan and Shub-Smale's result for real polynomials.
• The coefficient $ e_{\mathbb{R}}(i,n-1-i) $ decays exponentially in high dimensions, especially away from mid-dimensional Betti numbers.
• The empirical measure of index-i critical points on random real hypersurfaces converges weakly to $ \frac{1}{\sqrt{\pi}} e_{\mathbb{R}}(i,n-1-i) \, d\mathrm{vol}_h $ on $ \mathbb{R}X $ as $ d \to \infty $.
• Explicit values of $ e_{\mathbb{R}}(p,q) $ are computed for $ p+q \leq 3 $, including $ e_{\mathbb{R}}(1,0) = \frac{1}{\sqrt{2\pi}} $, $ e_{\mathbb{R}}(2,0) = \frac{1}{4}(\sqrt{2}-1) $, and $ e_{\mathbb{R}}(1,1) = \frac{1}{\sqrt{2}} $.
• The results are independent of the choice of normalized volume form on $ X $, relying only on the Kähler metric from the curvature form $ \omega $.