1. Introduction. To determine cohomology groups of holomorphic vector bundles or more general coherent analytic sheaves on complex manifolds is very important in several complex variables and complex geometry. For example, Cartan-Serre's theorem B and Kodaira's vanishing theorem are fundamental respectively in the studies of two classes of complex manifolds: Stein manifolds and projective algeb...

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In this paper, we will study the asymptotic geometry of 4-dimensional steady gradient Ricci solitons under condition that they dimension reduce to 3-manifolds. We show such either strongly a spherical space form S3/Γ or weakly 3-dimensional Bryant soliton. also soliton singularity models with nonnegative curvature outside compact set are Ricci-flat ALE 4-manifolds manifolds. As further applicat...

The holonomy of the Bismut connection on Vaisman manifolds is studied. We prove that if $$M^{2n}$$ endowed with a structure, then group contained in $${\text {U}}(n-1)$$ . compute explicitly this for particular types manifolds, namely, solvmanifolds and some classical Hopf manifolds.

2 Local symplectic, complex and Kähler geometry: a quick review 10 2.1 Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Symplectic manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Symplectic quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Complex manifolds . . . . . . . . . . . . . . ....

For k ≥ 2, we exhibit complete k-curvature homogeneous neutral signature pseudo-Riemannian manifolds which are not locally affine homogeneous (and hence not locally homogeneous). The curvature tensor of these manifolds is modeled on that of an indecomposible symmetric space. All the local scalar Weyl curvature invariants of these manifolds vanish. Dedicated to Professor Sekigawa on his 60th bir...

4 Super symmetric quantum field theories 32 4.1 Super manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.2 Super Riemannian structures on 1|1-manifolds . . . . . . . . . 36 4.3 Super Riemannian structures on 2|1-manifolds . . . . . . . . . 40 4.4 The super bordism category RB . . . . . . . . . . . . . . . . 46 4.5 Extending TV to a super category . . . . . . . . . . . . . . . . ...

We address the problem of computing the fundamental group of a symplectic S-manifold for non-Hamiltonian actions on compact manifolds, and for Hamiltonian actions on non-compact manifolds with a proper moment map. We generalize known results for compact manifolds equipped with a Hamiltonian S-action. Several examples are presented to illustrate our main results.