We determine a global structure of the moduli space of self-dual metrics on 3CP2 satisfying the following three properties: (i) the scalar curvature is of positive type, (ii) they admit a non-trivial Killing field, (iii) they are not conformal to the LeBrun’s self- dual metrics based on the ‘hyperbolic ansatz’. We prove that the moduli space of these metrics is isomorphic to an orbifold R3/G, where G is an involution of R3 having two-dimensional fixed locus. In particular, the moduli space is non-empty and connected. We also remark that Joyce’s self-dual metrics with torus symmetry appear as a limit of our self-dual metrics. Our proof of the result is based on the twistor theory. We first determine a defining equation of a projective model of the twistor space of the metric, and then prove that the projective model is always birational to a twistor space, by determining the family of twistor lines. In determining them, a key role is played by a classical result in algebraic geometry that a smooth plane quartic always possesses twenty-eight bitangents.

In [SW2], we defined a generalized mean curvature vector field on any almost Lagrangian submanifold with respect to a torsion connection on an almost K\"ahler manifold. The short time existence of the corresponding parabolic flow was established. In addition, it was shown that the flow preserves the Lagrangian condition as long as the connection satisfies an Einstein condition. In this article, we show that the canonical connection on the cotangent bundle of any Riemannian manifold is an Einstein connection (in fact, Ricci flat). The generalized mean curvature vector on any Lagrangian submanifold is related to the Lagrangian angle defined by the phase of a parallel (n, 0) form, just like the Calabi-Yau case. We also show that the corresponding Lagrangian mean curvature flow in cotangent bundles preserves the exactness and the zero Maslov class conditions. At the end, we prove a long time existence and convergence result to demonstrate the stability of the zero section of the cotangent bundle of spheres.

We prove the Poisson geometric version of the Local Reeb Stability (from foliation theory) and of the Slice Theorem (from equivariant geometry), which is also a generalization of Conn’s linearization theorem.

We construct Coleff–Herrera products and Bochner–Martinelli type residue currents associated with a tuple$f$of weakly holomorphic functions, and show that these currents satisfy basic properties from the (strongly) holomorphic case. This include the transformation law, the Poincaré–Lelong formula and the equivalence of the Coleff–Herrera product and the Bochner–Martinelli type residue current associated with$f$when$f$defines a complete intersection.

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