Nowhere Vanishing Harmonic 1-Form

• Nowhere vanishing harmonic 1-forms are differential 1-forms that are harmonic and nonzero at every point, encapsulating vital topological and geometric information.
• They are analyzed using techniques from global analysis and Kähler geometry, including explicit constructions on flat tori and perturbative methods.
• Their existence imposes strong rigidity and splitting results on manifolds, with obstructions arising from curvature conditions and topological constraints.

A nowhere vanishing harmonic 1-form is a global differential 1-form on a smooth manifold that is both harmonic—i.e., it lies in the kernel of the Hodge Laplacian—and has nonzero pointwise norm everywhere. Such objects encode deep information about the global topology and geometric structure of the underlying manifold. The existence, classification, and rigidity of nowhere vanishing harmonic 1-forms link major areas in global analysis, Kähler geometry, algebraic geometry, and topology.

1. Definitions and Basic Properties

Let $(M, g)$ be a Riemannian manifold. For $\omega \in \Omega^1(M)$, the Hodge Laplacian is defined as $\Delta = dd^* + d^*d$. The form $\omega$ is harmonic if

$$\Delta \omega = 0 \quad \Longleftrightarrow \quad d\omega = 0, \;\; d^*\omega = 0,$$

where $d$ is the exterior derivative and $d^*$ the $L^2$ adjoint. The form is nowhere vanishing harmonic if, in addition, $|\omega|_g(x) > 0$ for every $x \in M$ (Douglas et al., 2024).

On a compact Kähler manifold, every global holomorphic 1-form is harmonic with respect to the unique metric in its Kähler class, as both $\partial \omega = 0$ and $\bar{\partial} \omega = 0$, so $\Delta \omega = 0$ (Church, 26 Apr 2025).

2. Algebraic and Geometric Classification: Compact Kähler and Algebraic Manifolds

The most complete structural results pertain to compact Kähler manifolds and, more specifically, to smooth projective varieties over $\mathbb{C}$. For such varieties $X$, the space $H^0(X, \Omega_X^1)$ of global holomorphic 1-forms provides harmonic representatives, and the existence of nowhere vanishing forms is encoded in the algebraic and topological structure of $X$.

2.1 Characterization in Low Dimensions

Church proves Hao's conjecture in dimension $\leq 4$: Let $f: X \to A$ be a morphism from a smooth projective variety $X$ ($\dim X \leq 4$) to a simple abelian variety $A$. The following are equivalent:

• There exists $\omega \neq 0$ in $H^0(A, \Omega_A^1)$ such that $f^*\omega$ is nowhere vanishing on $X$.
• $f$ is a smooth morphism.

Explicit descriptions (classification by dimension) are given:

• If $\dim A \geq 4$ then $X$ is itself an abelian variety, $f$ is an isogeny.
• If $\dim A = 3$ then the Stein factorization gives a smooth isotrivial curve fibration followed by isogeny.
• If $\dim A = 2$, $X$ is a sequence of blow-ups along étale multisections of a smooth map $X^{min} \to A$, where $X^{min}$ is birational to a fiber bundle classified explicitly: either a quotient $(Z \times A') / G$, a conic bundle, or a del Pezzo fibration (Church, 26 Apr 2025).

2.2 Birational and Albanese Reductions

For smooth projective $4$-folds admitting a nowhere vanishing 1-form but whose Albanese does not admit an elliptic factor, Church shows that after finitely many blow-downs along nonvanishing surfaces, the variety becomes birational to a smooth isotrivial bundle over an abelian variety.

2.3 Projective, Kähler, and Topological Equivalence

For a smooth projective variety $X$, the following are equivalent (Dutta et al., 2021):

1. Existence of a nowhere vanishing holomorphic 1-form.
2. $X$ admits a smooth $S^1$-bundle structure (“circle bundle”).
3. For every finite étale cover $T:X'\to X$, the complex $(H^*(X', \mathbb{C}), \wedge T^* \omega)$ is exact in all degrees.
4. $\omega$ is not in the “microlocal bad locus” $T(SS(Ra_* \mathbb{C}_X)) \subset H^0(X,\Omega^1_X)$, as determined by the singular support of the pushforward sheaf under the Albanese map.

The set of holomorphic 1-forms on $X$ that vanish somewhere is always a finite union of linear subspaces in $H^0(X, \Omega^1_X)$. Hence, the existence of a nowhere vanishing holomorphic 1-form is equivalent to lying outside this finite collection.

