This paper treats the asymptotic problem of the variance of the volume of random real algebraic submanifolds. Let \(\mathcal{X}\) be a smooth complex projective manifold of dimension \(n\). Let \(\mathcal{L}\) be an ample holomorphic line bundle over \(\mathcal{X}\), and let \(\mathcal{E}\) be a rank-\(r\) holomorphic vector bundle over \(\mathcal{X}\) with \(r \in \{ 1, \dots, n \}\). Assume that the real locus \(M\) of \(\mathcal{X}\) is not empty. For any \(d \in \mathbb{N}\), \(\mathbb{R} H^0( \mathcal{X}, \mathcal{E} \otimes \mathcal{L}^d)\) denotes the space of real holomorphic sections of \(\mathcal{E} \otimes \mathcal{L}^d \to \mathcal{X}\), and \(\vert d V_s \vert\) denotes the Riemannian measure, considered as a positive Radon measure on \(M\). When \(s_d\) is a standard Gaussian vector in \(\mathbb{R} H^0( \mathcal{X}, \mathcal{E} \otimes \mathcal{L}^d )\), then \(\vert d V_{s_d} \vert\) is a random positive Radon measure on \(M\). The author studies the vanishing locus \(Z_{s_d}\) in \(M\) of a random real holomorphic section \(s_d\) of \(\mathcal{E} \otimes \mathcal{L}^d\).
For a test function \(\phi \in C^0(M)\) and every \(s \in \mathbb{R}H^0( \mathcal{X}, \mathcal{E} \otimes \mathcal{L}^d )\) vanishing transversally, the real random variable \[ \langle \vert d V_s \vert, \phi \rangle = \int_{x \in Z_s} \, \phi(x) \vert d V_s \vert \tag{1} \] is called linear statistics of degree \(d\) associated with \(\phi\). On the other hand, for \(\phi \in C^0(M)\), its continuity modulus \(\bar{\omega}_{\phi}\) is defined by \[ \bar{\omega}_{\phi} (\varepsilon) := \sup \{ \, \vert \phi(x) - \phi(y) \vert ; \,\, (x,y) \in M^2, \,\, \rho_g(x,y) \leqslant \varepsilon \, \} \tag{2} \] where \(\rho_g(\cdot, \cdot)\) stands for the geodesic distance on \((M, g)\) with a Riemannian metric \(g\). The first main result is as follows:
Theorem A. For \(\beta \in (0, 1/2)\), there exists \(C_{\beta} > 0\) such that, for all \(\alpha \in (0, \alpha_0)\), for all \(\phi \in C^0(M)\), the variance of the linear statistics admits the following asymptotics as \(d \to + \infty\): \[ \begin{split} Var(\langle \vert d V_{s_d} \vert , \phi \rangle) = d^{r - n/2} \left( \int_M \, \phi^2 \vert d V_M \vert \right) \frac{Vol(\mathbb{S}^{n-1})} {( 2 \pi )^r} \mathcal{I}_{n,r} \\ + \Vert \phi \Vert_{\infty}^2 \cdot O( d^{r - n/2- \alpha} ) + \Vert \phi \Vert_{\infty} \bar{\omega}_{\phi}(C_{\beta} d^{- \beta}) O(d^{r - n/2}) \end{split}\tag{3} \] where \[ \begin{split} \mathcal{I}_{n,r} = \frac{1}{2} \int_0^{+ \infty} \left\{ \frac{\mathbb{E} [ \vert det^{\perp}( X(t)) \vert \cdot \vert det^{\perp} (Y(t)) \vert ]}{( 1-e^{-t} )^{r/2}} \right. \\ - (2 \pi)^r \left. \left( \frac{Vol( \mathbb{S}^{n-r} )}{Vol ( \mathbb{S}^n )} \right)^2 \right\} t^{(n-2)/2} \, dt < + \infty. \end{split} \tag{4} \] This implies that, when \(r \in \{ 1, \dots, n-1 \}\), then we obtain an asymptotic of order \(d^{r - n/2}\), as \(d\) goes to infinity, for the variance of the linear statistics associated with \(Z_{s_d}\), including its volume. As a corollary, the following statement can be derived.
Corollary B. Let \(U \subset M\) be an open subset, then as \(d \to + \infty\) \[ \mathbb{P} (Z_{s_d} \cap U = \emptyset) = O(d^{- n/2}) \tag{5} \] holds.
This means that, given an open set \(U \subset M\), it can be shown that the probability that \(Z_{s_ d}\) does not intersect \(U\) is an order of \(d^{- n/2}\) when \(d\) goes to infinity. As a result of almost sure convergent, the author proves:
Theorem C. Assume \(n \geq 3\). Let \((s_d)_{d \in \mathbb{N}} \in \prod_{d \in \mathbb{N}} \mathbb{R} H^0(\mathcal{X}, \mathcal{E} \otimes \mathcal{L}^d)\) be a random sequence of sections. Then, \(d \nu\)-almost surely, the convergence \[ d^{- r/2} \langle \vert d V_{s_d} \vert , \phi \rangle \to \frac{Vol(\mathbb{S}^{n-r})}{Vol(\mathbb{S}^r)} \left(\int_M \, \phi \vert d V_{s_d} \vert \right) \tag{6} \] holds for \(\forall \phi \in C^0(M)\), as \(d \to + \infty\).
This result means that, when \(n \geq 3\), then the almost sure convergence can be verified for the linear statistics associated with a random sequence of sections of increasing degree. Furthermore, the framework given here contains the case of random real algebraic submanifolds of \(\mathbb{R}\mathbb{P}^n\), obtained as the common zero set of \(r\) independent Kostlan-Shub-Smale polunomials.
For other related works see, e.g., [D. Gayet and J.-Y. Welschinger, J. Inst. Math. Jussieu 14, No. 4, 673–702 (2015; Zbl 1326.32040)] for random real algebraic submanifolds, and also [the author, J. Funct. Anal. 270, No. 8, 3047–3110 (2016; Zbl 1349.58007)] for expected volume of random submanifolds.