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The vanishing point theorem is the principal theorem in the science of perspective. It says that the image in a picture plane π of a line L in space, not parallel to the picture, is determined by its intersection with π and its vanishing point. Some authors have used the phrase, "the image of a line includes its vanishing point".
In algebraic geometry, a vanishing theorem gives conditions for coherent cohomology groups to vanish. Andreotti–Grauert vanishing theorem; Bogomolov–Sommese vanishing theorem; Grauert–Riemenschneider vanishing theorem; Kawamata–Viehweg vanishing theorem; Kodaira vanishing theorem; Le Potier's vanishing theorem; Mumford vanishing theorem
Serre's vanishing theorem says that for any ample line bundle on a proper scheme over a Noetherian ring, and any coherent sheaf on , there is an integer such that for all , the sheaf is spanned by its global sections and has no cohomology in positive degrees.
For example, a Fourier series of sine and cosine functions, all continuous, may converge pointwise to a discontinuous function such as a step function. Carmichael's totient function conjecture was stated as a theorem by Robert Daniel Carmichael in 1907, but in 1922 he pointed out that his proof was incomplete. As of 2016 the problem is still open.
Kawamata–Viehweg vanishing theorem (algebraic geometry) Kodaira embedding theorem (algebraic geometry) Kodaira vanishing theorem (complex manifold) Lefschetz theorem on (1,1)-classes (algebraic geometry) Local invariant cycle theorem (algebraic geometry) Malgrange–Zerner theorem (complex analysis) Newlander–Niremberg theorem (differential ...
Kawamata–Viehweg vanishing theorem; Kawasaki's Riemann–Roch formula; Keel–Mori theorem; Kempf vanishing theorem; Kempf–Ness theorem; Kodaira embedding theorem; Kodaira vanishing theorem; Krivine–Stengle Positivstellensatz
Five color theorem; Five lemma; Fundamental theorem of arithmetic; Gauss–Markov theorem (brief pointer to proof) Gödel's incompleteness theorem. Gödel's first incompleteness theorem; Gödel's second incompleteness theorem; Goodstein's theorem; Green's theorem (to do) Green's theorem when D is a simple region; Heine–Borel theorem ...
Note that for large m the line bundle K M ⊗ L ⊗m is very ample by the standard Serre's vanishing theorem (and its complex analytic variant). Fujita conjecture provides an explicit bound on m, which is optimal for projective spaces.