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This is an audio version of the Wikipedia Article: https://en.wikipedia.org/wiki/Complex_affine_space 00:03:48 1 Affine structure 00:05:46 1.1 Affine functions 00:09:32 2 Low-dimensional examples 00:09:44 2.1 One dimension 00:11:09 2.2 Two dimensions 00:15:15 2.3 Four dimensions 00:16:52 3 Affine coordinates 00:18:01 4 Associated projective space 00:19:53 5 Structure group and automorphisms 00:20:30 6 Complex structure 00:20:47 7 Topologies 00:20:59 8 Sheaf of analytic functions 00:23:44 9 See also 00:24:24 10 References 00:27:59 Sheaf of analytic functions 00:30:02 See also Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago. Learning by listening is a great way to: - increases imagination and understanding - improves your listening skills - improves your own spoken accent - learn while on the move - reduce eye strain Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone. Listen on Google Assistant through Extra Audio: https://assistant.google.com/services/invoke/uid/0000001a130b3f91 Other Wikipedia audio articles at: https://www.youtube.com/results?search_query=wikipedia+tts Upload your own Wikipedia articles through: https://github.com/nodef/wikipedia-tts Speaking Rate: 0.7034991345960246 Voice name: en-GB-Wavenet-D "I cannot teach anybody anything, I can only make them think." - Socrates SUMMARY ======= 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 C n {\displaystyle \mathbb {C} ^{n}} 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 C n {\displaystyle \mathbb {C} ^{n}} , with the understanding that only affinely-invariant notions are ultimately meaningful. This usage is very common in modern algebraic geometry.
