Numerically Flatness: Geometry, Controls & Learning

• Numerical flatness is a structural property that combines positivity (nefness) with vanishing invariants (e.g., Chern classes) to ensure strong regularity and semistability.
• It connects classical geometric analysis in vector bundles with advanced applications in Higgs bundles, foliations, and lattice information theory.
• Applied in optimization and control theory, numerical flatness underlies invariant metrics in deep learning and flatness-preserving discretization for trajectory planning.

Numerical flatness is a structural property arising in multiple areas of mathematics and mathematical physics, notably in complex geometry, the theory of vector bundles, geometric control theory, and the analysis of optimization landscapes in machine learning. It always encodes, at a geometric or algebraic level, a combination of two features: a positivity (often “nefness”) and a vanishing condition (most often for certain Chern classes, spectral invariants, or similar quantities), usually leading to strong regularity, semistability, and splitting properties for the objects involved.

1. Classical Notions: Numerically Flat Vector Bundles

Let $X$ be a compact Kähler manifold with Kähler form %%%%1%%%%. A holomorphic vector bundle $E\to X$ is called nef if the tautological line bundle $\mathcal O_{\mathbb P(E)}(1)\to\mathbb P(E)$ is nef. It is called numerically flat if both $E$ and its dual $E^*$ are nef. Demailly–Peternell–Schneider (DPS) established several equivalent characterizations for numerically flat bundles on compact Kähler manifolds:

• $E$ is numerically flat if and only if there exists a filtration by subbundles $0 = E_0 \subsetneq \cdots \subsetneq E_m = E$ such that each quotient $E_k/E_{k-1}$ is Hermitian flat (carries a unitary flat connection).
• All Chern classes $c_i(E)$ vanish in cohomology.
• $c_1(E)=0$ and $E$ is nef, equivalently $E$ is polystable of slope zero and $c_2(E)=0$.
• For every morphism $f:C\to X$ from a smooth projective curve, $f^*E$ is semistable of degree zero (Druel et al., 2024).

On non-Kähler compact complex manifolds with Hermitian metrics satisfying the Gauduchon and Astheno–Kähler conditions ($d\omega^{n-1} = d\omega^{n-2} = 0$), Li, Nie, and Zhang showed that numerical flatness admits a full analog:

• $E$ is numerically flat if and only if it is $\omega$-nef with vanishing $c_1(E)\cdot [\omega^{n-1}]$, and semistable with vanishing $c_2(E)\cdot [\omega^{n-2}]$, and admits a filtration with Hermitian flat quotients.
• $E$ admits an approximate Hermitian flat structure: for every $\epsilon > 0$, there exists a smooth Hermitian metric $H_\epsilon$ such that $\|F_{H_\epsilon}\|_{L^\infty} \to 0$ as $\epsilon \to 0$ (Li et al., 2019).

The case of Fujiki manifolds (bimeromorphic images of Kähler manifolds) extends the above algebraic-geometric criteria via pullback arguments and direct images for principal $G$-bundles. For holomorphic principal $G$-bundles, numerical flatness of the adjoint bundle corresponds to nefness of the relative anti-canonical bundle on the associated flag bundle (Biswas, 2021).

2. Numerical Flatness for Higgs Bundles

Let $X$ be a smooth projective variety over an algebraically closed field of characteristic zero, and $(E, \varphi)$ a Higgs bundle. Numerical flatness in this context generalizes the notion for plain vector bundles via the theory of Higgs–Grassmannians:

• $(E, \varphi)$ is Higgs-nef (H-nef) if $\det E$ is nef, and all universal quotient Higgs bundles $(Q_{s, E}, \rho_s^*\varphi)$ obtained via Higgs–Grassmannian schemes are H-nef by induction.
• $(E, \varphi)$ is H-numerically flat (H-nflat) if both $(E, \varphi)$ and $(E^*, \varphi^*)$ are H-nef (Capasso, 29 Dec 2025).

