A nilmanifold is a homogenous space of a nilpotent Lie group. If M is a compact symplectic nilmanifold, then any 1-periodic Hamiltonian system on M has at least dim(M) + 1 contractible periodic orbits with period 1. This provides an aarmative answer to the Arnold conjecture for such manifolds. The proof uses the techniques of rational homotopy theory, and in particular an extension of the ratio...

A class of examples of Riemannian metrics with holonomy G2 on compact 7-manifolds was constructed by the author in [Ko] using a certain ‘generalized connected sum’ of two asymptotically cylindrical manifolds with holonomy SU(3). We consider, on each of the two initial SU(3)-manifolds, a fibration arising from a Lefschetz pencil of K3 surfaces. The gluing of the two K3 fibrations yields a coasso...

In this paper we study compact Sasaki manifolds in view of transverse Kähler geometry and extend some results in Kähler geometry to Sasaki manifolds. In particular we define integral invariants which obstruct the existence of transverse Kähler metric with harmonic Chern forms. The integral invariant f1 for the first Chern class case becomes an obstruction to the existence of transverse Kähler m...

We extend Turaev’s definition of torsion invariants of 3-dimensional manifolds equipped with non-singular vector fields, by allowing (suitable) tangency circles to the boundary, and manifolds with non-zero Euler characteristic. We show that these invariants apply in particular to (the exterior of) Legendrian links in contact 3-manifolds. Our approach uses a combinatorial encoding of vector fiel...

We study a class of asymptotically cylindrical Ricci-flat Kähler metrics arising on quasiprojective manifolds. Using the Calabi–Yau geometry and analysis and the Kodaira–Kuranishi–Spencer theory and building up on results of N.Koiso, we show that under rather general hypotheses any local asymptotically cylindrical Ricci-flat deformations of such metrics are again Kähler, possibly with respect t...

This article is devoted to a study of flat orbifold vector bundles. We construct bijection between the isomorphic classes proper bundles and equivalence representations fundamental groups base orbifolds. establish Bismut-Zhang like anomaly formula for Ray-Singer metric on determine line cohomology compact with coefficients in an bundle. show that analytic torsion acyclic unitary bundle equal va...

In this paper, we study stability for harmonic foliations on locally conformal Kähler manifolds with complex leaves. We also discuss instability for harmonic foliations on compact submanifolds immersed in Euclidean spaces and compact homogeneous spaces.

We prove a positive mass theorem for spaces which asymptotically approach a flat Euclidean space times a Calabi-Yau manifold (or any special honolomy manifold except the quaternionic Kähler). This is motivated by the very recent work of Hertog-HorowitzMaeda [HHM]. In general relativity, isolated gravitational systems are modelled by asymptotically flat spacetimes. The spatial slices of such spa...

We couple n copies of N = (2, 0) scalar multiplets to a gauged N = (2, 0) supergravity in 2 + 1 dimensions which admits AdS 3 as a vacuum. The scalar fields are charged under the gauged R-symmetry group U (1) and parametrize certain Kahler manifolds with compact or non-compact isometries. The radii of these manifolds are quantized in the compact case, but arbitrary otherwise. In the compact cas...