We show the boundedness of B-pluricanonical representations of lc log Calabi-Yau pairs in dimension $2$. As applications, we prove the boundedness of indices of slc log Calabi-Yau pairs up to dimension $3$ and that of non-klt lc log Calabi-Yau pairs in dimension $4$.
We exhibit examples of pairs $(X,D)$ where $X$ is a smooth projective variety and $D$ is an anticanonical reduced simple normal crossing divisor such that the deformations of $(X,D)$ are obstructed. These examples are constructed via toric geometry.
We prove a structure theorem for the Albanese maps of varieties with Q-linearly trivial log canonical divisors. Our start point is the action of a nonlinear algebraic group on a projective variety.
We prove that rationally connected Calabi--Yau 3-folds with kawamata log terminal (klt) singularities form a birationally bounded family, or more generally, rationally connected $3$-folds of $epsilon$-CY type form a birationally bounded family for $e
psilon>0$. Moreover, we show that the set of $epsilon$-lc log Calabi--Yau pairs $(X, B)$ with coefficients of $B$ bounded away from zero is log bounded modulo flops. As a consequence, we deduce that rationally connected klt Calabi--Yau $3$-folds with mld bounded away from $1$ are bounded modulo flops.
We establish a relative spannedness for log canonical pairs, which is a generalization of the basepoint-freeness for varieties with log-terminal singularities by Andreatta--Wisniewski. Moreover, we establish a generalization for quasi-log canonical pairs.
We generalize the cohomological mirror duality of Borcea and Voisin in any dimension and for any number of factors. Our proof applies to all examples which can be constructed through Berglund-H{u}bsch duality. Our method is a variant of the so-called
Landau-Ginzburg/Calabi-Yau correspondence of Calabi-Yau orbifolds with an involution that does not preserve the volume form. We deduce a version of mirror duality for the fixed loci of the involution, which are beyond the Calabi-Yau category and feature hypersurfaces of general type.