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The purpose of this note is twofold. First, we give a quick proof of Ballico-Chiantini's theorem stating that a Fano or Calabi-Yau variety of dimension at least 4 in codimension two is a complete intersection. Second, we improve Barth-Van…
We determine the structure of the BPS algebra of 2-Calabi-Yau Abelian categories for which the stack of objects admits a good moduli space. We prove that this algebra is isomorphic to the positive part of the enveloping algebra of a…
We construct examples of complete Calabi-Yau metrics on smoothings of 3-dimensional Calabi-Yau cones that are not products of lower-dimensional Calabi-Yau cones and that have orbifold singularities away from the vertex.
We introduce a notion generalizing Calabi-Yau structures on A-infinity algebras and categories, which we call pre-Calabi-Yau structures. This notion does not need either one of the finiteness conditions (smoothness or compactness) which are…
We study irreducible subvarieties of the universal hypersurface $\mathcal{X}/B$ of degree $d$ and dimension $n$. We prove that when $d$ is sufficiently large, a degree $kd$ subvariety $Z$ which dominates $B$ comes from intersection with a…
We consider a generalization of Calabi-Yau structures in the context of $\alpha$-Sasakian manifolds. We study deformations of a special class of Legendrian submanifolds and classify invariant contact Calabi-Yau structures on 5-dimensional…
In this note we speculate about the structure of maximal product subvarieties in moduli stacks of Calabi-Yau manifolds. We discuss examples for quintic hypersurfaces in the four dimensional projective space.
We present a classification algorithm for Calabi-Yau complete intersections arising from nef-partitions in fake weighted projective spaces, allowing us to determine all such complete intersections up to dimension five. Furthermore, we…
The Green-Griffiths-Lang conjecture stipulates that for every projective variety $X$ of general type over ${\mathbb C}$, there exists a proper algebraic subvariety of $X$ containing all non constant entire curves $f:{\mathbb C}\to X$. Using…
We prove Homological Mirror Symmetry for a smooth d-dimensional Calabi-Yau hypersurface in projective space, for any d > 2 (for example, d = 3 is the quintic three-fold). The main techniques involved in the proof are: the construction of an…
In this paper, we prove that in any projective manifold, the complements of general hypersurfaces of sufficiently large degree are Kobayashi hyperbolic. We also provide an effective lower bound on the degree. This confirms a conjecture by…
We consider fibrations by abelian surfaces and K3 surfaces over a one dimensional base that are Calabi-Yau and we obtain dual fibrations that are derived equivalent to the original fibration. Finally, we relate the problem to mirror…
In [Ann. of Math. 169 (2009)], Min Ru proved a second main theorem for algebraically nondegenerate holomorphic curves in complex projective varieties intersecting fixed hypersurface targets. In this paper, by introducing a new proof method…
In this second part of the work, we correct the flaw which was left in the proof of the main Theorem in the first part. This affects only a small part of the text in this first part and two consecutive papers. Yet, some additional arguments…
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…
First, we classify Calabi-Yau threefolds with infinite fundamental group by means of their minimal splitting coverings introduced by Beauville, and deduce that the nef cone is a rational simplicial cone and any rational nef divisor is…
We collect evidence that the notion of path-ordered non-abelian integration admits an extension to two dimensions. We propose the corresponding notion of non-abelian 2-form along the lines of Lie algebroid theory and argue it is an…
We produce the family of Calabi-Yau hypersurfaces $X_{n}$ of $(\mathbb{P}^{1})^{n+1}$ in higher dimension whose inertia group contains non commutative free groups. This is completely different from Takahashi's result \cite{ta98} for…
In 2015, Van den Bergh showed that complete 3-Calabi-Yau algebras over an algebraically closed field of characteristic 0 are equivalent to Ginzburg dg algebras associated with quivers with potential. He also proved the natural…
We show that on every non-$G_2$ complex symmetric space of rank two, there are complete Calabi-Yau metrics of Euclidean volume growth with prescribed horospherical singular tangent cone at infinity, providing the first examples of affine…