Related papers: Deligne's conjecture and mirror symmetry
We study the predictions of mirror symmetry for the 1-parameter family of Calabi-Yau 3-folds $\tilde{X}$ with hodge numbers $h^{11}=31,h^{21}=1$ constructed in \cite{BN}. We calculate the Picard-Fuchs differential equation associated to…
The aim of this article is to report on recent progress in understanding mirror symmetry for some non-complete intersection Calabi-Yau threefolds. We first construct four new smooth non-complete intersection Calabi-Yau threefolds with…
It is frequently possible to produce new Calabi-Yau threefolds from old ones by a process of allowing the complex structure to degenerate to a singular one, and then performing a resolution of singularities. (Some care is needed to ensure…
Given two Calabi--Yau threefolds which are believed to constitute a mirror pair, there are very precise predictions about the enumerative geometry of rational curves on one of the manifolds which can be made by performing calculations on…
In this paper, we present an application of mirror symmetry to arithmetic geometry. The main result is the computation of the period of a mixed Hodge structure, which lends evidence to its expected motivic origin. More precisely, given a…
We survey mirror symmetry of Calabi-Yau manifolds from the perspective of families of Calabi-Yau manifolds and their period integrals. Special emphasis is laid on distinguished properties of the hypergeometric series of Gel'fand, Kapranov,…
In this paper, we verify a part of the Mirror Symmetry Conjecture for Schoen's Calabi-Yau 3-fold, which is a special complete intersection in a toric variety. We calculate a part of the prepotential of the A-model Yukawa couplings of the…
Calabi-Yau manifolds are important objects in algebraic geometry and in theoretical physics. A hypothesis of mirror symmetry is that Calabi-Yau manifolds of dimension 3 come in pairs, with the Hodge numbers of one manifold mirroring the…
We study mirror symmetric pairs of Calabi--Yau manifolds over finite fields. In particular we compute the number of rational points of the manifolds as a function of the complex structure parameters. The data of the number of rational…
This paper is the first arising from our project announced in math.AG/0211094, "Affine manifolds, log structures, and mirror symmetry." We aim to study mirror symmetry by studying the log structures of Illusie-Fontaine and Kato on…
We consider certain families of Calabi-Yau orbifolds and their mirror partners constructed from Fermat hypersurfaces in weighted projective 4-spaces. Our focus is the topological mirror symmetry. There are at least three known ingredients…
The geometric aspects of mirror symmetry are reviewed, with an eye towards future developments. Given a mirror pair (X,Y) of Calabi-Yau threefolds, the best-understood mirror statements relate certain small corners of the moduli spaces of X…
As a continuation of \lianyaufour, we study modular properties of the periods, the mirror maps and Yukawa couplings for multi-moduli Calabi-Yau varieties. In Part A of this paper, motivated by the recent work of Kachru-Vafa, we degenerate a…
In this note we construct conifold transitions between several Calabi-Yau threefolds given by Pfaffians in weighted projective spaces and Calabi-Yau threefolds appearing as complete intersections in toric varieties. We use the obtained…
We describe how to find period integrals and Picard-Fuchs differential equations for certain one-parameter families of Calabi-Yau manifolds. These families can be seen as varieties over a finite field, in which case we show in an explicit…
Studying the mirror symmetry of a Calabi-Yau threefold $X$ of the Reye congruence in $\mP^4$, we conjecture that $X$ has a non-trivial Fourier-Mukai partner $Y$. We construct $Y$ as the double cover of a determinantal quintic in $\mP^4$…
Assuming the standard framework of mirror symmetry, a conjecture is formulated describing how the diffeomorphism group of a Calabi-Yau manifold Y should act by families of Fourier-Mukai transforms over the complex moduli space of the mirror…
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 discuss some numerical parts of the mirror conjecture. For any 3 - dimensional Calabi - Yau manifold author introduces a generalization of the Casson invariant known in 3 - dimensional geometry, which is called Casson -…
We study mirror symmetry of Calabi-Yau manifolds within the framework of the Gauss-Manin system. Applying the flat coordinates to the Gauss-Manin system for the periods, we derive differential equations for the mirror map in addition to the…