Complex Geometry and String Theory

String theory is a rather new field of theoretical physics. It appeared only twenty years ago to describe phenomenology of strong interactions of elementary particles, and until recently, it has been developing rather slowly. This is because string theory has encountered a number of difficulties that were not easy to overcome; in particular, this theory contained the so-called anomalies that hindered construction of self-consistent string theory. Especially, anomalies led to the breakdown of symmetry properties of this theory after its quantization.