We describe the potential functions for N= 4B supersymmetric quantum mechanics with D (2, 1; ) symmetry.

Using deformations of singular twistor spaces, a generalisation of the connected sum construction appropriate for quaternionic manifolds is introduced. This is used to construct examples of quaternionic manifolds which have no quaternionic symmetries and leads to examples of quaternionic manifolds whose twistor spaces have arbitrary algebraic dimension.

In this paper, we show that any compact Kähler manifold homotopic to a compact Riemannian manifold with negative sectional curvature admits a Kähler–Einstein metric of general type. Moreover, we prove that, on a compact symplectic manifold X homotopic to a compact Riemannian manifold with negative sectional curvature, for any almost complex structure J compatible with the symplectic form, there is no non-constant J-holomorphic entire curve f:C→X.

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We prove that the first Chern form of the moduli space of polarized Calabi-Yau manifolds, with the Hodge metric or the Weil-Petersson metric, represent the first Chern class of the canonical extensions of the tangent bundle to the compactification of the moduli space with normal crossing divisors.