Landau–Ginzburg mirror symmetry conjecture

Mirror Conjecture Mirror Symmetry for Hyperkk Ahler Manifolds

We prove the Mirror Conjecture for Calabi-Yau manifolds equipped with a holomorphic symplectic form. Such man-ifolda are also known as complex manifolds of hyperkk ahler type. We obtain that a complex manifold of hyperkk ahler type is Mirror dual to itself. The Mirror Conjecture is stated (following Kontsevich, ICM talk) as the equivalence of certain algebraic structures related to variations o...

Doran–Harder–Thompson Conjecture via SYZ Mirror Symmetry: Elliptic Curves

We prove the Doran–Harder–Thompson conjecture in the case of elliptic curves by using ideas from SYZ mirror symmetry. The conjecture claims that when a Calabi– Yau manifold X degenerates to a union of two quasi-Fano manifolds (Tyurin degeneration), a mirror Calabi–Yau manifold of X can be constructed by gluing the two mirror Landau– Ginzburg models of the quasi-Fano manifolds. The two crucial i...

Log Mirror Symmetry and Local Mirror Symmetry

We study Mirror Symmetry of log Calabi-Yau surfaces. On one hand, we consider the number of “affine lines” of each degree in P\B, where B is a smooth cubic. On the other hand, we consider coefficients of a certain expansion of a function obtained from the integrals of dx/x ∧ dy/y over 2chains whose boundaries lie on Bφ, where {Bφ} is a family of smooth cubics. Then, for small degrees, they coin...

On the Mirror Conjecture

In 1989, B. Greene, a postdoc of Yau at the time, and R. Plesser constructed an important example of mirror manifolds. Using physics argument, they proved the first case of mirror symmetry for Calabi-Yau threefolds. Their result is considered fundamental, though its proof lacks the precision of a mathematical proof, hence the result does not constitute a mathematical theorem. Two years later, C...

Mirror Symmetry and Localization