Related papers: Every projective Oka manifold is elliptic

We generalize our elliptic characterization of Oka manifolds to Oka maps. The generalized characterization can be considered as an affirmative answer to the relative version of Gromov's conjecture. As an application, we unify previously…

We give the following positive answer to Gromov's question (in "Oka's principle for holomorphic sections of elliptic bundles", J. Amer. Math. Soc. 2, 851-897 (1989), 3.4.(D), page 881). THEOREM: If every holomorphic map from a compact…

We prove a homotopy theorem for sheaves. Its application shortens and simplifies the proof of many Oka principles such as Gromov's Oka principle for elliptic submersions.

The basic result of Oka theory, due to Gromov, states that every continuous map $f$ from a Stein manifold $S$ to an elliptic manifold $X$ can be deformed to a holomorphic map. It is natural to ask whether this can be done for all $f$ at…

We will give a new proof for the Gromov's theorem on almost flat manifolds, which is an inductive proof on dimension.

It was conjectured by Bott-Grove-Halperin that a compact simply connected Riemannian manifold $M$ with nonnegative sectional curvature is rationally elliptic. We confirm this conjecture under the stronger assumption that $M$ has entire…

We prove that the Chekhov-Eynard-Orantin recursion on the mirror curve of $\mathbb{P}^1$ encodes all genus equivariant open Gromov-Witten invariants of $(\mathbb{P}^1, \mathbb{R}\mathbb{P}^1)$. This result can be viewed as an all genus…

In this paper we proved the following: \emph{Let $K, L\subset \mathbb R^3$ be two $O$-symmetric convex bodies with $L\subset \emph{int} K$ strictly convex. Suppose that from every $x$ in $\emph{bd} K$ the graze $\Sigma(L,x)$ is a planar…

We show that a complex projective manifold X which satisfies the Gromov's h-principle is `special', and raise some questions about the reverse implication, the extension to the quasi-K\" ahler case, and the relationships of these properties…

In this paper, we prove Gromov's flat corner domination conjecture in all dimensions. As a consequence, we answer positively the Stoker conjecture for convex Euclidean polyhedra in all dimensions. By applying the same techniques, we also…

Over the past decade, the class of Oka manifolds has emerged from Gromov's seminal work on the Oka principle. Roughly speaking, Oka manifolds are complex manifolds that are the target of "many" holomorphic maps from affine spaces. They are…

In this paper, we gave some explicit relations between absolute and relative Gromov-Witten invariants. We proved that a symplectic manifold is symplectic rationally connected if it contains a positive divisor symplectomorphic to $P^n$.

This paper provides an introduction to exploded manifolds. The category of exploded manifolds is an extension of the category of smooth manifolds with an excellent holomorphic curve theory. Each exploded manifold has a tropical part which…

Gromov, in his seminal 1989 paper on the Oka principle, proved that every continuous map from a Stein manifold into an elliptic manifold is homotopic to a holomorphic map. Previously we have shown that, given a continuous map $X \to…

We survey on algebraically elliptic varieties in the sense of Gromov.

We prove that all elliptic curves defined over real quadratic fields are modular.

In this paper, we establish the convergence for Gromov-Witten invariant of elliptic orbifold $\mathbb{P}^1$ with type $(3,3,3), (4,4,2)$ and $(6,3,2)$. We also prove the mirror theorems of Gromov-Witten theory for those orbifolds and FJRW…

We prove that the universal covering space of a complex projective manifold is holomorphically convex provided its fundamental group is linear.

In this note, we prove that every fibre space structures of a projective irreducible symplectic manifold is a lagrangian fibration.

Orbifold groupoids have been recently widely used to represent both effective and ineffective orbifolds. We show that every orbifold groupoid can be faithfully represented on a continuous family of finite dimensional Hilbert spaces. As a…