Global Classification of Generic Multi‐Vector Fields of Top Degree

Abstract

Abstract. For any compact oriented manifold M , we show that that the top degree multi-vector fields transverse to the zero section of ∧ top T M are classified, up to orientation preserving diffeomorphism, in terms of the topology of the arrangement of its zero locus and a finite number of numerical invariants. The group governing the infinitesimal deformations of such multi-vector fields is computed, and an explicit set of generators exhibited. For the spheres S n , a correspondence between certain isotopy classes of multi-vector fields and classes of weighted signed trees is established.

Deformations of log-symplectic structures

“…, we see that Corollary 1 extends the classification of generic Nambu structures of top degree from [10] to the case of non-orientable manifolds M . In the orientable case, fixing orientations on M and on the components of Z, we have that a generic Nambu structure w of top degree with singular locus Z is determined, up to orientation-preserving diffeomorphisms, by the regularized volume of μ := w −1 and by the volumes of the components Z i of Z computed with the aid of ι ξZ ω| Zi .…”

“…is a multi-vector field of top degree on M , which intersects the zero-section of ∧ m T M transversally at Z. These structures are called generic Nambu structures of top degree and were studied in [10]. In Corollary 2, we show that the b-geometric Moser argument implies the main result from [10].…”

“…Proof of Lemma 1. Recall from [10] the definition of the regularized volume: namely, let ω be a b-top-form on a compact oriented b-manifold (N, W ). Its regularized volume is defined by the limit…”

“…The b-geometric version of the Moser lemma from [12], which plays a fundamental role in our proof, is stated in Lemma 2 in the more general setting of 1-, 2-and top-forms. In the case of top-forms, we obtain a simple proof (Corollary 2) of the classification of generic multi-vector fields of top degree from [10]. In Section 3, we compute the Poisson cohomology of log-symplectic structures (Proposition 1) and relate this to Theorem 1.…”

Poisson Structures and Their Normal Forms

“…See [296,303] for some generalizations of the above results to the case when Σ is not necessarily closed and S may be a singular level of f , and [240] for a generalization of Theorem 2.5.19 to the case of multi-vector fields of top degree on a manifold. This number is called the regularized volume of (Σ, Λ) and is an invariant of Λ.…”