Talk:Tautological line bundle

In complex geometry (see for example the book "Complex Geometry" by Huybrechts) the canonical line bundle of a complex manifold $X$ with complex dimension $n$ is just $\Wedge^n T^{(1,0) \ *}X$ (i.e. the top exterior power of the dual of the holomorphic tangent space). Notice that this is a complex line bundle, hence each fiber is a 2-dimensional real space.

In complex geometry, a tautological line bundle over a complex projective space is the dual of what is being called the canonical line bundle here. At least this is the definition found in Huybrechts' book. Perhaps a warning note should be placed in this article. —The preceding unsigned comment was added by 155.198.157.113 (talk • contribs) 19:48, 19 July 2006 (UTC)


 * That concept is described at canonical bundle. -- Fropuff 06:54, 14 February 2007 (UTC)

Name change proposal
I propose we move this article to tautological line bundle and then either redirect canonical line bundle to canonical line bundle or make it into a disambig page. I would prefer the former. We can put a disambig notice at the top of the canonical bundle article. -- Fropuff 06:54, 14 February 2007 (UTC)


 * I agree. Geometry guy 00:47, 14 May 2007 (UTC)


 * Yes, definitely. Also the term canonical line bundle is ambiguous even within algebraic geometry.  There's the Canonical sheaf to deal with (which, it just so happens, is also a line bundle).  I think tautological is a better term for what the present article deals with, since it really isn't canonical at all: it depends on the structure of projective space.  Silly rabbit 19:04, 25 May 2007 (UTC)

Correct definition
The definition of the tautological bundle is missing the subset $$\lbrace(x,0):x\in\mathbb{CP}^{n}\rbrace$$ since the vector $$0$$ don't belong to any class. NeoBeowulf (talk) 14:57, 29 September 2014 (UTC)