We investigate mirror symmetry for toric Calabi-Yau manifolds from the perspective of the SYZ conjecture. Starting with a non-toric special Lagrangian torus fibration on a toric Calabi-Yau manifold $X$, we construct a complex manifold $\check{X}$ using T-duality modified by quantum corrections. These corrections are encoded by Fourier transforms of generating functions of certain open Gromov-Witten invariants. We conjecture that this complex manifold $\check{X}$, which belongs to the Hori-Iqbal-Vafa mirror family, is inherently written in canonical flat coordinates. In particular, we obtain an enumerative meaning for the (inverse) mirror maps, and this gives a geometric reason for why their Taylor series expansions in terms of the Kähler parameters of X have integral coefficients. Applying the results in "A formula equating open and closed Gromov-Witten invariants and its applications to mirror symmetry," to appear in Pacific J. Math., and "A relation for Gromov-Witten invariants of local Calabi-Yau threefolds," to appear in Math. Res. Lett., we compute the open Gromov-Witten invariants in terms of local BPS invariants and give evidences of our conjecture for several 3-dimensional examples including $K_{\mathbb{P}^2}$ and $K_{\mathbb{P}^1}$.

Let $X$ be a compact connected strongly pseudoconvex $CR$ manifold of real dimension $2_{n − 1}$ in $\mathbb{C}^N$. It has been an interesting question to find an intrinsic smoothness criteria for the complex Plateau problem. For $n\ge 3$ and $N = n+1$, Yau found a necessary and sufficient condition for the interior regularity of the Harvey– Lawson solution to the complex Plateau problem by means of Kohn–Rossi cohomology groups on $X$ in 1981. For $n = 2$ and $N \ge n + 1$, the problem has been open for over 30 years. In this paper we introduce a new CR invariant $g^{(1,1)}(X)$ of $X$. The vanishing of this invariant will give the interior regularity of the Harvey–Lawson solution up to normalization. In the case $n = 2$ and $N = 3$, the vanishing of this invariant is enough to give the interior regularity.

In this note we prove a conjecture of Kashiwara, which states that the Euler class of a coherent analytic sheaf $\mathcal{F}$ on a complex manifold $X$ is the product of the Chern character of $\mathcal{F}$ with the Todd class of $X$. As a corollary, we obtain a functorial proof of the Grothendieck–Riemann–Roch theorem in Hodge cohomology for complex manifolds.

Let $(M, h)$ be a compact 4-dimensional Einstein manifold, and suppose that $h$ is Hermitian with respect to some complex structure $J$ on $M$. Then either $(M, J, h)$ is Kähler-Einstein, or else, up to rescaling and isometry, it is one of the following two exceptions: the Page metric on $\mathbb{CP}_2\#\overline{\mathbb{CP}}_2$, or the Einstein metric on $\mathbb{CP}_2\#2\overline{\mathbb{CP}}_2$ discovered in "On conformally Kähler, Einstein manifolds," J. Amer. Math. Soc. 21 (2008), pp. 1137–1168.

The Cauchy problem for the homogeneous real/complex Monge–Ampère equation (HRMA/HCMA) arises from the initial value problem for geodesics in the space of Kähler metrics equipped with the Mabuchi metric. This Cauchy problem is believed to be ill-posed and a basic problem is to characterize initial data of (weak) solutions which exist up to time $T$. In this article, we use a quantization method to construct a subsolution of the HCMA on a general projective variety, and we conjecture that it solves the equation for as long as the unique solution exists. The subsolution, called the “quantum analytic continuation potential,” is obtained by (i) Toeplitz quantizing the Hamiltonian flow determined by the Cauchy data, (ii) analytically continuing the quantization, and (iii) taking a certain logarithmic classical limit. We then prove that in the case of torus invariant metrics (where the HCMA reduces to the HRMA) the quantum analytic continuation potential coincides with the well-known Legendre transform potential, and hence solves the equation as long as it is smooth. In the sequel, "The Cauchy problem for the homogeneous Monge–Ampère equation, II. Legendre transform," Adv. Math., 228 (2011), 2989–3025, it is proved that the Legendre transform potential ceases to solve the HCMA once it ceases to be smooth. The results here and in the sequels in particular characterize the initial data of smooth geodesic rays.

In this article, we study local holomorphic isometric embeddings from $\mathbb{B}^n$ into $\mathbb{B}^{N_1} × • • • × \mathbb{B}^{N_m}$ with respect to the normalized Bergman metrics up to conformal factors. Assume that each conformal factor is smooth Nash algebraic. Then each component of the map is a multi-valued holomorphic map between complex Euclidean spaces by the algebraic extension theorem derived along the lines of Mok, and Mok and Ng. Applying holomorphic continuation and analyzing real analytic subvarieties carefully, we show that each component is either a constant map or a proper holomorphic map between balls. Applying a linearity criterion of Huang, we conclude the total geodesy of non-constant components.