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...

In this article and in its sequel we propose the study of certain discretizations of geometric evolution equations as an approach to the study of the existence problem of some elliptic partial differential equations of a geometric nature as well as a means to obtain interesting dynamics on certain infinite-dimensional spaces. We illustrate the fruitfulness of this approach in the context of the...

We show that conformal manifolds in $d\geq 3$ field theories with at least 4 supercharges are K\"ahler-Hodge, thus extending to 3d ${\cal N}=2$ and 4d N}=1$ similar results previously derived for N}=4$ various types of 2d SCFTs. Conformal SCFTs equipped a holomorphic line bundle L}$, which encodes the operator mixing under marginal deformations. Using perturbation theory superconformal Ward ide...

We study the orientifold truncation that arises when compactifying type II string theory on Calabi-Yau orientifolds with O3/O7-planes, in the context of supergravity. We look at the N=2 to N=1 reduction of the hypermultiplet sector of N=2 supergravity under the truncation, for the case of very special quaternionic-Kähler target space geometry. We explicitly verify the Kähler structure of the tr...

We construct left invariant quaternionic contact (qc) structures on Lie groups with zero and non-zero torsion and with non-vanishing quaternionic contact conformal curvature tensor, thus showing the existence of non-flat quaternionic contact manifolds. We prove that the product of the real line with a seven dimensional manifold, equipped with a certain qc structure, has a quaternionic Kähler me...

In this paper, we derive a partial result related to a question of Professor Yau: “Does a simply-connected complete Kähler manifold M with negative sectional curvature admit a bounded non-constant holomorphic function?” Main Theorem. Let M be a simply-connected complete Kähler manifold M with negative sectional curvature ≤ −1 and S∞(M) be the sphere at infinity of M . Then there is an explicit ...

In this note, we will show that the fundamental group of any negatively δ-pinched (δ > 14) manifold can’t be the fundamental group of a quasi-compact Kähler manifold. As a consequence of our proof, we also show that any nonuniform lattice in F4(−20) cannot be the fundamental group of a quasi-compact Kähler manifold. The corresponding result for uniform lattices was proved by Carlson and Hernánd...

It is shown that the Hermitian-symmetric space CP1 × CP1 × CP1 and the flag manifold F1,2 endowed with any left invariant metric admit no compatible integrable almost complex structures (even locally) different from the invariant ones. As an application it is proved that any stable harmonic immersion from F1,2 equipped with an invariant metric into an irreducible Hermitian symmetric space of co...

We review our present understanding of heterotic compactifications on non-Kähler complex manifolds with torsion. Most of these manifolds can be obtained by duality chasing a consistent F-theory compactification in the presence of fluxes. We show that the duality map generically leads to non-Kähler spaces on the heterotic side, although under some special conditions we recover Kähler compactific...