Generalized Local Product Structure

• Generalized local product structure is a framework in geometry and analysis that locally decomposes complex structures into simpler, lower-dimensional products using gauge-equivalent transformations.
• Local splitting theorems in generalized complex geometry demonstrate that unique product representations—combining holomorphic Poisson and symplectic parts—exist under controlled deformation methods.
• The concept extends to symmetric differentials, contact and Dirac–Jacobi geometries, and dynamical measures, offering robust tools for deformation theory and singularity classification.

A generalized local product structure is a unifying concept arising in several distinct areas of geometry and analysis, characterizing situations where a geometric, analytic, or dynamical structure, while possibly highly nontrivial globally, admits a local presentation as a gauge-equivalent product of simpler, lower-dimensional structures. Central examples include generalized complex manifolds, closed symmetric 2-differentials, Weyl and conformal products, generalized (CRF) structures, and dynamical measures. The precise nature of "local product" and "generalized" is context-dependent, but the common theme is the uniqueness and stability—modulo gauge transformations or additional singularities—of such local decompositions.

1. Foundational Definition and Algebraic Realization

In the formalism of generalized geometry, a generalized local product structure is best understood via the generalized tangent bundle $\mathbb{T}M = TM \oplus T^*M$ equipped with the neutral pairing of signature $(n,n)$. Here, a generalized almost product structure is an orthogonal endomorphism $P$ with $P^2 = \mathrm{Id}$, whose $+1$ and $-1$ eigenbundles $L$ and $L'$ are maximally isotropic (self-dual) and complementary subbundles: $\mathbb{T}M = L \oplus L'.$ A structure is a (true) generalized local product structure if both $L$ and $(n,n)$0 are involutive under the Dorfman bracket, equivalent to the vanishing of the odd-parity component $(n,n)$1 of the de Rham differential. Up to $(n,n)$2–field transforms (automorphisms induced by closed $(n,n)$3-forms), local decompositions are, in principle, unique and exist in a neighborhood of any regular point where appropriate symmetries hold (Aldi et al., 2017).

2. Local Splitting Theorems in Generalized Complex Geometry

A central analytic result is the local splitting theorem for generalized complex manifolds. Let $(n,n)$4 be a $(n,n)$5-dimensional generalized complex manifold and $(n,n)$6 a point where the underlying real Poisson tensor $(n,n)$7 has rank $(n,n)$8. Then, there exists a neighborhood of $(n,n)$9 gauge-equivalent to the product

$P$0

where $P$1 carries a holomorphic Poisson structure and $P$2 a standard symplectic form. The generalized complex structure takes block-diagonal form, with the upper-left block corresponding to the holomorphic Poisson part and the lower-right to the symplectic part. The corresponding isomorphism class of holomorphic Poisson structures is a local invariant independent of the choice of gauge, and the product form is unique up to $P$3–field transformations and diffeomorphisms (Bailey et al., 2014, Bailey, 2012).

The proof employs Nash–Moser-type iteration and Maurer–Cartan deformation theory, converging rapidly via smoothing operators and controlled error estimates. The key insight is that integrability reduces the problem to holomorphic Poisson geometry in local charts, with local obstructions vanishing due to the triviality of Dolbeault cohomology in small balls.

3. Gauge Transformations, $P$4–field Actions, and Uniqueness

Gauge equivalence is central in the theory, with $P$5–field transforms acting orthogonally on $P$6: $P$7 Two generalized complex structures $P$8 and $P$9 are gauge-equivalent if there exists a diffeomorphism and a closed $P^2 = \mathrm{Id}$0 such that $P^2 = \mathrm{Id}$1. The matrix characterization of this equivalence is explicit; all local product representatives are related in this manner, and the Poisson structure component is unique up to local holomorphic equivalence (Bailey et al., 2014).

Analogous $P^2 = \mathrm{Id}$2–field actions appear in other generalized settings, such as the omni-Lie algebroid of generalized contact bundles, where real Atiyah $P^2 = \mathrm{Id}$3-form $P^2 = \mathrm{Id}$4–field transforms bring the structure into canonical local product forms (Schnitzer et al., 2017).

