arXiv subjects
Publications and source records attributed to Michele Ancona.
We prove the existence of a non-linear recursive relation for the volume of the moduli space of hyperbolic spheres with conical points or geodesic boundaries. This relation generalizes a result by Zograf, where the same was derived for cusps.
In this note, we answer positively a question of Yau by proving the existence of closed minimal surfaces with negative induced curvature in any sphere of large dimension. The proof follows the strategy of Song, applying it to closed Riemann surfaces with large automorphism groups, and obtaining almost hyperbolic minimal surfaces.
In this paper, we provide a lower bound for the Cheeger constant and the spectral gap for random complex curves in $\C P^2$. The complex curve is endowed with the restriction of the ambient Fubini-Study metric, and the probability measure is the Gaussian measure induced by the $\mathscr{L}^2$-Hermitian product on the space of complex homogeneous polynomialsof degree $d$ in $3$ variables. The proof relies on our previous bounds for the systole and the curvature of random complex curves, together with an isoperimetric inequality for small ovals on complex curves. More generally, we establish such lower bounds for random complex curves within complex projective manifolds.
Let $f:\mathbb{R}^d \to \mathbb{R}^k$ be a smooth centered stationary Gaussian field and $\mathcal{B} \subset \mathbb{R}^d$ be a bounded Borel set. In this paper, we determine the asymptotics as $R \to \infty$ of all the cumulants of the $(d-k)$-dimensional volume of $f^{-1}(0) \cap R\mathcal{B}$. When $k=1$, we obtain similar asymptotics for the number of critical points of $f$ in $R\mathcal{B}$. Our main hypotheses are some regularity and non-degeneracy of the field, as well as mild integrability conditions on the first derivatives of its covariance kernel. As corollaries of these cumulants estimates, we deduce a strong Law of Large Numbers and a Central Limit Theorem for the nodal volume (resp.~the number of critical points) of a regular and non-degenerate enough field whose covariance decays fast enough at infinity. Our results hold more generally for a one-parameter family $(f_R)$ of Gaussian fields admitting a stationary local scaling limit as $R \to \infty$, for example Kostlan polynomials in the large degree limit. They also hold for the random measures of integration over the vanishing locus of $f_R$ as $R \to +\infty$.
For any integers $n\geq 2$ and $1\leq r\leq n-1$ satisfying $3r\geq 2n-1$, we show that the expected volume fraction of a random degree $d$ complex submanifold of $\C\mathbb{P}^n$ of codimension $r$ where the bisectional holomorphic curvature (for the induced ambient metric) is negative tends to one when $d$ goes to infinity. Here, the probability measure is the natural one associated with the Fubini--Study metric. We provide similar estimates for the holomorphic sectional curvature, the Ricci curvature, and the scalar curvature. Our results hold more generally for random submanifolds within any complex projective manifold.
In this paper, we study the curvature properties of random complex plane curves. We bound from below the probability that a uniform proportion of the area of a random complex degree $d$ plane curve has a curvature smaller than $-d/8$. Our lower bound is uniform, in the sense that it does not depend on $d$. We also provide uniform upper bounds for similar probabilities. These results extend to random complex curves of projective surfaces equipped with an ample line bundle. This paper can be viewed as a sequel of [1], where other metric statistics were given. On a larger time scale, it joins the general program initiated in [11] of understanding random complex hypersurfaces of projective manifolds.
For any integer $n\geq 2$, we prove that for any large enough integer $d$, with large probability the injectivity radius of a random degree $d$ complex hypersurface in $\C P^n$ is larger than $d^{-\frac{1}2(3n+2)}$. Here the hypersurface is endowed with the restriction of the ambient Fubini-Study metric, and the probability measure is induced by the Fubini-Study $L^2$-Hermitian product on the space of homogeneous complex polynomials of degree $d$ in $(n+1)$-variables. We also prove that with high probability, the sectional curvatures of the random hypersurface are bounded by $d^{\frac{3}2(n+2)}$, and that its spectral gap is bounded below by $\exp(-d^{\frac{1}4(3n+15)})$. These results extend to random submanifolds of higher codimension in any complex projective manifold. Independently, we prove that the diameter of a degree $d$ divisor is bounded by $Cd^3$, which generalizes and amends the bound given in~\cite{feng1999diameter} for planar curves.
