A non-exact l.c.s. family on a Sol mapping torus

Definitions and mathematical statement

The source asks for further locally conformally symplectic structures representing nonzero Lichnerowicz classes. Its abbreviation "c.s.s." is genuine; the supplied verification identifies it with the modern l.c.s. terminology. In the AIM sign convention, fix a closed 1-form $\omega$, let ${\,\mathrm{d}}_\omega={\,\mathrm{d}}+\omega\wedge$, and define \[ H_\omega^j(M)=\ker{\,\mathrm{d}}_\omega/\operatorname{im}{\,\mathrm{d}}_\omega. \] An l.c.s. form is a nondegenerate 2-form $\Omega$ with ${\,\mathrm{d}}_\omega\Omega=0$, equivalently ${\,\mathrm{d}}\Omega=-\omega\wedge\Omega$. The question seeks constructions with $[\Omega]_\omega\ne0$, not merely with nonexact Lee form. In the alternative convention ${\,\mathrm{d}}\Omega=\theta\wedge\Omega$, the Lee form is $\theta=-\omega$.

The supplied aid gives a compact four-dimensional family. Let $A\in\operatorname{SL}(2,{\mathbb{Z}})$ have positive hyperbolic eigenvalues $\lambda,\lambda^{-1}$, $\lambda>1$, put $\kappa=\log\lambda$, and form its Sol mapping torus $N_A$ and $M=N_A\times S^1$. Use a global coframe $(\alpha,\beta,\gamma,\eta)$ satisfying \[ {\,\mathrm{d}}\alpha=-\kappa\alpha\wedge\gamma,\qquad {\,\mathrm{d}}\beta=\kappa\beta\wedge\gamma,\qquad {\,\mathrm{d}}\gamma={\,\mathrm{d}}\eta=0. \] For $\omega=-\kappa\gamma$, and constants $u,v,w\in{\mathbb{R}}$ with $uv\ne0$, set \[ \Omega_{u,v,w}=u\alpha\wedge\eta+v\beta\wedge\gamma-w\kappa\gamma\wedge\eta. \] The reported identities are \[ {\,\mathrm{d}}_\omega\Omega_{u,v,w}=0,\qquad \Omega_{u,v,w}^2=2uv\,\alpha\wedge\beta\wedge\gamma\wedge\eta, \] \[ [\Omega_{u,v,w}]_\omega=u[\alpha\wedge\eta]_\omega, \quad \beta\wedge\gamma=\frac1{2\kappa}{\,\mathrm{d}}_\omega\beta, \quad-\kappa\gamma\wedge\eta={\,\mathrm{d}}_\omega\eta. \] Thus $u$ controls the cohomology class, $uv$ nondegeneracy, and $w$ an exact shear.

The oppositely twisted test form $\Psi=\beta\wedge\gamma$ satisfies ${\,\mathrm{d}}_{-\omega}\Psi=0$ and \[ \int_M\Omega_{u,v,w}\wedge\Psi =u\int_M\alpha\wedge\beta\wedge\gamma\wedge\eta\ne0. \] If $\Omega={\,\mathrm{d}}_\omega\xi$, the integrand would equal ${\,\mathrm{d}}(\xi\wedge\Psi)$, contradicting Stokes on the closed manifold. This proves the claimed nonzero twisted class for the stated family. Compatible almost complex structures can be chosen for its nondegenerate form, as the source remarks; integrability or a full holomorphic-curve theory is not thereby supplied.

The construction is an example answering the existence request, not a classification of all l.c.s. structures with nonzero Lichnerowicz class. The source's solved-in-literature label is not accompanied by a full external bibliographic reference.

Short English statement

Construct locally conformally symplectic forms with nonzero twisted cohomology class. The supplied aid gives a Sol-mapping-torus family and an opposite-twist pairing that proves nonexactness. It separates cohomology, nondegeneracy and an exact deformation parameter.

Sources

• Question 1.37, AIM workshop Holomorphic curves in contact geometry: https://aimath.org/WWN/contactgeom2/contactgeom2.pdf. The source uses "c.s.s." and notes compatible almost complex structures; its verification identifies preceding Question 1.35 as defining local conformal symplectic structures.
• Local registry data/unsolvedmath/problems.json, permanent ID 20002095, code catalogue entry; canonical aim-geometry-notes.json, notes[432]; updated 14 August 2026.
• No detailed external reference for the construction is supplied; the original literature field is empty. Attempt 1: partial result, problem status at run solved, rigor label proved, novelty confidence low; current solved and solved-in-literature labels. The machine-generated aid is not peer reviewed and requires independent verification before citation.