International Journal of Mathematics

We present a novel $C^0$-characterization of symplectic embeddings and diffeomorphisms in terms of Lagrangian embeddings. Our approach is based on the shape invariant, which was discovered by Sikorav and Eliashberg, intersection theory and the displacement energy of Lagrangian submanifolds, and the fact that non-Lagrangian submanifolds can be displaced immediately. This characterization gives rise to a new proof of $C^0$-rigidity of symplectic embeddings and diffeomorphisms. The various manifestations of Lagrangian rigidity that are used in our arguments come from $J$-holomorphic curve methods. An advantage of our techniques is that they can be adapted to a $C^0$-characterization of contact embeddings and diffeomorphisms in terms of coisotropic (or pre-Lagrangian) embeddings, which in turn leads to a proof of $C^0$-rigidity of contact embeddings and diffeomorphisms. We give a detailed treatment of the shape invariants of symplectic and contact manifolds, and demonstrate that shape is often a natural language in symplectic and contact topology. We consider homeomorphisms that preserve shape, and propose a hierarchy of notions of Lagrangian topological submanifold. Moreover, we discuss shape-related necessary and sufficient conditions for symplectic and contact embeddings, and define a symplectic capacity from the shape.

Every Finsler metric on a differential manifold induces a spray. The converse is not true. Therefore, it is one of the most fundamental problems in spray geometry to determine whether a spray is induced by a Finsler metric which is regular, but not necessary positive definite. This problem is called inverse problem. This paper discuss inverse problem of sprays with scalar curvature. In particular, we show that if such a spray $G$ on a manifold is of vanishing $H$-curvature, but $G$ has not isotropic curvature, then $G$ is not induced by any (not necessary positive definite) Finsler metric. We also find infinitely many sprays on an open domain $U\subset {ℝ}^n$ with scalar curvature and vanishing $H$-curvature, but these sprays have no isotropic curvature. This contrasts sharply with the situation in Finsler geometry.

We introduce a concept of blown-up Čech cohomology for coherent sheaves of homological dimension $\le 1$ and some quasi-coherent sheaves on a nonsingular real affine variety. Its construction involves a directed set of multi-blowups. We establish, in particular, long exact cohomology sequence and Cartan’s Theorem B. Finally, some applications are provided, including universal solution to the first Cousin problem (after blowing up).

We study the Neumann problem for complex special Lagrangian equations with critical phase. A priori derivative estimates up to order two are established. And we obtain the existence and uniqueness theorem by the method of continuity.

We prove that O’Grady’s birational maps [K. G O’Grady, The weight-two Hodge structure of moduli spaces of sheaves on a K3 surface, J. Algebr. Geom. 6(4) (1997) 599–644] between moduli of sheaves on an elliptic K3 surface can be interpreted as intermediate wall-crossing (wall-hitting) transformations at so-called totally semistable walls, studied by Bayer and Macrì [A. Bayer and E. Macrì, MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations, Inventiones Mathematicae 198(3) (2014) 505–590]. As a key ingredient, we describe the first totally semistable wall for ideal sheaves of $n$ points on the elliptic $K3$. As an application, we give new examples of strange duality isomorphisms, based on a result of Marian and Oprea [A. Marian and D. Oprea, Generic strange duality for K3 surfaces, with an appendix by Kota Yoshioka, Duke Math. J. 162(8) (2013) 1463–1501].

We study in this paper the local rigidity proprieties of deformation parameters of the natural action of a discontinuous group $\mathrm{\Gamma }\subset G_n$ acting on a homogeneous space $G_n/H$, where $H$ stands for a closed subgroup of the Heisenberg motion group $G_n:={𝕌}_n⋉{ℍ}_n$. That is, the parameter space admits a locally rigid (equivalently a strongly locally rigid) point if and only if $\mathrm{\Gamma }$ is finite. Moreover, Calabi–Markus’s phenomenon and the question of existence of compact Clifford–Klein forms are also studied.