# 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 functor of infinitesimal smooth deformations of a smooth and proper Calabi--Yau variety $X$ is unobstructed, meaning that any infinitesimal deformation can be lifted along any thickening. The same is true when we deform not only a Calabi--Yau variety, but a pair $(X,\mathcal{L})$ of a Calabi--Yau variety together with a line bundle. In logarithmic geometry, we replace the smooth Calabi--Yau variety with a log smooth space over a log point $S_0$. By previous work, we know already that the log smooth deformation functor of a proper log Calabi--Yau is unobstructed; in this talk, I will report on work in progress showing that log smooth deformations of a pair of a log smooth log Calabi--Yau $f_0: X_0 \rightarrow S_0$ together with a line bundle $\mathcal{L}_0$ are unobstructed as well.