Complex manifold in the context of K3 surface

In mathematics, the complex projective plane, usually denoted ⁠${\displaystyle \mathbb {P} ^{2}(\mathbb {C} )}$⁠ or ⁠${\displaystyle \mathbb {CP} ^{2},}$⁠ is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three complex coordinates

where, however, the triples differing by an overall rescaling are identified:

In mathematics, complex dimension usually refers to the dimension of a complex manifold or a complex algebraic variety. These are spaces in which the local neighborhoods of points (or of non-singular points in the case of a variety) are modeled on a Cartesian product of the form ${\displaystyle \mathbb {C} ^{d}}$ for some ${\displaystyle d}$, and the complex dimension is the exponent ${\displaystyle d}$ in this product. Because ${\displaystyle \mathbb {C} }$ can in turn be modeled by ${\displaystyle \mathbb {R} ^{2}}$, a space with complex dimension ${\displaystyle d}$ will have real dimension ${\displaystyle 2d}$. That is, a smooth manifold of complex dimension ${\displaystyle d}$ has real dimension ${\displaystyle 2d}$; and a complex algebraic variety of complex dimension ${\displaystyle d}$, away from any singular point, will also be a smooth manifold of real dimension ${\displaystyle 2d}$.

However, for a real algebraic variety (that is a variety defined by equations with real coefficients), its dimension refers commonly to its complex dimension, and its real dimension refers to the maximum of the dimensions of the manifolds contained in the set of its real points. The real dimension is not greater than the dimension, and equals it if the variety is irreducible and has real points that are nonsingular.For example, the equation ${\displaystyle x^{2}+y^{2}+z^{2}=0}$ defines a variety of (complex) dimension 2 (a surface), but of real dimension 0 — it has only one real point, (0, 0, 0), which is singular.

In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may be generalized to categories with more structure than smooth manifolds, such as complex manifolds, or (in the form of cotangent sheaf) algebraic varieties or schemes. In the smooth case, any Riemannian metric or symplectic form gives an isomorphism between the cotangent bundle and the tangent bundle, but they are not in general isomorphic in other categories.