Resonant Vector Bundles in Dynamics & Geometry

• Resonant vector bundles are vector bundles with nontrivial resonance among eigenvalues that create obstructions in analytic parameterizations in both dynamical systems and algebraic geometry.
• They are constructed using techniques such as high-order Taylor expansions, normal form transformations, and validated numerics to resolve singular resonant coefficients.
• Applications include spectral stability analysis of PDE pulse solutions and the classification of resonance varieties in vector bundles, offering rigorous stability certificates.

A resonant vector bundle is a vector bundle—frequently arising over invariant manifolds in dynamical systems—whose associated parameterization, normal form structure, or cohomological invariants exhibit nontrivial “resonances”; that is, degeneracies or singularities given by specific algebraic relations among eigenvalues or cup products. Such resonance presents both fundamental theoretical obstructions and computational challenges, specifically in the context of analytic parameterizations and spectral stability analysis for partial differential equations, and in the classification and configuration space of vector bundles arising from algebraic geometry.

1. Resonant Vector Bundles in Dynamical Systems

In the context of spatial dynamics and the paper of pulse solutions to nonlinear PDEs such as the Swift-Hohenberg equation, resonant vector bundles appear as unstable or stable bundles over invariant manifolds. The spectral stability of localized solutions is governed by the linearized operator $\mathcal{L}$; eigenvalue resonance conditions naturally emerge in Hamiltonian (reversible) systems. Given eigenvalues $\{\mu_i\}$ of the linearized operator, bundle resonance is defined by the existence of nontrivial integral relations of the form

$\sum_k \alpha_k \mu_k + \mu - \mu_j = 0,$

where $\alpha_k \in \mathbb{Z}$, and this degeneracy renders the parameterization recursion for the bundle singular at certain multi-indices (Definition 2.5, (Beck et al., 28 Oct 2025)).

The geometry underpinning the existence and computation of these bundles is directly linked to the theory of conjugate points: topological invariants counting parameter values where evolving Lagrangian subspaces intersect a reference plane, which in turn enumerate unstable eigenvalues without explicit spectral calculation.

2. Mathematical Obstructions Due to Resonance

Resonant vector bundles fundamentally alter the solvability and analytic structure of the bundle parameterization problem. In particular, the classical parameterization method for analytic, invariant manifolds and their bundles—using power series expansions and recursive homological equations (see Cabré-Fontich-de la Llave)—breaks down due to resonance:

• At resonant multi-indices, the recursion denominators vanish, rendering the coefficients either indeterminate or singular.
• Attempts to maintain both analyticity and invariance fail: in general, a bundle can be made analytic but not invariant, or invariant but not analytic; both simultaneously can only be realized in the non-resonant case.

This dichotomy is resolved by prioritizing analyticity (essential for validated numerics), relaxing the invariance constraint, and introducing an upper-triangular normal form $\mathcal{A}(\sigma)$ that absorbs the obstructions via additional (moduli) parameters in the power series expansion (Section 2.4, (Beck et al., 28 Oct 2025)).

3. Parameterization and Validated Numerics for Resonant Bundles

The theoretical framework for constructing analytic parameterizations of resonant vector bundles is formalized through the bundle conjugacy equation:

$$DG(P(\sigma)) W(\sigma) = D W(\sigma) \Omega^s \sigma + W(\sigma) \mathcal{A}(\sigma),$$

with $DG(P(\sigma))$ the flow Jacobian, $\Omega^s$ the diagonal eigenvalue matrix, and $\mathcal{A}(\sigma)$ a normal form matrix polynomial encoding resonance effects.

Recursion for Taylor coefficients at each order is governed by a homological equation:

$$(\alpha \cdot \mu + \mu_j - \mu_i) \tilde{w}_\alpha^{i,j} + a_\alpha^{i,j} = \tilde{s}_\alpha^{i,j}$$

• Non-resonant terms: solved directly for $\tilde{w}$, with $a=0$.
• Resonant terms: $\tilde{w}=0$, solve for $a$ to match $\tilde{s}$. $\mathcal{A}(\sigma)$ thus accumulates only at resonant degrees, yielding an upper-triangular structure.

Ensuring the reliability of this local analytic construction over large (possibly infinite) domains is accomplished using a posteriori validation via radii-polynomial estimates and contraction-mapping arguments (Theorem 3.6, (Beck et al., 28 Oct 2025)). This is essential for computer-assisted proofs: truncating the Taylor expansion at finite order $N$ (with $N$ as high as $35$ in examples), with the analytic tail handled via interval arithmetic and operator-norm bounds.

