Related papers: Another way to enumerate rational curves with toru…

This is an expanded version of the author's talk at the third annual meeting of International Consortium of Chinese Mathematicians held at USTC in December 2020. In this expository article, we give a survey on joint works with Hosono, Lian,…

Mirror Symmetry for Calabi-Yau hypersurfaces in toric varieties is by now well established. However, previous approaches to it did not uncover the underlying reason for mirror varieties to be mirror. We are able to calculate explicitly…

We use Lagrangian torus fibrations on the mirror $X$ of a toric Calabi-Yau threefold $\check X$ to construct Lagrangian sections and various Lagrangian spheres on $X$. We then propose an explicit correspondence between the sections and line…

For a del Pezzo surface of degree $\geq 3$, we compute the oscillatory integral for its mirror Landau-Ginzburg model in the sense of Gross-Hacking-Keel [Mark Gross, Paul Hacking, and Sean Keel, "Mirror symmetry for log Calabi-Yau surfaces…

We give a mathematical account of a recent string theory calculation which predicts the number of rational curves on the generic quintic threefold. Our account involves the interpretation of Yukawa couplings in terms of variations of Hodge…

We use the mirror theorem for toric Deligne-Mumford stacks, proved recently by the authors and by Cheong-Ciocan-Fontanine-Kim, to compute genus-zero Gromov-Witten invariants of a number of toric orbifolds and gerbes. We prove a mirror…

We show that a smooth projective complex manifold of dimension greater than two endowed with an elliptic fiber space structure and with finite fundamental group always contains a rational curve, provided its canonical bundle is relatively…

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…

For an arbitrary smooth n-dimensional Fano variety $X$ we introduce the notion of a small toric degeneration. Using small toric degenerations of Fano n-folds $X$, we propose a general method for constructing mirrors of Calabi-Yau complete…

We consider a class of Calabi-Yau compactifications which are constructed as a complete intersection in weighted projective space. For manifolds with one K\"ahler modulus we construct the mirror manifolds and calculate the instanton sum.

We apply mirror symmetry to the problem of counting holomorphic rational curves in a Calabi-Yau threefold X with Z_3 x Z_3 Wilson lines. As we found in Part A [hep-th/0703182], the integral homology group H_2(X,Z)=Z^3 + Z_3 + Z_3 contains…

We provide an enumerative meaning of the mirror maps for toric Calabi-Yau orbifolds in terms of relative Gromov-Witten invariants of the toric compactifications. As a consequence, we obtain an equality between relative Gromov-Witten…

The point is to compare the mathematical meaning of the ``number of rational curves on a Calabi-Yau threefold'' to the meaning ascribed to the same notion by string theorists.

Let $Y$ be a smooth projective threefold and let $f:Y\to X$ be a birational map with $Rf_*\mathcal{O}_Y=\mathcal{O}_X$. When $Y$ is Calabi-Yau, Bryan-Steinberg defined enumerative invariants associated to such maps called $f$-relative…

Given a brane tiling, that is, a bipartite graph on a torus, we can associate with it a singular 3-Calabi-Yau variety. Using the brane tiling, we can also construct all crepant resolutions of the above variety. We give an explicit toric…

We study an example of complete intersection Calabi-Yau threefold due to Libgober and Teitelbaum arXiv:alg-geom/9301001, and verify mirror symmetry at a cohomological level. Direct computations allow us to propose an analogue to the…

We extend the discussion of mirror symmetry, Picard-Fuchs equations, instanton-corrected Yukawa couplings, and the topological one-loop partition function to the case of complete intersections with higher-dimensional moduli spaces. We will…

We give geometric explanations and proofs of various mirror symmetry conjectures for $T^{n}$-invariant Calabi-Yau manifolds when instanton corrections are absent. This uses fiberwise Fourier transformation together with base Legendre…

In this paper, we extend our result in [3] to hypersurfaces of any smooth projective variety $Y$. Precisely we let $X_0$ be a generic hypersurface of $Y$ and $c_0:\mathbf P^1\to X_0$ be a generic birational morphism to its image, i.e.…

Gross, Hacking, and Keel have constructed mirrors of log Calabi-Yau surfaces in terms of counts of rational curves. Using $q$-deformed scattering diagrams defined in terms of higher genus log Gromov-Witten invariants, we construct…