Let $S$ be a K3 surface and $\pi:S\rightarrow B$ be a SLAG $T^2$-fibration. I am struggling with a statement that Fiberwise dualization does not change the topology of $S$. …

I am learning about mirror symmetry and having trouble understanding Givental's I- and J-functions. For example the J-function for the quintic threefold $X$ is defined by the formu …

Let $E=\mathbb{C}/(\mathbb{Z}+\tau\mathbb{Z})$ be an alliptic curve. Define a holomorphic 1-form $dz=dx+idy$ and a Kahler form $\omega=dx\wedge dy$. How can one prove that special …

A toy SYZ mirror symmetry is described as follows. Let $X$ be a $n$-dimensional compact manifold. The contangent bundle $T^*X$ has a natural symplectic structure locally given by t …

I'm working on a project and I've used the Picard-Fuchs equation at a maximally unipotent monodromy point for a certain 1-dimensional family of Calabi-Yau 3-folds to calculate the …

Hi there, I have some questions about the mirror symmetry of hyperkahler manifold and K3 surface. The well-known result said: the mirror symmetry for K3 surface is just given by …

Mirror symmetry relates two Calabi-Yau threefolds with mirrored Hodge diamonds. Since Calabi-Yau threefold is Kahler, this naive correspondence does not hold for rigid Calabi-Yau t …

I have a question on maximally unipotent monodromy point (or large complex structure limit) of the family of polarized K3 surfaces $(X,L)$. It is known that the moduli space of suc …

Donaldson-Thomas invariants are the (virtual) Euler characteristics of moduli spaces of elements of the derived category of coherent sheaves (with some fixed Chern class, satisfyin …

It seems to me, that a typical science often has simple and important examples whose formulation can be understood (or at least there are some outcomes that can be understood). So …

The Gross-Siebert program is said to be an algebraic analog of the SYZ conjecture and they used toric degeneration to construct a mirror dual of Calabi-Yau varieties. It seems like …

In a fabulous paper (http://annals.math.princeton.edu/wp-content/uploads/annals-v166-n2-p01.pdf) Bridgeland showed that locally finite stability conditions on a triangulated catego …

I saw the stringy moduli space mentioned in a few papers but with little no explanation. I vaguely understand it is supposed to be the moduli space of complex structures on the mir …

There is a well known class of Calabi-Yau (3 dimensional) pairs constructed by Batyrev. These are resolutions of Calabi-Yau hypersurfaces in reflexive polytops of dimension 4. Que …

Here is what I know about Landau-Ginzburg models: It is an important player in the story of mirror symmetry. It involves "potentials" which are functions of complex varibles, whic …