The notion of stability plays a central role in complex and algebraic geometry.It was introduced by D. Mumford [5] and F. Takemoto [10] for the study of the moduli space of holomorphic vector bundles; S. Kobayshi and M. Lύbke found that for irreducible bundles the existence of a Hermitian-Einstein metric is a sufficient condition for stability, and a major achievement of the theory has consisted in the work of M. Narashimhan and C. Seshadri for algebraic curves, S. Donaldson in the case of algebraic manifolds, K. Uhlenbeck and S.T. Yau for general Kahler manifolds (easily extended to regularized Hermitian n-manifolds, i.e., whose Kahler form η satisfies ddη n ~ι = 0) proving the existence of a Hermitian-Einstein connection on stable holomorphic vector bundles ([6], [1], [12]). Further generalization to Higgs bundles can be found in [2] and [9].These results have made the tools of differential geometry available to complex and algebraic geometry, leading to several important applications, e.g., a much more extensive comprehension of Bogomolov-Gieseker type inequalities and the characterization of flat vector bundles. On the other hand, a general theory of the existence of holomorphic structures on complex bundles is far from being understood, and therefore it is very natural to try to extend the differential geometric characterization of stability to complex bundles with an unnecessarily integrable almost complex structure.The first main result of the present paper is the following. Theorem 0.1. Assume a complex vector bundle over a compact almost Hermitian regularized manifold is equipped with a stable almost complex structure. Then it admits a Hermitian-Einstein connection.The notion of stability which we consider is the following: we require that μ(F) < μ(E) holds for any J-holomorfic subbundle F C E which

We characterize the special Lagrangian submanifolds of a generalized Calabi–Yau manifold, with vanishing Maslov class. Then, we carefully describe several examples, including a non-Kähler generalized Calabi–Yau manifold foliated by special Lagrangian submanifolds.

It has been known for some time now that not every compact kfihler manifold of positive first Chern class admits a k/ihler-einstein metric, or even a k/ihler metric of constant scalar curvature. This is due to structure theorems of Matsushima and Lichnerowicz on the algebra of holomorphic vector fields on M. that is, the (1, 0)-component of the gradient of the scalar curvature is a holomorphic vector field. The problem of finding extremal metrics is quite natural but quite difficult. Extremal metrics should be easier to find than k/ihler-einstein metrics or metrics of constant scalar curvature. Nevertheless, Calabi has proved some (weaker) structure theorems for the algebra of holomorphic vector fields on an M with an extremal kfihler metric, and M. Levine 1-8] has shown that these conditions are sufficient to obstruct the existence of an extremal metric on some M with the "wrong kinds" of algebras. In a different direction, Futaki has studied the very interesting interrelationship between the algebra of holomorphic vector fields and the given k/ihler class [m] which was fixed in the definition above.In this note, we give examples of ruled surfaces M which have no non-trivial holomorphic vector fields, and yet which admit no extremal kfihler metric in a specifically given k/ihler class. For such an example, an extremal metric would * Partially supported by the National Science Foundation (LISA) 404 D. Burns and P. de Bartolomeis necessarily be a metric of constant scalar curvature, and the obstruction found here in new in that context as well. The obstruction involves the borderline semi-stability properties of hermitian vector bundles with hermite-einstein connections (cf., e.g., [7,9]). We came across these examples as an empirical off-shoot of our work on the integrability of twistor spaces over four-manifolds (cf. [2]). We have not been able to digest a simple general principle from the calculations, but it is clear that the borderline stability properties play the key role.Acknowledgement. The authors would like to thank E. Calabi for the interest he has shown in this work.