Calabi–Yau m-folds (M,J, ω,Ω) are compact complex manifolds (M,J) of complex dimension m, equipped with a Ricci-flat Kähler metric g with Kähler form ω, and a holomorphic (m, 0)-form Ω of constant length |Ω| = 2. Using Algebraic Geometry and Yau’s solution of the Calabi Conjecture, one can construct them in huge numbers. String Theorists (a species of theoretical physicist) are very interested ...

The modularity conjecture for Calabi–Yau manifolds predicts that every Calabi–Yau manifold should be modular in the sense that its L– series coincides with the L–series of some automorphic form(s). The case of rigid Calabi–Yau threefolds was (almost) solved by Dieulefait and Manoharmayum in [7, 6]. On the other hand in the non–rigid case it is even not clear which automorphic forms should appea...

In 1975, Yau [Y] proved, by way of a gradient estimate, that a complete manifold M with non-negative Ricci curvature must satisfy the strong Liouville property for harmonic functions. The strong Liouville property (Liouville property) asserts that any positive (bounded) harmonic function defined on M must be identically constant. In 1980, Cheng [C] generalized the gradient estimate to harmonic ...

In this paper, we consider gradient estimates on complete noncompact Riemannian manifolds (M, g) for the following general heat equation ut = ∆V u+ au log u+ bu where a is a constant and b is a differentiable function defined onM×[0,∞). We suppose that the Bakry-Émery curvature and the N -dimensional Bakry-Émery curvature are bounded from below, respectively. Then we obtain the gradient estimat...

Borcea-Voisin threefolds are Calabi-Yau manifolds. They are constructed by Borcea ([B]) and Voisin ([V]) in the construction of mirror manifolds. In [SYZ], Strominger, Yau and Zaslow propose a geometric construction of mirror manifolds using special Lagrangian tori (called SYZ-construction below). Using degenerate Calabi-Yau metrics Gross and Wilson show that SYZ-construction works for any Borc...

This thesis is concerned with higher cluster tilting objects in generalized higher cluster categories and tropical friezes associated with Dynkin diagrams. The generalized cluster category arising from a suitable 3-Calabi-Yau differential graded algebra was introduced by C. Amiot. It is Hom-finite, 2-Calabi-Yau and admits a canonical cluster-tilting object. In this thesis, we extend these resul...

We study a duality that relates the T /Z2 orientifold with N = 2 flux to standard fluxless Calabi-Yau compactifications of type IIA string theory. Using the duality map, we show that the Calabi-Yau manifolds that arise are abelian surface (T ) fibrations over P . We compute a variety of properties of these threefolds, including Hodge numbers, intersection numbers, discrete isometries, and H1(X,...

In this paper we study several issues related to the generation of superpotential induced by background Ramond-Ramond fluxes in compactification of Type IIA string theory on Calabi-Yau four-folds. Identifying BPS solitons with D-branes wrapped over calibrated submanifolds in a Calabi-Yau space, we propose a general formula for the superpotential and justify it comparing the supersymmetry condit...

Calabi–Yau m-folds (M,J, ω,Ω) are compact complex manifolds (M,J) of complex dimension m, equipped with a Ricci-flat Kähler metric g with Kähler form ω, and a holomorphic (m, 0)-form Ω of constant length |Ω| = 2. Using Algebraic Geometry and Yau’s solution of the Calabi Conjecture, one can construct them in huge numbers. String Theorists (a species of theoretical physicist) are very interested ...