Talk:Positive form

Latest comment: 2 years ago by 2001:1711:FA4B:E5C0:388B:6291:6B8F:C58B in topic Contradiction between 2. and 3.

Please check the signs, I tend to mix them up. Tiphareth 17:13, 11 April 2007 (UTC)Reply

Should Condition 1 for a real positive (1,1)-form not be that is the imaginary part of a hermitian form, not ? (The imaginary part of is , not .) -- Hiferator (talk) 20:11, 30 March 2015 (UTC)Reply

According to Lazarsfeld's "Positivity in Algebraic Geometry I", for some positive-defninite hermitian metric . So I will change that.
Lazarsfeld seems to be using a slightly stronger notion of positivity. (He also states an equivalent property to 2. with strictly positive .) Is there a canonical way to differentiate the two, e.g. calling one strictly positive? I think both notions should be mentioned in the article. -- Hiferator (talk) 20:52, 30 March 2015 (UTC)Reply
By the way, is there a particular reason for writing instead of for the imaginary unit? To me using looks more clear. -- Hiferator (talk) 21:06, 30 March 2015 (UTC)Reply

A down to earth interpretation

edit

Is this statement true?

Suppose that a divisor   on   generates a positive line bundle  . Then if   is any homology class of   that can be represented by a embedded Riemannian surface (perhaps with singularities), then   is greater or equal to zero.

in case this question is true, is this the main point of positive line bundles? That is, that if it has a divisor representing it, then it will have positive intersection with any other class that is represented by a complex submfld.

ELSE:

If L is represented by a divisor, then   where   is a complex subvariety. Hence, if we want to know if   for some divisor   the first thing to check is if   is positive since every   is so.

is this the point of positive line bundles? 14:49, 27 February 2008 (UTC) — Preceding unsigned comment added by 155.198.157.118 (talk)

The second of the above statements is False. I capitalized "false" because there are easy counterexamples. Blow up any complex surface, say  , at a point. Let E be the exceptional set. Let L=[E]. Then  .
I think the usual definition of positive is that   is represented by a positive definite, ie Kähler form  , rather than just positive semi-definite. —Preceding unsigned comment added by 76.24.20.200 (talk) 08:39, 10 October 2008 (UTC)Reply
'"I think the usual definition of positive is that c_1(L) is represented by a positive definite"' -- this is correct, thanks for pointing this out. My error. Fixed. Tiphareth (talk) 07:33, 12 October 2008 (UTC)Reply

Contradiction between 2. and 3.

edit

In the definition of a positive (1,1)-form, conditions 2 and 3 do not seem to be equivalent — they should either both use   or they should both use  . -- 2001:1711:FA4B:E5C0:388B:6291:6B8F:C58B (talk) 17:59, 9 January 2022 (UTC)Reply