Calculus of variations in complex vector bundles

Complex calculus of variations

In this article, we present a detailed study of the complex calculus of variations introduced in [4]. This calculus is analogous to the conventional calculus of variations, but is applied here to C functions in C. It is based on new concepts involving the minimum and convexity of a complex function. Such an approach allows us to propose explicit solutions to complex Hamilton-Jacobi equations, i...

Weak Equivalence Classes of Complex Vector Bundles

For any complex vector bundle Ek of rank k over a manifold Mm with Chern classes ci ∈ H2i(Mm, Z) and any non-negative integers l1, · · · , lk we show the existence of a positive number p(m, k) and the existence of a complex vector bundle Êk over Mm whose Chern classes are p(m, k) · li · ci ∈ H2i(Mm, Z). We also discuss a version of this statement for holomorphic vector bundles over projective a...

Differential Geometry of Complex Vector Bundles

Complex Vector Bundles and Jacobi Forms

The elliptic genus (EG) of a compact complex manifold was introduced as a holomorphic Euler characteristic of some formal power series with vector bundle coefficients. EG is an automorphic form in two variables only if the manifold is a Calabi–Yau manifold. In physics such a function appears as the partition function of N = 2 superconformal field theories. In these notes we define the modified ...

On the classification of complex vector bundles of stable rank

Abstract. One describes, using a detailed analysis of Atiyah–Hirzebruch spectral sequence, the tuples of cohomology classes on a compact, complex manifold, corresponding to the Chern classes of a complex vector bundle of stable rank. This classification becomes more effective on generalized flag manifolds, where the Lie algebra formalism and concrete integrability conditions describe in constru...