We define a notion of multijet for functions on $\mathbb{R}^n$, which extends the classical notion of jets in the sense that the multijet of a function is defined by contact conditions at several points. For all $p \geq 1$ we build a vector bundle of $p$-multijets, defined over a well-chosen compactification of the configuration space of $p$ distinct points in $\mathbb{R}^n$. As an application, we prove that the linear statistics associated with the zero set of a centered Gaussian field on a Riemannian manifold have a finite $p$-th moment as soon as the field is of class~$\mathcal{C}^p$ and its $(p-1)$-jet is nowhere degenerate. We prove a similar result for the linear statistics associated with the critical points of a Gaussian field and those associated with the vanishing locus of a holomorphic Gaussian field.
We investigate the failure of Bézout's Theorem for two symplectic surfaces in $\mathbb{C}\mathrm{P}^2$ (and more generally on an algebraic surface), by proving that every plane algebraic curve $C$ can be perturbed in the $\mathscr{C}^1$-topology to an arbitrarily close smooth symplectic surface $C_ε$ with the property that the cardinality $\#C_ε\cap Z_d$ of the transversal intersection of $C_ε$ with an algebraic plane curve $Z_d$ of degree $d$, as a function of $d$ can grow arbitrarily fast. As a consequence we obtain that, although Bézout's Theorem is true for pseudoholomorphic curves with respect to the same almost complex structure, it is "arbitrarily false" for pseudoholomorphic curves with respect to different (but arbitrarily close) almost-complex structures (we call this phenomenon "instability of Bézout's Theorem").
We study the zeros of sections of the form $T_k s_k$ of a large power $L^{\otimes k} \to M$ of a holomorphic positive Hermitian line bundle over a compact K\''ahler manifold $M$, where $s_k$ is a random holomorphic section of $L^{\otimes k}$ and $T_k$ is a Berezin-Toeplitz operator, in the limit $k \to +\infty$. In particular, we compute the second order approximation of the expectation of the distribution of these zeros. In a ball of radius of order $k^{-\frac{1}{2}}$ around $x \in M$, assuming that the principal symbol $f$ of $T_k$ is real-valued and vanishes transversally, we show that this expectation exhibits two drastically different behaviors depending on whether $f(x) = 0$ or $f(x) \neq 0$. These different regimes are related to a similar phenomenon about the convergence of the normalized Fubini-Study forms associated with $T_k$: they converge to the K\''ahler form in the sense of currents as $k\rightarrow + \infty$, but not as differential forms (even pointwise). This contrasts with the standard case $f=1$, in which the convergence is in the $\mathscr{C}^{\infty}$-topology. From this, we are able to recover the zero set of $f$ from the zeros of $T_k s_k$.
Given a real algebraic variety $X$ of dimension $n$, a very ample divisor $D$ on $X$ and a smooth closed hypersurface $Σ$ of $\mathbf{R}^n$, we construct real algebraic hypersurfaces in the linear system $|mD|$ whose real locus contains many connected components diffeomorphic to $Σ$. As a consequence, we show the existence of real algebraic hypersurfaces in the linear system $|mD|$ whose Betti numbers grow by the maximal order, as $m$ goes to infinity. As another application, we recover a result by D. Gayet on the existence of many disjoint lagrangians with prescribed topology in any smooth complex hypersurface of $\mathbf{C}\mathbf{P}^n$. The results in the paper are proved more generally for complete intersections. The proof of our main result uses probabilistic tools.
We study the number of real roots of a Kostlan random polynomial of degree $d$ in one variable. More generally, we are interested in the distribution of the counting measure of the set of real roots of such a polynomial. We compute the asymptotics of the central moments of any order of these random variables, in the large degree limit. As a consequence, we prove that these quantities satisfy a strong Law of Large Numbers and a Central Limit Theorem. In particular, the real roots of a Kostlan polynomial almost surely equidistribute as the degree diverges. Moreover, the fluctuations of the counting measure of this random set around its mean converge in distribution to the Gaussian White Noise. More generally, our results hold for the real zeros of a random real section of a line bundle of degree d over a real projective curve, in the complex Fubini--Study model.
