Related papers: Birational complexity and dual complexes

In this article, we study the geometry of log Calabi-Yau pairs $(X,B)$ of index one and birational complexity zero. Firstly, we propose a conjecture that characterizes such pairs $(X,B)$ in terms of their dual complex and the rationality of…

Let $(X,B)$ be a log Calabi-Yau pair of dimension $n$, index one, and birational complexity $c$. We show that $(X,B)$ has a crepant birational model that admits a tower of Mori fiber spaces of which at least $n-c$ are conic fibrations.…

We study the birational complexity of log Calabi-Yau $3$-folds. For such a pair $(X,B)$ of index one and coregularity zero, we show that $c_{\rm bir}(X,B)\in \{0,2,3\}$. Further, we prove that $(X,B)$ has a log Calabi-Yau crepant birational…

A Calabi-Yau pair of index one and complexity zero is toric. Furthermore, a Calabi-Yau pair of index one and complexity one is of cluster type. In this article, we study Calabi-Yau pairs of index one and complexity two. We develop machinery…

We introduce a new invariant of $G$-varieties, the dual complex, which roughly measures how divisors in the complement of the free locus intersect. We show that the top homology group of this complex is an equivariant birational invariant…

We construct BCOV invariant for Calabi-Yau pairs. The construction covers the classical BCOV invariant and certain equivariant BCOV invariant. The BCOV invariant obtained is expected to be well-behaved under birational equivalence.

This article deals with the study of the birational transformations of the projective complex plane which leave invariant an irreducible algebraic curve. We try to describe the state of art and provide some new results on this subject.

A log Calabi--Yau pair consists of a proper variety $X$ and a divisor $D$ on it such that $K_X+D$ is numerically trivial. A folklore conjecture predicts that the dual complex of $D$ is homeomorphic to the quotient of a sphere by a finite…

In this article, we give a characterization of log Calabi--Yau pairs of complexity zero and arbitrary index. As an application, we show that a log Calabi--Yau pair of birational complexity zero admits a crepant birational model which is a…

The complexity of a pair $(X,B)$ is an invariant that relates the dimension of $X$, the rank of the group of divisors, and the coefficients of $B$. If the complexity is less than one, then $X$ is a toric variety. We prove that if the…

These notes contain a brief introduction to the construction of toric Calabi--Yau hypersurfaces and complete intersections with a focus on issues relevant for string duality calculations. The last two sections can be read independently and…

In this paper we show that the dual complex of a dlt log Calabi-Yau pair $(Y, \Delta)$ on a Mori fibre space $\pi: Y \to Z$ is a finite quotient of a sphere, provided that either the Picard number of $Y$ or the dimension of $Z$ is $\leq 2$.…

We study the structures of klt Calabi--Yau pairs. We show that the discrepancies of log centers of all klt Calabi--Yau varieties with fixed dimension are in a finite set. As a corollary, we show that the index of 4-dimensional non-canonical…

We will survey some aspects of the smooth topology, algebraic geometry, symplectic geometry and contact geometry of anti-canonical pairs in complex dimension two.

We study a new object that can be attached to an abelian variety or a complex torus: the invariant Brauer group, as recently defined by Yang Cao. Over the field of complex numbers this is an elementary abelian 2-group with an explicit upper…

We develop a framework that allows one to describe the birational geometry of Calabi-Yau pairs $(X,D)$. After establishing some general results for Calabi-Yau pairs $(X,D)$ with mild singularities, we focus on the special case when…

In this note, we study quasi-Albanese morphisms for log canonical Calabi-Yau pairs and obtain several structural results. As an application, we prove a characterization of toric pairs.

We develop some methods to construct normal crossing varieties whose dual complexes are two-dimensional, which are smoothable to Calabi--Yau threefolds. We calculate topological invariants of smoothed Calabi--Yau threefolds and show that…

Birational Calabi-Yau threefolds in the same deformation family provide a `weak' counterexample to the global Torelli problem, as long as they are not isomorphic. In this paper, it is shown that deformations of certain desingularized…

We introduce new invariants in equivariant birational geometry and study their relation to modular symbols and cohomology of arithmetic groups.