arXiv subjects
Publications and source records attributed to Grigory Papayanov.
We prove that the cohomology algebra of a conilpotent Lie coalgebra is generated in degree 1 as an A-infinity algebra. By dualizing, the same is true about cohomology algebras of finite dimensional nilpotent Lie algebras. In the process, we provide a proof of the conilpotent version of the dual Poincare-Birkhoff-Witt theorem.
In \cite{Goto}, Ryushi Goto has constructed the deformation space for a manifold equipped with a collection of closed differential forms and showed that in some important cases (Calabi-Yau, $G_2$- and $Spin(7)$-structures) this deformation space is smooth. This result unifies the classical Bogomolov-Tian-Todorov and Joyce theorems about unobstructedness of deformations. Using the work of Fiorenza and Manetti, we show that this deformation space could be obtained as the deformation space associated to a certain $L_{\infty}$-algebra. We also show that for Calabi-Yau, $G_2$- and $Spin(7)$-structures this $L_{\infty}$-algebra is homotopy abelian. This gives a new proof of Goto's theorem.
A Hermitian symplectic manifold is a complex manifold endowed with a symplectic form $ω$, for which the bilinear form $ω(I\cdot,\cdot)$ is positive definite. In this work we prove $dd^c$-lemma for 1- and (1,1)-forms for compact Hermitian symplectic manifolds of dimension 3. This shows that Albanese map for such manifolds is well-defined and allows one to prove Kählerness if the dimension of the Albanese image of a manifold is maximal.