Equivalent characterizations include:

• For every morphism $f:C \to X$ from a smooth curve, $f^*(E,\varphi)$ is semistable with $\int_C f^* c_1(E) = 0$.
• $(E, \varphi)$ admits a filtration by Higgs subbundles with stable, degree-zero, H-nflat quotients (“pseudostability”).
• All pullbacks to curves are semistable of degree 0. Furthermore, in the ordinary vector bundle case, H-nflatness coincides with classical numerical flatness, entailing vanishing of all Chern classes. For general Higgs bundles, the vanishing of $c_2(\End E)$ in the curve-semistable case is conjectural in higher rank but established in rank two and some special geometries.

3. Applications in Foliations and Holomorphic Poisson Geometry

A holomorphic foliation $\mathcal F$ of dimension $p$ on a compact Kähler manifold $X$ is numerically flat if the subbundle $T_{\mathcal F} \subset T_X$ is numerically flat:

• $T_{\mathcal F}$ and $\Omega^1_{\mathcal F} = (T_{\mathcal F})^*$ are nef; $c_i(T_{\mathcal F}) = 0$ for all $i$.
• $T_{\mathcal F}$ is polystable of slope zero and carries a unitary flat connection (Druel et al., 2024).

Key structural consequences include:

• Existence of a complementary foliation $T_X = T_{\mathcal F} \oplus T_{\mathcal G}$.
• The universal cover $\widetilde{X}$ splits as $\mathbb{C}^p \times Y$ with $T_{\mathcal F}$ tangent to $\mathbb{C}^p$.
• The canonical bundle $\omega_{\mathcal F}$ is torsion.
• Leaves are uniformized by Euclidean spaces; closures are (finite étale quotients of) equivariant compactifications of abelian Lie groups.

In the theory of holomorphic Poisson manifolds, numerically flat foliations underlie a global Weinstein splitting theorem for Poisson structures. For rank-2 Poisson structures, the universal cover splits as a product of symplectic and complementary Poisson manifolds, provided the minimal leaf dimension is two, and this holds for all projective $X$ up to dimension five.

4. Numerical Flatness in Flatness Measures for Optimization

In deep learning, “flatness” of a minimum in the loss landscape has been heuristically linked to good generalization. Classical measures such as the spectral norm or trace of the Hessian are not invariant to parameterization: rescaling weights layer-wise in ReLU networks can change the Hessian spectrum without changing the function. This undermines the theoretical significance of such measures.

A reparameterization-invariant flatness measure is defined by, for each layer $l$,

• $$\kappa^l(w) = \|\mathrm{vec}(w_l)\|^2 \lambda_{\max}^{(l)}$$,
• $$\kappa^l_\tau(w) = \|\mathrm{vec}(w_l)\|^2 \operatorname{Tr}(H^{(l)}(w))$$,

where $H^{(l)}(w)$ is the Hessian (w.r.t. the vectorized weights for that layer), and $\lambda_{\max}^{(l)}$ its largest eigenvalue (Petzka et al., 2019). These measures are strictly invariant under all layer-wise weight rescalings that leave the network function unchanged, and correlate strongly with generalization error across architectures and reparameterizations. Network-wide summaries (e.g., $\max_l\kappa^l$ and $\sum_l\kappa^l$) provide robust, invariant sharpness metrics.

5. Numerical Flatness in Control Theory and Discretization

Differential flatness of a control system refers to the existence of an output map $y$ such that states and inputs can be algebraically parameterized by $y$ and its finite derivatives. Difference flatness generalizes this to discrete-time systems, requiring parametrization in terms of a discrete sequence of outputs and forward shifts.

It is established that standard numerical discretization schemes (Euler, etc.) do not in general preserve flatness. Explicitly, there exist continuous-time flat systems whose Euler discretization is not difference-flat for any nonzero step size.

