Related papers: Birational complexity and dual complexes
This paper contains some applications of Bridgeland-Douglas stability conditions on triangulated categories, and Joyce's work on counting invariants of semistable objects, to the study of birational geometry. We introduce the notion of…
Mutations of the cluster variables generating the cluster algebra of type $A^{(2)}_2$ reduce to a two-dimensional discrete integrable system given by a quartic birational map. The invariant curve of the map is a singular quartic curve, and…
Using three different approaches, we analyze the complexity of various birational maps constructed from simple operations (inversions) on square matrices of arbitrary size. The first approach consists in the study of the images of lines,…
We study the geometry of a birational map between an intersection of a web of quadrics in seven-dimensional complex projective space that contains a plane and the double octic branched along the discriminant of the web.
Let X be a smooth complex variety and Y be a closed subvariety of X, or more generally, a closed subscheme of X. We are interested in invariants attached to the singularities of the pair (X, Y). We discuss various methods to construct such…
This is the author's PhD thesis. Two main sections address various aspects of mirror symmetry for compact Calabi-Yau threefolds and the roles that classically modular varieties play in string theory compactifications. The main results…
Let $Y$ be a smooth projective threefold and let $f:Y\to X$ be a birational map with $Rf_*\mathcal{O}_Y=\mathcal{O}_X$. When $Y$ is Calabi-Yau, Bryan-Steinberg defined enumerative invariants associated to such maps called $f$-relative…
In this paper we discuss recent progress on the modularity of Calabi-Yau varieties. We focus mostly on the case of surfaces and threefolds. We will also discuss some progress on the structure of the L-function in connection with mirror…
We review briefly the characteristic topological data of Calabi--Yau threefolds and focus on the question of when two threefolds are equivalent through related topological data. This provides an interesting test case for machine learning…
We investigate the open mirror symmetry of certain non-complete intersection Calabi- Yau 3-folds, so called pfaffian Calabi-Yau. We perform the prediction of the number of disk invariants of several examples by using the direct integration…
We give an explicit example of log Calabi-Yau pairs that are log canonical and have a linearly decreasing Euler characteristic. This is constructed in terms of a degree two covering of a sequence of blow ups of three dimensional projective…
It is known that a maximal intersection log canonical Calabi-Yau surface pair is crepant birational to a toric pair. This does not hold in higher dimension: this paper presents some examples of maximal intersection Calabi-Yau pairs that…
The most impressively prolific exploration of superstring models (aiming for our physical reality) has been focused on worldsheet-supersymmetric gauged linear sigma models and the closely associated complex-algebraic toric geometry. Mirror…
Most of Calabi-Yau manifolds that have been considered by physicists are complete intersection Calabi-Yau manifolds of toric varieties or some quotients of product types. Purpose of this paper is to introduce a different and rather new kind…
We define a topological invariant of complex projective plane curves. As an application, we present new examples of arithmetic Zariski pairs.
We investigate the birational geometry of Deligne-Mumford stacks and define new birational invariants in this context.
In this article, we study mirror symmetry for pairs of singular Calabi--Yau manifolds which are double covers of toric manifolds. Their period integrals can be seen as certain `fractional' analogues of those of ordinary complete…
In this article, we summarize combinatorial description of complete intersection Calabi-Yau threefolds in Hibi toric varieties. Such Calabi-Yau threefolds have at worst conifold singularities, and are often smoothable to non-singular…
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…
The integrable systems associated with Seiberg-Witten geometry are considered both from the Hitchin-Donagi-Witten gauge model and in terms of intermediate Jacobians of Calabi-Yau threefolds. Dual pairs and enhancement of gauge symmetries…