Logarithmically smooth deformations of strict normal crossing logarithmically symplectic varieties
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Abstract
In this thesis we give a definition of the term logarithmically symplectic variety; to be precise, we distinguish even two types of such varieties. The general type is a triple $(f,\nabla,\omega)$ comprising a log smooth morphism $f\colon X\to\mathrm{Spec}\kappa$ of log schemes together with a flat log connection $\nabla\colon L\to\Omega^1_f\otimes L$ and a ($\nabla$-closed) log symplectic form $\omega\in\Gamma(X,\Omega^2_f\otimes L)$. We define the functor of log Artin rings of log smooth deformations of such varieties $(f,\nabla,\omega)$ and calculate its obstruction theory, which turns out to be given by the vector spaces $H^i(X,B^\bullet_{(f,\nabla)}(\omega))$, $i=0,1,2$. Here $B^\bullet_{(f,\nabla)}(\omega)$ is the class of a certain complex of $\mathcal{O}_X$-modules in the derived category $\mathrm{D}(X/\kappa)$ associated to the log symplectic form $\omega$. The main results state that under certain conditions a log symplectic variety can, by a flat deformation, be smoothed to a symplectic variety in the usual sense. This may provide a new approach to the construction of new examples of irreducible symplectic manifolds.