Globality, Lee-cover volume, and mapping-torus resonance for lcs forms

Definitions and mathematical statement

Question 1.36 asks when a conformally symplectic structure is global, stating that this is equivalent to exactness of a one-form denoted $\omega$, and asking for other conditions. The paired aid interprets this as the Lee form. Let $M^{2n}$ be connected, $2n\geq4$, and let $\Omega$ be nondegenerate with closed Lee form $\theta$, using the source convention \[ d\Omega=-\theta\wedge\Omega,\qquad d_\theta=d+\theta\wedge. \] Then $d_\theta\Omega=0$. Global conformality means there is a smooth $f$ with $e^f\Omega$ symplectic. The reported equivalent tests are \[ \theta=df\text{ globally},\qquad [\theta]=0\in H^1(M;\mathbb R),\qquad \rho_\theta(\gamma)=\int_\gamma\theta=0\quad\forall\gamma\in\pi_1(M). \] These use the specified sign and distinguish the Lee form from the nondegenerate two-form.

The aid gives two additional closed-manifold tests. Let $\pi:\widetilde M\to M$ be the minimal Lee cover, corresponding to $\ker\rho_\theta$, choose $df=\pi^*\theta$, and put $\widetilde\omega=e^f\pi^*\Omega$. For closed $M$, globality is equivalent to finite total symplectic volume of $(\widetilde M,\widetilde\omega)$. Nonzero Lee periods produce deck homotheties, whose iterates force infinite total volume. Also no finite cover can globalize a strict lcs form, since finite-cover pullback is injective on real first cohomology.

On a closed manifold, twisted exactness $\Omega=d_\theta\eta$ obstructs globality: if $\theta=df$, then $e^f\Omega=d(e^f\eta)$ would be an exact symplectic form on a closed manifold. This implication is not asserted on a noncompact base. Nonvanishing of $[\Omega]\in H^2_\theta(M)$, however, is not sufficient for globality. The source historically mentions examples of Banyaga with nonzero Lichnerowicz class, without a full reference.

For a mapping torus \[ T_\varphi=(N\times[0,1])/((x,1)\sim(\varphi(x),0)), \] with compact fiber $N$ and Lee class $b[dt]$, choose the representative $\theta=b\,dt$. The aid reports a twisted Wang/mapping-cone description governed by \[ e^b\varphi^*-I. \] In real cohomology its degree-two part has the short exact form \[ 0\longrightarrow\operatorname{coker}(e^b\varphi^*-I|_{H^1(N)}) \longrightarrow H^2_{b\,dt}(T_\varphi) \longrightarrow\ker(e^b\varphi^*-I|_{H^2(N)}) \longrightarrow0. \] Consequently a nonzero $[\Omega]$ requires $e^{-b}$ in the monodromy spectrum on $H^1(N;\mathbb R)$ or $H^2(N;\mathbb R)$. Reversing the gluing or twisted-differential convention changes the dictionary, so the signs are not dropped.

The aid reports an explicit hyperbolic $T^3$ mapping-torus example realizing this resonance with a nonzero twisted lcs class and nonzero Lee class. Its matrix and form are not supplied here, so it is retained as an attributed counterexample to sufficiency of twisted-class nonvanishing, not reconstructed. These are globality and resonance tests, not a classification of all conformally symplectic structures.

Short English statement

The supplied aid interprets globality as vanishing Lee class. On closed manifolds it adds a minimal-Lee-cover volume test and a twisted-exactness obstruction. Its mapping-torus formula imposes a sign-sensitive monodromy resonance for nonzero Lichnerowicz class, while a reported strict example shows that such nonvanishing alone does not imply globality.

Sources

• Local source record: catalogue entry, catalogue entry; canonical aim-geometry-notes.json, notes[431], Question 1.36. Title: "Globality, Lee-cover volume, and mapping-torus resonance for lcs forms."
• AIM workshop list Holomorphic curves in contact geometry, supplied reference https://aimath.org/WWN/contactgeom2/contactgeom2.pdf. The supplied verification locates the text in sixth-day notes; the embedded page number is an extraction artifact. This draft does not access a PDF or HTML source.
• Original attribution to Banyaga for nonzero Lichnerowicz-class examples, without a full citation. Attempt 1 supplies the Lee-cover/finite-cover tests, twisted Wang sequence and hyperbolic torus mapping-torus example; the latter's explicit data are not reproduced.
• Registry dated 14 August 2026: "partially_solved," "PARTIAL-PROGRESS," statement status "exact"; attempt partial, rigor "proved," novelty confidence low. The machine-generated, non-peer-reviewed aid requires independent verification before citation.