HOLOMORPHY AND SEMIHOLOMORPHY

Abstract

Connection is one of the most important mathematical notions. Donaldson’s theory of autodual connections and instantons shows that this notion is especially important for algebraic geometry. The classical Donaldson theorem states that the following notions are equivalent: autodual connection for Kahler’s metric over an algebraic surface and stability via class of Kahler’s metric of holomorphic bundles of the same topological type. This example is very visual. We hope that the consideration of a new object (connection spaces) is justified and will lead us to results that have long been expected. This article can be considered as a small illustration of such a connection in higher dimensions over complex manifolds of special type. In December 1996, in his lectures at the conference “Gauge theories” in Cambridge, S. Donaldson suggested a program of complexification of “classical” gauge theory in dimensions 2, 3, and 4. A group of mathematicians from Oxford (P. Thomas and K. Lewis) is working on this program under his supervision. A preliminary decription of basic constructions is given in the preprint by S. Donaldson and P. Thomas, Gauge theory in higher dimensions. A part of this theory is a generalization of the theory of autodual connections on 4-dimensional Riemann manifolds. Examining this construction, we note that it is possible to apply it to the proof of Hodge’s conjecture for 4-dimensional Calabi–Yau manifolds, for instance, for the product of K3-surfaces. In this article, we present some results about semiholomorphic structures on complex bundles over 4-dimensional Calabi–Yau manifolds and investigate the connection of them with the usual holomorphic structures. Traditionally, an algebraic 4-dimensional variety with a trivial canonical class is called a Calabi–Yau manifold. But this can lead to ambiguity. For instance, in the further constructions we do not use a Ricci flat metric on these manifolds, but use only a complex orientation, i.e., a trivial canonical bundle. We are not able to overcome a stable tradition introduced by physicists, who persistently call the variety with a trivial canonical class a Calabi–Yau manifold. In many aspects, the new theory is a “complexification” of the classical one. This can be seen especially clearly at the level of Lagrangians. Therefore, we cover that in a special section. The visuality of Lagrange’s approach in analytic tasks (adopted from physical arrangments of the Yang–Mills theory) indeed clarifies the situation and leads to interesting results with small effort. This method has been used in the classical Donaldson theory. So first we recall some basic construction of it. Let X be a smooth orientable compact 4-dimensional manifold. The choice of the Riemann metric g and the orientation defines the real Hodge operator