Related papers: Log canonical $3$-fold complements

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…

We prove the rationality of the K\"ahler cone and the positivity of $c_2(X)$, if $X$ is a Calabi-Yau-threefold with $\rho(X)=2$ and has an embedding into a ${\bb P}^n$-bundle over ${\bb P}^m$ in the cases $(n,m)=(1,3),(3,1)$. The case…

We ask about the simply connected compact smooth 6-manifolds which can support structures of Calabi-Yau threefolds. In particular, we study the interesting case of Calabi-Yau threefolds $X$ with second betti number 3. We have a cup-product…

We study various examples of Calabi-Yau threefolds over finite fields. In particular, we provide a counterexample to a conjecture of K. Joshi on lifting Calabi-Yau threefolds to characteristic zero. We also compute the p-adic cohomologies…

In a recent preprint, Chi Li proved that aymptotically conical complex manifolds with regular tangent cone at infinity admit holomorphic compactifications (his result easily extends to the quasiregular case). In this short note, we show…

We show that for each fixed dimension $d\geq 2$, the set of $d$-dimensional klt elliptic varieties with numerically trivial canonical bundle is bounded up to isomorphism in codimension one, provided that the torsion index of the canonical…

Calabi-Yau algebras are particularly symmetric differential graded algebras. There is a construction called `Calabi-Yau completion' which produces a canonical Calabi-Yau algebra from any homologically smooth dg algebra. Homologically smooth…

We study type III contractions of Calabi-Yau threefolds containing a ruled surface over a smooth curve. We discuss the conditions necessary for the image threefold to by smoothable. We describe the change in Hodge numbers caused by this…

We consider smooth complete intersection Calabi-Yau 3-folds in minuscule Schubert varieties, and study their mirror symmetry by degenerating the ambient Schubert varieties to Hibi toric varieties. We list all possible Calabi-Yau 3-folds of…

We provide a fine classification of rigid three-dimensional torus quotients with isolated canonical singularities, up to biholomorphism and diffeomorphism. This complements the classification of Calabi-Yau 3-folds of type $\rm{III}_0$,…

We prove three fundamental properties of counting holomorphic cylinders in log Calabi-Yau surfaces: positivity, integrality and the gluing formula. Positivity and integrality assert that the numbers of cylinders, defined via virtual…

In this paper, we establish a structure theorem for projective klt pairs $(X,\Delta)$ with nef anti-log canonical divisor; specifically, we prove that, up to replacing $X$ with a finite quasi-\'etale cover, $X$ admits a locally trivial…

We prove the existence of asymptotically cylindrical (ACyl) Calabi-Yau 3-folds starting with (almost) any deformation family of smooth weak Fano 3-folds. This allow us to exhibit hundreds of thousands of new ACyl Calabi-Yau 3-folds;…

A primitive Calabi-Yau threefold is a non-singular Calabi-Yau threefold which cannot be written as a crepant resolution of a singular fibre of a degeneration of Calabi-Yau threefolds. These should be thought as the most basic Calabi-Yau…

Let $(X/Z,B+A)$ be a $\Q$-factorial dlt pair where $B,A\ge 0$ are $\Q$-divisors and $K_X+B+A\sim_\Q 0/Z$. We prove that any LMMP$/Z$ on $K_X+B$ with scaling of an ample$/Z$ divisor terminates with a good log minimal model or a Mori fibre…

Motivated by S-duality modularity conjectures in string theory, we define new invariants counting a restricted class of 2-dimensional torsion sheaves, enumerating pairs $Z\subset H$ in a Calabi-Yau threefold X. Here H is a member of a…

Consider a log Calabi-Yau pair $(X,D)$ consisting of a smooth del Pezzo surface $X$ of degree $\geq 3$ and a smooth anticanonical divisor $D$. We prove a correspondence between genus zero logarithmic Gromov-Witten invariants of $X$…

We show an example of non-simply connected non-liftable Calabi-Yau threefold over an algebraically closed field of characteristic $3$. It is constructed from a simply connected example by S.~Schr\"oer.

We show that crystals with the properties of crystalline cohomology of ordinary Calabi-Yau threefolds in characteristic p>0, exhibit a remarkable similarity with the well known structure on the cohomology of complex Calabi-Yau threefolds…

Let X and Y be two smooth projective n-dimensional algebraic varieties X and Y over C with trivial canonical line bundles. We use methods of p-adic analysis on algebraic varieties over local number fields to prove that if X and Y are…