[Paper Review] A computation of invariants of a rational self-map

Ekaterina Amerik|ArXiv.org|Jul 26, 2007

Mathematical Dynamics and Fractals2 references3 citations

TL;DR

This paper establishes the algebraic stability of a rational self-map $f$ on the Fano variety $X = \mathcal{F}(V)$ of lines on a general smooth cubic fourfold $V \subset \mathbb{P}^5$, computes its dynamical degrees as $\lambda_1 = 7$, $\lambda_2 = 31$, $\lambda_3 = 28$, $\lambda_4 = 16$, and proves that $f$ is cohomologically hyperbolic with $\lambda_2$ as the dominant degree. The computation relies on resolving the indeterminacy locus via blow-up and analyzing the induced action on cohomology using intersection theory and Chern classes.

ABSTRACT

I compute the dynamical degrees in C. Voisin's example of a rational self-map of the variety of lines on a cubic fourfold.

Motivation & Objective

• To establish the algebraic stability of a rational self-map $f$ on the Fano variety $X = \mathcal{F}(V)$ of lines on a general smooth cubic fourfold $V \subset \mathbb{P}^5$.
• To compute the dynamical degrees of $f$ using the action of $f^*$ on cohomology, particularly in $H^4(X, \mathbb{Z})$.
• To analyze the geometry of the indeterminacy locus $S \subset X$, consisting of lines of the second kind, and resolve $f$ via blow-up along $S$.
• To prove that $f$ is cohomologically hyperbolic, with $\lambda_2 = 31$ as the maximal dynamical degree.
• To show that the cohomology classes $H$ (hyperplane class) and $\Delta$ (class of the surface of lines of the second kind) span an invariant subspace under $f^*$, and to compute the eigenvalues of $f^*$ on this subspace.

Proposed method

• Resolve the rational self-map $f$ by blowing up its indeterminacy locus $S$, the surface of lines of the second kind, to obtain a morphism $g$ on the blow-up $\tilde{X}$.
• Use the projective bundle structure of the exceptional divisor $E$ over $S$ to compute Chern classes and intersection numbers via exact sequences involving tautological line bundles.
• Compute the pullback $f^*H = 7H$ using the pushforward of $g^*H$ under the projection $\pi: \tilde{X} \to X$, leveraging the geometry of the universal family of lines.
• Derive $f^*H^2 = 4H^2 + 45\Delta$ by analyzing $\pi_*(\pi^*H \cdot E)$ and $\pi_*(E^2)$, using the fact that $E$ restricts to $\mathcal{O}_E(-1)$ and $\pi_*E^2 = -S = -(H^2 - \Delta)$.
• Compute $f^*\Delta = 31\Delta$ by expressing $g^*\Delta = c_2(g^*U_X)$ via two exact sequences involving $P$ and $g^*U_X$, and then pushing forward.
• Compute $f^*H^3 = 28H^3$ by expanding $g^*H^3 = (7\pi^*H - 3E)^3$, and evaluating $\pi_*(\pi^*H^2 E)$, $\pi_*(\pi^*H E^2)$, and $\pi_*(E^3)$ using the projective bundle formula and Chern class identities.

Experimental results

Research questions

• RQ1 Is the rational self-map $f$ on the Fano variety $X = \mathcal{F}(V)$ of lines on a general cubic fourfold algebraically stable, i.e. does $(f^n)^* = (f^*)^n$ hold for all $n$?
• RQ2 What are the dynamical degrees $\lambda_l(f)$ of the self-map $f$, and which degree dominates?
• RQ3 How does the action of $f^*$ on $H^4(X, \mathbb{Z})$ decompose, and what is its spectral behavior on the invariant subspace spanned by $H^2$ and $\Delta$?
• RQ4 What is the geometric origin of the eigenvalues of $f^*$, particularly the values 7, 31, and 28?
• RQ5 Does the self-map $f$ preserve the Lagrangian property of subvarieties, and how is this reflected in the cohomology class of $\Delta$?

Key findings

• The rational self-map $f$ on $X = \mathcal{F}(V)$ is algebraically stable, meaning $(f^n)^* = (f^*)^n$ for all $n$, which allows the dynamical degrees to be computed via the spectral radii of $f^*$.
• The dynamical degrees of $f$ are $\lambda_1 = 7$, $\lambda_2 = 31$, $\lambda_3 = 28$, and $\lambda_4 = 16$, with $\lambda_2$ being the dominant degree.
• The pullback action of $f^*$ on cohomology satisfies $f^*H = 7H$, $f^*H^2 = 4H^2 + 45\Delta$, $f^*\Delta = 31\Delta$, and $f^*H^3 = 28H^3$, computed via intersection theory on the blow-up $\tilde{X}$.
• The cohomology class $\Delta$ is an eigenvector of $f^*$ with eigenvalue 31, reflecting the fact that $f$ preserves the Lagrangian surface $S$ of lines of the second kind.
• The decomposition of $H^4(X, \mathbb{Z})$ into orthogonal Hodge substructures implies that $f^*$ acts as a homothety on the irreducible components, supporting the self-adjointness of $f^*$ with respect to the intersection form.
• The computation of $\pi_*(E^3)$ as $-\frac{35}{4}H^3$ via Chern classes and the projective bundle formula is essential to obtaining $f^*H^3 = 28H^3$.