18 Facts About Complex geometry

In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers.

In particular, complex geometry is concerned with the study of spaces such as complex manifolds and complex algebraic varieties, functions of several complex variables, and holomorphic constructions such as holomorphic vector bundles and coherent sheaves.

Application of transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis.

Complex geometry sits at the intersection of algebraic geometry, differential geometry, and complex analysis, and uses tools from all three areas.

Additionally, the extra structure of complex geometry allows, especially in the compact setting, for global analytic results to be proven with great success, including Shing-Tung Yau's proof of the Calabi conjecture, the Hitchin–Kobayashi correspondence, the nonabelian Hodge correspondence, and existence results for Kahler–Einstein metrics and constant scalar curvature Kahler metrics.

Complex geometry has significant applications to theoretical physics, where it is essential in understanding conformal field theory, string theory, and mirror symmetry.

Broadly, complex geometry is concerned with spaces and geometric objects which are modelled, in some sense, on the complex plane.

Complex geometry is different in flavour to what might be called real geometry, the study of spaces based around the geometric and analytical properties of the real number line.

However, complex geometry is not typically seen as a particular sub-field of differential geometry, the study of smooth manifolds.

In practice, complex geometry sits in the intersection of differential geometry, algebraic geometry, and analysis in several complex variables, and a complex geometer uses tools from all three fields to study complex spaces.

Typical directions of interest in complex geometry involve classification of complex spaces, the study of holomorphic objects attached to them, and the intimate relationships between complex geometric objects and other areas of mathematics and physics.

Complex geometry is concerned with the study of complex manifolds, and complex algebraic and complex analytic varieties.

Complex geometry Fano variety is a complex algebraic variety with ample anti-canonical line bundle .

Special examples of sheaves used in complex geometry include holomorphic line bundles, holomorphic vector bundles, and coherent sheaves.

Examples of vanishing theorems in complex geometry include the Kodaira vanishing theorem for the cohomology of line bundles on compact Kahler manifolds, and Cartan's theorems A and B for the cohomology of coherent sheaves on affine complex varieties.

Complex geometry makes use of techniques arising out of differential geometry and analysis.

Classification in complex and algebraic geometry often occurs through the study of moduli spaces, which themselves are complex manifolds or varieties whose points classify other geometric objects arising in complex geometry.

Complex geometry is concerned not only with complex spaces, but other holomorphic objects attached to them.