Given a real projective variety $X$ and $m$ ample line bundles $L_1,\dots L_m$ on $X$ also defined over $\mathbb{R}$, we study the topology of the real locus of the complete intersections defined by global sections of $L_1^{\otimes d}\oplus\cdots\oplus L^{\otimes d}_m$. 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 $L_1^{\otimes d}\oplus\dots\oplus L^{\otimes d}_m$ is isotopic to the real locus of a complete intersection of smaller degree.
Let $f:\mathbb{R} \to \mathbb{R}$ be a stationary centered Gaussian process. For any $R>0$, let $ν_R$ denote the counting measure of $\{x \in \mathbb{R} \mid f(Rx)=0\}$. In this paper, we study the large $R$ asymptotic distribution of $ν_R$. Under suitable assumptions on the regularity of $f$ and the decay of its correlation function at infinity, we derive the asymptotics as $R \to +\infty$ of the central moments of the linear statistics of $ν_R$. In particular, we derive an asymptotics of order $R^\frac{p}{2}$ for the $p$-th central moment of the number of zeros of $f$ in $[0,R]$. As an application, we derive a functional Law of Large Numbers and a functional Central Limit Theorem for the random measures~$ν_R$. More precisely, after a proper rescaling, $ν_R$ converges almost surely towards the Lebesgue measure in weak-$*$ sense. Moreover, the fluctuation of $ν_R$ around its mean converges in distribution towards the standard Gaussian White Noise. The proof of our moments estimates relies on a careful study of the $k$-point function of the zero point process of~$f$, for any $k \geq 2$. Our analysis yields two results of independent interest. First, we derive an equivalent of this $k$-point function near any point of the large diagonal in~$\mathbb{R}^k$, thus quantifying the short-range repulsion between zeros of $f$. Second, we prove a clustering property which quantifies the long-range decorrelation between zeros of $f$.
Given an ample real Hermitian holomorphic line bundle $L$ over a real algebraic variety $X$, the space of real holomorphic sections of $L^{\otimes d}$ inherits a natural Gaussian probability measure. We prove that the probability that the zero locus of a real holomorphic section $s$ of $L^{\otimes d}$ defines a maximal hypersurface tends to $0$ exponentially fast as $d$ goes to infinity. This extends to any dimension a result of Gayet and Welschinger valid for maximal real algebraic curves inside a real algebraic surface. The starting point is a low degree approximation property which relates the topology of the real vanishing locus of a real holomorphic section of $L^{\otimes d}$ with the topology of the real vanishing locus a real holomorphic section of $L^{\otimes d'}$ for a sufficiently smaller $d'<d$. Such a statement is inspired by a recent work of Diatta and Lerario.
Given a closed Riemann surface $Σ$ equipped with a volume form $ω$, we construct a natural probability measure on the space $\mathcal{M}_d(Σ)$ of degree $d$ branched coverings from $Σ$ to the Riemann sphere $\mathbb{C}\mathbb{P}^1.$ We prove a large deviations principle for the number of critical points in a given open set $U\subset Σ$: given any sequence $ε_d$ of positive numbers, the probability that the number of critical points of a branched covering deviates from $2d\cdot\textrm{Vol}(U)$ more than $ε_d$ is smaller than $\exp(-C_Uε^3_d d)$, for some positive constant $C_U$. In particular, the probability that a covering does not have any critical point in a given open set goes to zero exponential fast with the degree.
In this paper, we construct a natural probability measure on the space of real branched coverings from a real projective algebraic curve $(X,c_X)$ to the projective line $(\mathbb{C}\mathbb{P}^1,\textrm{conj})$. We prove that the space of degree $d$ real branched coverings having "many" real branched points (for example more than $\sqrt{d}^{1+α}$, for any $α>0$) has exponentially small measure. In particular, maximal real branched coverings, that is real branched coverings such that all the branched points are real, are exponentially rare.
We give an asymptotic probabilistic real Riemann-Hurwitz formula computing the expected real ramification index of a random covering over the Riemann sphere. More generally, we study the asymptotic expected number and distribution of critical points of a random real Lefschetz pencil over a smooth real algebraic variety.