4. Generalized Local Product Structure in Analytic, Symplectic, and Contact Contexts

The generalized local product structure paradigm encompasses:

• Symmetric 2-differentials: On complex surfaces, a closed symmetric $P^2 = \mathrm{Id}$5-differential $P^2 = \mathrm{Id}$6 of rank $P^2 = \mathrm{Id}$7 (locally factoring into two transverse foliations) decomposes locally as $P^2 = \mathrm{Id}$8 with closed holomorphic $P^2 = \mathrm{Id}$9-forms $+1$0 off a "breakdown" locus $+1$1. Even at points of $+1$2 where such a decomposition fails to be directly holomorphic, Theorem 2.6 of (Bogomolov et al., 2014) assures that a generalized decomposition exists: $+1$3 still splits as a product of (possibly multi-valued) closed $+1$4-forms whose singularities are controlled—specifically, exponential of meromorphic functions with pole orders bounded by the local contact order of foliations. This provides a generalized closed local product structure at all points.
• Contact and Dirac–Jacobi Geometry: In generalized contact bundles (the odd-dimensional analogue of generalized complex structures), local product decompositions correspond to either a contact factor $+1$5 a complex factor or a symplectic factor $+1$6 a gauge-complex factor near regular points. Dirac–Jacobi theory and gauge transformations via real Atiyah $+1$7-forms ensure the uniqueness and integrability of these local splittings (Schnitzer et al., 2017).
• Locally Conformally Product Structures: In Riemannian geometry, an LCP structure is given by a non-flat, closed Weyl connection with reducible holonomy. Kourganoff's splitting theorem demonstrates that the universal cover is locally isometric to $+1$8, where $+1$9 is flat and $-1$0 is irreducible, non-flat. Adapted metrics allow the Lee form to vanish on the flat factor, and the existence of such a structure is preserved under suitable products and similarity group actions (Flamencourt, 2022).

5. Dynamical Systems and Local Product Structures for Measures

Local product structure also plays a crucial role in thermodynamic formalism and ergodic theory. For equilibrium states $-1$1 of geodesic flows on negatively curved surfaces (or surfaces with cone singularities), there exist local "flow boxes" where $-1$2 decomposes into product conditionals on local unstable and center-stable manifolds: $-1$3 with Radon–Nikodym derivatives bounded above and below by uniform constants. The construction employs the "Bowen bracket," conditional measure regularity, and holonomy properties. This product structure enables the proof of the Bernoulli property for these measure-preserving flows (Call et al., 2024).

6. Deformation Theory and Spinorial Formalism

Infinitesimal deformations of generalized local product structures are governed by Lie algebroid cohomology and pure spinor techniques. For a generalized CRF-structure with eigenbundle $-1$4, deformations are characterized by sections $-1$5 satisfying

$-1$6

where $-1$7 and $-1$8 are the components of the de Rham differential associated to $-1$9 and $L$0. The full infinitesimal moduli space is thus identified within the subcomplex annihilated by $L$1, restricting to those deformations preserving the local-product splitting (Aldi et al., 2017).

The pure spinor approach also yields a $L$2-bigrading on forms, with the product structure’s factors associated to annihilators of pure spinor lines. In the integrable case, the de Rham differential splits as $L$3, encoding the local product geometry.

7. Canonical Structures, Analytic Loci, and Applications

The generalized local product structure framework allows the identification of canonical analytic or geometric loci—for instance, the complex locus of a generalized complex manifold, defined as the zero-locus of the Poisson tensor. By local normal form, this zero-locus inherits a well-defined structure of a complex analytic space, and these local analytic models glue via biholomorphisms on overlapped charts. The existence and uniqueness of product structures up to gauge act as foundational tools across geometry, analysis, and dynamics—enabling both classification results and rigorous descriptions of singularity and deformation phenomena (Bailey et al., 2014, Bogomolov et al., 2014, Flamencourt, 2022).