Related papers: An Introduction to Counting Orbifolds
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…
In a previous study, we constructed a family of elliptic Calabi-Yau 4-folds possessing a geometric structure that allowed them to be split into a pair of rational elliptic 4-folds. In the present study, we introduce a method of classifying…
The paper investigates some aspects of the geometry and the arithmetic of a non-rigid Calabi-Yau threefold. Particular emphasis is given to the study of its L-function L(H^3,s) and the Galois representation.
We compute the asymptotic number of monic trace-one integral polynomials with Galois group $C_3$ and bounded height. For such polynomials we compute a height function coming from toric geometry and introduce a parametrization using the…
In this paper, we investigate the space of certain weak stability conditions on the triangulated category of D0-D2-D6 bound states on a smooth projective Calabi-Yau 3-fold. In the case of a quintic 3-fold, the resulting space is interpreted…
We construct several examples of higher-dimensional Calabi-Yau manifolds and prove their modularity.
The Gopakumar-Vafa invariants are numbers defined as certain linear combinations of the Gromov-Witten invariants. We prove that the GV invariants of a toric Calabi-Yau threefold are integers and that the invariants for high genera vanish.…
Extending the usual $\mathbf{C}^{\ast r}$ actions of toric manifolds by allowing asymmetries between the various $\mathbf{C}^{\ast}$ factors, we build a class of non commutative (NC) toric varieties $\mathcal{V}%_{d+1}^{(nc)}$. We construct…
The first part of this paper discusses general procedures for finding numerical approximations to distinguished Kahler metrics, such as Calabi-Yau metrics, on complex projective manifolds. These procedures are closely related to ideas from…
We give some concrete examples of Calabi-Yau 3-manifolds with complex multiplication.
An orientifold of Type-IIB theory on a $K3$ realized as a $Z_2$ orbifold is constructed which corresponds to F-theory compactification on a Calabi-Yau orbifold with Hodge numbers $(51, 3)$. The T-dual of this model is analogous to an…
We consider type IIA string theory on a Calabi-Yau 2-fold with D6-branes wrapping 2-cycles in the 2-fold. We find a complete set of conditions on the supergravity solution for any given wrapped brane configuration in terms of SU(2)…
We count tilings of the $n \times m$ rectangular grid, cylinder, and torus with arbitrary tile sets up to arbitrary symmetries of the square and rectangle, along with cyclic shifting of rows and columns. This provides a unifying framework…
We use the gluing method to give a refined description of the collapsing Calabi-Yau metrics on Calabi-Yau 3-folds admitting a Lefschetz K3 fibration.
We develop a theory of Gopakumar-Vafa (GV) invariants for a Calabi-Yau threefold (CY3) $X$ which is equipped with an involution $\imath$ preserving the holomorphic volume form. We define integers $n_{g,h}(\beta) $ which give a virtual count…
We compute a complete set of isomorphism classes of cubic fourfolds over $\mathbb{F}_2$. Using this, we are able to compile statistics about various invariants of cubic fourfolds, including their counts of points, lines, and planes; all…
The BKMP conjecture (2006-2008), proposed a new method to compute closed and open Gromov-Witten invariants for every toric Calabi-Yau 3-folds, through a topological recursion based on mirror symmetry. So far, this conjecture had been…
We introduce a class of 4-dimensional crystal melting models that count the BPS bound state of branes on toric Calabi-Yau 4-folds. The crystalline structure is determined by the brane brick model associated to the Calabi-Yau 4-fold under…
We introduce orbifolds from the classical point of view, using charts, and present orbifold versions of elementary objects from Algebraic Topology, such as the fundamental group, coverings and Euler characteristic; Differential…
Let $X$ be a complex four-dimensional compact Calabi-Yau manifold equipped with a K\"ahler form $\omega$ and a holomorphic four-form $\Omega$. Under certain assumptions, we define Donaldson-Thomas type deformation invariants by studying the…