A complex symplectic structure on a Lie algebra h is an integrable complex structure J with a closed non-degenerate (2, 0)-form. It is determined by J and the real part of the (2, 0)-form. Suppose that h is a semi-direct product g V, and both g and V are Lagrangian with respect to and totally real with respect to J. This note shows that g V is its own weak mirror image in the sense that the associated differential Gerstenhaber algebras controlling the extended deformations of and J are isomorphic. The geometry of (, J) on the semi-direct product g V is also shown to be equivalent to that of a torsion-free flat symplectic connection on the Lie algebra g. By further exploring a relation between (J, ) with hypersymplectic Lie algebras, we find an inductive process to build families of complex symplectic algebras of dimension 8 n from the data of the 4 n-dimensional ones.
It is shown that an HKT space with closed parallel potential 1-form has D (2, 1; 1) symmetry. Every locally conformally hyper-Khler manifold generates this type of geometry. The HKT spaces with closed parallel potential 1-form arising in this way are characterized by their symmetries and an inhomogeneous cubic condition on their torsion.
This article provides a complete description of the differential Gerstenhaber algebras of all nilpotent complex structures on any real six-dimensional nilpotent algebra. As an application, we classify all pseudo-Khlerian complex structures on six-dimensional nilpotent algebras whose differential Gerstenhaber algebra is quasi-isomorphic to that of the symplectic structure. In a weak sense of mirror symmetry, this gives a classication of pseudo-Khler structures on six-dimensional nilpotent algebras whose mirror images are themselves.
The geometry arising from Michelson & Strominger's study of =4<i>B</i> supersymmetric quantum mechanics with superconformal <i>D</i>(2, 1; )-symmetry is a hyperKhler manifold with torsion (HKT) together with a special homothety. It is shown that different parameters are related via changes in potentials for the HKT target spaces. For 0, 1, we describe how each such HKT manifold <i>M</i> <sup> <i>4m</i> </sup> is derived from a space <i>N</i> <sup> <i>4m4</i> </sup> which is quaternionic Khler with torsion and carries an Abelian instanton.
Weak mirror symmetry relates a manifold with complex structure to another manifold equipped with a symplectic structure through a quasiisomorphism of associated differential Gerstenhaber algebras. The two manifolds are then mirror partners. In this paper we consider the analogous problem on Lie algebras. In particular we show that the semi-direct product of a Lie algebra equipped with a torsion-free flat connection with itself is a mirror partner of a semi-direct product of the same Lie algebra with its dual space. For nilpotent algebras this analysis on Lie algebras can be applied to the compact quotients of the underlying nilpotent group. We classify mirror pairs among 6-dimensional nilpotent Lie algebras that have the semi-direct product structure as well as mirror pairs admitting the more involved special Lagrangian structure, namely compatible complex and symplectic structures on the same space.
We present the extended Kuranishi space for a Kodaira surface as a nontrivial example to Kontsevich and Barannikov's extended deformation theory. We provide a non-trivial example of Hertling-Manin's weak Frobenius manifold. In addition, we find that Kodaira surface is its own mirror image in the sense of Merkulov. The calculations of extended deformation and the weak Frobenius structure are based on Merkulov's perturbation method. Our computation of cohomology is done in the context of compact nilmanifolds.
We generalize the Kodaira Embedding Theorem and Chow's Theorem to the context of families of complex supermanifolds. In particular, we show that every family of super Riemann surfaces is a family of projective superalgebraic varieties.
In , Hitchin proved that a compact K/ihlerian twistor space is in fact a projective algebraic manifold. Moreover, it is the twistor space associated to either the Euclidean 4-sphere S 4 or the complex projective plane CF 2 with Fubini-Study metric. The twistor spaces are 3 and the flag of lines in~ p2 respectively. In [10, 11], the author proved the existence of self-dual metric with positive scalar curvature on the connected-sums of two or three copies of the complex projective planes. Their twistor spaces are Moish6zon spaces. In fact, they are the small resolution of the intersection of two quadrics in~ ps with four nodes and the double covering of CP 3 branched along a quartic with thirteen nodes. Recently, the joint work of Donaldson and Friedman  produced a general procedure to construct new twistor spaces and hence self-dual manifolds. In this article, we shall follow the spirit of Hitchin's work and prove the
The concept of a selfdual connection on a fourdimensional Riemannian manifold is generalized to the 4ndimensional case of any quaternionic Khler manifold. The generalized selfdual connections are minima of a modified YangMills functional. It is shown that our definitions give a correct framework for a mapping theory of quaternionic Khler manifolds. The mapping theory is closely related to the construction of YangMills fields on such manifolds. Some monopolelike equations are discussed.
If M is a quaternionic manifold and P is an S 1-instanton over M, then Joyce constructed a hypercomplex manifold we call(M) over M. These hypercomplex manifolds admit a U (2)-action of a special type permuting the complex structures. We show that up to double covers, all such hypercomplex manifolds arise in this way. Examples, including that of a hypercomplex structure on SU (3), show the necessity of including double covers of(M).