The Bogomolov--Tian--Todorov theorem for log smooth pairs $(X_0/S_0,\mathcal{L}_0)
The Bogomolov--Tian--Todorov theorem for log smooth pairs $(X_0/S_0,\mathcal{L}_0)
The celebrated Bogomolov--Tian--Todorov theorem states that the functorÌýof infinitesimal smooth deformations of a smooth and proper Calabi--YauÌývariety $X$ is unobstructed, meaning that any infinitesimal deformationÌýcan be lifted along any thickening. The same is true when we deform notÌýonly a Calabi--Yau variety, but a pair $(X,\mathcal{L})$ of aÌýCalabi--Yau variety together with a line bundle. In logarithmicÌýgeometry, we replace the smooth Calabi--Yau variety with a log smoothÌýspace over a log point $S_0$. By previous work, we know already that theÌýlog smooth deformation functor of a proper log Calabi--Yau isÌýunobstructed; in this talk, I will report on work in progress showingÌýthat log smooth deformations of a pair of a log smooth log Calabi--YauÌý$f_0: X_0 \rightarrow S_0$ together with a line bundle $\mathcal{L}_0$Ìýare unobstructed as well.