4. Applications: Conjugate Point Detection and Spectral Stability

The analytic parameterization of resonant bundles is central to the rigorous computation of conjugate points, which in turn count unstable eigenvalues of $\mathcal{L}$. In the setting of the Swift-Hohenberg equation:

• The subspaces $E^{u}_-(x; 0)$ associated with the unstable bundle along the stable manifold must be transported accurately across large domains.
• Near spatial infinity, the system is governed by its linearization, for which resonance imposes obstructions to classical analytic/invariant bundles; only the new parameterization approach yields computationally tractable and rigorous enclosures.
• The presence or absence of conjugate points (i.e., intersections with a sandwich plane) provides a topological certificate of pulse solution (in)stability, circumventing the need for direct spectral computation, and is validated numerically within the analytic framework above ((Beck et al., 28 Oct 2025), Section 3).

5. Resonant Vector Bundles in Algebraic Geometry and Resonance Varieties

Resonance phenomena also arise in algebraic geometry, in connection with vector bundles on curves and their resonance loci. The resonance variety $\mathbb{R}(V,K)$ associated to a pair $(V,K)$, where $K \subseteq \bigwedge^2 V$, is

$$\mathbb{R}(V, K) := \left\{ [a] \in \mathbb{P}V^\vee : \exists\, b\in V^\vee,\, 0\neq a\wedge b\in K^\perp \right\}.$$

If $K$ is the kernel of a determinant (cup product) map $d_2: \bigwedge^2 H^0(X,E) \to H^0(X,\wedge^2E)$ for a bundle $E$ on $X$, the resonance is realized as the union of projectivized spaces of global sections of saturated sub-line bundles of $E$.

A central result is the identification of obstructions: not all linear unions in projective space can occur as resonance varieties of vector bundles—there exists a tight link to Quot scheme stratifications and the geometry of pencils (Theorem 3.4, Proposition 3.7, (Aprodu et al., 10 Oct 2025)). In particular, for universal quotient bundles on Grassmannians, specific configurations, such as the disjoint union of 14 lines in $\mathbb{P}^5$ associated to a Mukai bundle on a genus 8 curve, are exactly realized as resonance loci.

6. Low-Dimensional and Explicit Constructions

Examples in low dimensions showcase the sharp structural constraints:

• For $n=4$, resonance for the restricted universal quotient bundle yields $\mathbb{P}^1 \times \mathbb{P}^1$ in $\mathbb{P}^3$,
• For $n=5$, the variety of resonance for a generic elliptic curve bundle is a ruled surface,
• For $n=6$, every configuration of 14 disjoint lines in $\mathbb{P}^5$ arises as the resonance of a Mukai bundle on a canonical genus 8 curve (Aprodu et al., 10 Oct 2025).

The resonance locus is invariably a disjoint union of projective spaces, reflecting the stratified geometric nature of pencils on the underlying bundles. Configurations involving intersecting subspaces, such as those arising in right-angled Artin group resonance, are precluded.

7. Computational and Analytical Advancements

The analytic and computational innovations applicable to resonant vector bundles include:

• The capability of validated Taylor expansions of very high order using interval arithmetic, for large regions of phase space.
• The fusion of non-resonant and resonant analytic charts via composite constructions, enabling full coverage of the stable/unstable manifolds and their relevant bundles.
• The establishment of a general algorithm, via Lemma 2.8 and Theorem 3.6 (Beck et al., 28 Oct 2025), for the recursive construction of Taylor coefficients and normal forms, along with validated enclosures guaranteeing analytic continuation.
• The recognition of the necessity to prioritize analyticity in bundle construction for the purpose of rigorous computer-assisted proof, despite the sacrifice of strict invariance.

Challenge	Resonance Effects/Obstructions	Analytic/Computational Resolution
Pulse stability via conjugate pts	Non-existence of analytic invariant bundles	Analytic but non-invariant parameterization
High-order Taylor expansions	Fails at resonant coefficients	Normal form $\mathcal{A}(\sigma)$ absorbs resonance
Validated numerics over large domains	Analyticity/invariance tension	Radii polynomial validation, interval arithmetic
Realization of resonance varieties	Restrictions from Quot/flatt. stratification	Only disjoint unions of projective subspaces possible

Resonant vector bundles thus play a pivotal role at the intersection of infinite-dimensional dynamical systems, rigorous numerics, and algebraic geometry. The combined algebraic, topological, and computational framework for their construction, analysis, and application constitutes a robust and general approach, both for the rigorous determination of spectral stability in PDEs and for the classification of resonance phenomena in the moduli of vector bundles.