Complex affine space

Complex manifolds

Affine geometry, broadly speaking, is the study of the geometrical properties of lines, planes, and their higher dimensional analogs, in which a notion of "parallel" is retained, but no metrical notions of distance or angle are. Affine spaces differ from linear spaces (that is, vector spaces) in that they do not have a distinguished choice of origin. So, in the words of Marcel Berger, "An affine space is nothing more than a vector space whose origin we try to forget about, by adding translations to the linear maps." Accordingly, a complex affine space, that is an affine space over the complex numbers, is like a complex vector space, but without a distinguished point to serve as the origin. Affine geometry is one of the two main branches of classical algebraic geometry, the other being projective geometry. A complex affine space can be obtained from a complex projective space by fixing a hyperplane, which can be thought of as a hyperplane of ideal points "at infinity" of the affine space. To illustrate the difference (over the real numbers), a parabola in the affine plane intersects the line at infinity, whereas an ellipse does not. However, any two conic sections are projectively equivalent. So a parabola and ellipse are the same when thought of projectively, but different when regarded as affine objects. Somewhat less intuitively, over the complex numbers, an ellipse intersects the line at infinity in a pair of points while a parabola intersects the line at infinity in a single point. So, for a slightly different reason, an ellipse and parabola are inequivalent over the complex affine plane but remain equivalent over the (complex) projective plane. Any complex vector space is an affine space: all one needs to do is forget the origin (and possibly any additional structure such as an inner product). For example, the complex n-space can be regarded as a complex affine space, when one is interested only in its affine properties (as opposed to its linear or metrical properties, for example). Since any two affine spaces of the same dimension are isomorphic, in some situations it is appropriate to identify them with , with the understanding that only affinely-invariant notions are ultimately meaningful. This usage is very common in modern algebraic geometry. (Wikipedia).

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Complex affine space

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