The pseudo-Riemannian manifold M = (M, g), n ≥ 2 is paraquaternionic Kähler if hol(M) ⊂ sp(n,R)⊕sp(1, R). If hol(M) ⊂ sp(n, R), than the manifold M is called para-hyperKähler. The other possible definitions of these manifolds use certain parallel para-quaternionic structures in End(TM), similarly to the quaternionic case. In order to relate these different definitions we study para-quaternionic...

Let (M,ω0) be a compact Kähler manifold of dimension n and D a divisor of M with at most normal crossing singularities . Let D = ∪i=1Di , where the Di are smooth divisors of M . Denote M \ D by M , called a quasi-compact Kähler manifold. j : M → M is the inclusion mapping. According to [3], one can construct a complete Kähler metric ω on M which is a Poincaré-like metric near the divisor and of...

The scalars of a N = 1 supersymmetric σ-model in 4 dimensions parametrize a Kähler manifold. This Kähler manifold often takes the form of a tensor product of bundles over a basis manifold. When one of these bundles is a line bundle, the corresponding Kähler form has to satisfy a quantization condition when the topology of the submanifold is non-trivial. This results in a consistency restriction...

We prove a simple, explicit formula for the mass of any asymptotically locally Euclidean (ALE) Kähler manifold, assuming only the sort of weak fall-off conditions required for the mass to actually be well-defined. For ALE scalar-flat Kähler manifolds, the mass turns out to be a topological invariant, depending only on the underlying smooth manifold, the first Chern class of the complex structur...

We prove that for every compact Kähler manifold X the cup product H ∗(X, TX)⊗ H ∗(X,Ω∗X ) → H ∗(X,Ω∗−1 X ) can be lifted to an L∞-morphism from the Kodaira-Spencer differential graded Lie algebra to the suspension of the space of linear endomorphisms of the singular cohomology of X. As a consequence we get an algebraic proof of the principle “obstructions to deformations of compact Kähler manif...

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...

Let M = M̃/Γ be a Kähler manifold, where Γ ' π1(M) and where M̃ is the universal Kähler cover. Let (L, h) → M be a positive Hermitian holomorophic line bundle. We first prove that the L Szegő projector Π̃N for L holomorphic sections on the lifted bundle L̃ → M̃ is related to the Szegő projector for H(M,L ) by Π̂N (x, y) = ∑ γ∈Γ ̃̂ ΠN (γ · x, y). We apply this result to give a simple proof of Napier’s t...

A Riemannian manifold is said to be Kähler if the holonomy group is contained in U(n). It is quaternion Kähler if the holonomy group is contained in Sp(n)Sp(1). It is known that quaternion Kähler manifolds of dimension ≥ 8 are Einstein, so the scalar curvature s splits these manifolds according to whether s > 0, s = 0 or s < 0. Ricci flat quaternion Kähler manifolds include hyperkähler manifold...