dimC

dimC is a notation encountered in several branches of mathematics to indicate a dimension taken over the complex numbers. Because it is used in different contexts, its exact meaning is context dependent, though the common idea is the size or complexity of a structure when viewed as a complex-number object.

In linear algebra, dimC(V) denotes the complex dimension of a vector space V over the field of complex numbers C. If V has complex dimension d, then as a real vector space its dimension is 2d. This notation emphasizes that the basis and linear operations are defined with complex scalars.

In complex analysis and algebraic geometry, dimC is used to denote the complex dimension of a complex manifold or a complex-analytic variety. For a complex manifold, the complex dimension is the dimension of its tangent space when viewed as a complex vector space; the corresponding real dimension is twice the complex dimension. This concept contrasts with real dimension, which is used for real manifolds or when the complex structure is not essential.

In category theory and homological algebra, dim C can refer to the Rouquier dimension of a triangulated category C. This invariant measures the minimal number of steps required to generate the category from a single object using cones, shifts, and taking direct summands. It provides a sense of the category’s structural complexity and has applications in representation theory and algebraic geometry.

Because dimC arises in multiple areas, its precise definition should be inferred from the surrounding mathematical context. See also: dimension, dim, dimR, dimC in complex geometry, and Rouquier dimension.