3. Obstructions and Vanishing Theorems

The existence of nowhere vanishing harmonic 1-forms is often obstructed by:

• Curvature conditions: On a complete noncompact manifold $(M, g)$ whose Ricci curvature (or more generally, the curvature operator) is bounded below by $-a\,q$, $a < n$ and $(P(q))$ holds, all $L^2$ harmonic 1-forms must vanish unless the rigidity case is achieved [(Vieira, 2014); (Cao, 2022)].
• Topological obstructions: E.g., Betti number $b_1(M)$ must be positive; on $\mathbb{R}P^3$ or connected sums like $\mathbb{R}P^3\#(S^1\times S^2)$, no nowhere vanishing harmonic 1-forms can exist (Douglas et al., 2024).

The only escape from such vanishing theorems is the rigidity/splitting case: when the spectral lower bound for the Laplacian is exactly attained, $M$ splits isometrically as $\mathbb{R} \times N^{n-1}$, and the harmonic 1-form is a multiple of $dt$, which is nowhere zero.

4. Analytic Techniques and Explicit Constructions

4.1 Torus and Perturbative Constructions

On compact flat tori $T^n$ and metrics close to flat, explicit harmonic 1-forms are given by the coordinate differentials $dx^i$. For small $C^1$ perturbations of the metric, existence of nowhere vanishing harmonic 1-forms persists: one constructs a solution $\xi$ to $\Delta^g \xi = d^*_g (dx^1)$, then sets $\alpha = dx^1 - d\xi$, which is harmonic and remains nowhere vanishing when the perturbation is sufficiently small (Deep, 25 Dec 2025).

4.2 Numerical and Computer-Assisted Methods

For more complicated geometries, e.g., the real loci of Calabi-Yau threefolds, numerical methods using neural network ansatz (for both the metric and the form) have been used to produce approximate nowhere-vanishing harmonic 1-forms. Under suitable analytic control (uniform lower bound on pointwise norm, $C^1$ closeness of approximate metric to genuine Ricci-flat metric), these numerics translate into rigorous existence via perturbative results (Douglas et al., 2024).

5. Extensions, Multivalued and $\mathbb{Z}_2$-Harmonic 1-Forms

The framework of nowhere vanishing harmonic 1-forms extends to multivalued forms, notably $\mathbb{Z}_2$-harmonic forms, arising on manifolds with codimension 2 branch loci and nontrivial monodromy. A topological criterion via cyclic branched covers and symmetry (e.g., existence of a nontrivial odd cohomology class fixed by a $\mathbb{Z}_3$-action) guarantees the existence of nondegenerate, hence nowhere vanishing (away from branch locus), $\mathbb{Z}_2$-harmonic 1-forms on 3-manifolds, with applications to link complements and rational homology spheres (He, 2022).

6. Summary Table: Key Structural Results

Setting	Existence Condition	Structure/Obstruction
Compact Kähler X	$\omega\in H^0(X,\Omega^1_X)$, $Z(\omega)=\emptyset$	X isometrically fibered; classified via Albanese, see (Church, 26 Apr 2025, Dutta et al., 2021)
Noncompact M, Ric $\geq -a\,q$	$L^2$ harmonic 1-form	Vanishes unless $M$ splits as $\mathbb{R}\times N$ (Vieira, 2014)
Flat tori $T^n$ or $C^1$–small $g$	$dx^i$ or perturbation	Always exists, explicit analytic estimates (Deep, 25 Dec 2025)
Real loci of CY	$b_1>0$, numerical lower norm bound	Existence or nonexistence varies; requires analytic input (Douglas et al., 2024)
$\mathbb{Z}_2$ harmonic, 3-mfds	$\mathbb{Z}_3$-symmetry criterion	Nondegenerate multivalued form exists for generic metric (He, 2022)

7. Applications and Broader Connections

• Fibrations and morphisms: Nowhere vanishing harmonic/holomorphic 1-forms correspond to fibrations over circles or abelian varieties; their existence controls algebraic-geometric and topological structure, e.g., in the Minimal Model Program and Albanese classification.
• Gauge theory and Calabi-Yau geometry: Nonvanishing harmonic forms facilitate constructions in $G_2$-geometry via neck-stretching and in gauge theory where they model defect/branched structures.
• Vanishing theorems as rigidity: Curvature-based vanishing results provide powerful rigidity properties—if a nowhere vanishing harmonic 1-form exists, the global geometry of $M$ is extremely restricted.

This synthesis reflects both the breadth and rigidity phenomena associated with nowhere vanishing harmonic 1-forms in modern geometry, as seen in contemporary research (Church, 26 Apr 2025, Deep, 25 Dec 2025, Douglas et al., 2024, He, 2022, Cao, 2022, Vieira, 2014, Dutta et al., 2021).