A theory of flatness-preserving discretization has been developed:

• Starting from a flat continuous-time system, one lifts to a chain-of-integrator model via endogenous feedback linearization.
• Discretize in these coordinates using a tangent-bundle map $D'$, typically $D'(z,\rho) = (z, z + \rho)$.
• Pull back via the linearizing diffeomorphism to original coordinates, yielding an implicit first-order accurate discrete-time scheme that remains difference-flat, preserving all flat output-based parametrization and trajectory generation capabilities (Jindal et al., 14 Nov 2025).

6. Flatness Factor in Lattice Information Theory

In information theory, the flatness factor $\epsilon_\Lambda(\sigma)$ of a lattice $\Lambda$ at noise standard deviation $\sigma$ quantifies how close the periodized lattice Gaussian is to uniform over the fundamental domain:

$$\epsilon_\Lambda(\sigma) = \max_{x \in \mathbb{R}^n} | g_n(\Lambda + x; \sigma) \cdot \nu_\Lambda - 1 |.$$

Here, $g_n$ is the $n$-dimensional Gaussian, and $\nu_\Lambda$ is the lattice volume. Flatness factor upper bounds eavesdropping success and information leakage in wiretap lattice codes. The expected flatness factor under fading, $E_H[\epsilon_{H \Lambda}(\sigma)]$, provides a practical metric for code selection and performance ranking.

For moderate to high dimension, the theta series $\Theta_\Lambda$ admits efficient numerical approximation via truncated sums and incomplete gamma function integrals, enabling rapid Monte Carlo estimation of $E_H[\epsilon_{H \Lambda}(\sigma)]$ for design and analysis (Barreal et al., 2016).

7. Numerical Flatness in Geometric Control and Optimization

Flatness in geometric control—the property that all trajectories and controls are determined by a global output and its derivatives—lends itself to direct trajectory planning. However, constructing a globally valid flat output for complex robotic systems is nontrivial.

Recent work frames the search for equivariant flat outputs as an optimization problem over sections of a principal bundle, constrained via Riemannian geometry, Lie group symmetry, and orthogonality of the horizontal distribution induced by the mechanical connection. The problem is transcribed to a sparse quadratic program, solved numerically to recover the globally flat output, as confirmed in benchmark systems (planar rocket, aerial manipulator) (Welde et al., 2023). This approach enables the construction of flat outputs where no closed-form analysis is available.

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References Table

Area	Notion of Flatness	Key Result / Equivalent Criteria	Reference
Holomorphic vector bundles on Kähler/non-Kähler manifolds	Numerically flat bundle	Nef + $c_1 = 0$ + polystable + $c_2 = 0$ + filtration with flat quotients	(Druel et al., 2024, Li et al., 2019, Biswas, 2021)
Higgs bundles	H-numerically flat (H-nflat)	H-nef + vanishing $c_1$ + curve semistability + pseudostability	(Capasso, 29 Dec 2025)
Holomorphic foliations	N.f. tangent bundle	Hermitian flat + torsion canonical + splitting + Euclidean uniformization	(Druel et al., 2024)
Deep learning optimization	Invariant flatness of minima	Scale-invariant Hessian-weight metric $\kappa$ correlates with generalization	(Petzka et al., 2019)
Control theory/discretization	Discrete flatness-preserving schemes	Tangent-lifted discretization conjugate to chain of integrators	(Jindal et al., 14 Nov 2025)
Lattice information theory (Gaussian channels)	Flatness factor	Periodization deviation quantify security bounds and rank lattice performance	(Barreal et al., 2016)
Geometric nonlinear control on Lie/manifold systems	Constructed via numerical optimization	QP-based solution for globally valid, equivariant flat outputs	(Welde et al., 2023)

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Numerical flatness thus encapsulates, across domains, a nexus of geometric, spectral, and algebraic regularity, often admitting mutually reinforcing characterizations (nefness, vanishing Chern classes, flat filtrations, and invariance properties) that bridge structure theorems, stability analysis, and concrete computational design.