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Recently, Masuda-Sato and Precup-Sommers independently proved an LLT versionof the Shareshian-Wachs conjecture which says that the Frobeniuscharacteristics of the cohomology of the twin manifolds of regular semisimpleHessenberg varieties are unicellular LLT polynomials. The purpose of this paperis to study the geometry of twin manifolds and we prove that they are relatedby explicit blowups and fiber bundle maps. Upon taking their cohomology, weobtain a direct proof of the modular law which establishes the LLTShareshian-Wachs conjecture.

Geometry of the twin manifolds of regular semisimple Hessenberg varieties and unicellular LLT polynomials