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What is the meaning of Holomorphic Sphere $ S^2$ with (-1)- self intersection number in intersection theory. Can we draw such sphere?

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The self intersection of a manifold is a topological invariant related to how this manifold is embedded (or immersed) in a surrounding space. You move a copy of the manifold a little and count the number of intersection points with the original manifold. To move the manifold you need to construct an initial velocity field, which is a normal vector field. The index sum of such a normal vector field is then the intersection number.

If you are dealing with real vector fields you can only count index mod 2, it's the odd/even part that is invariant and not the total number of zeros since they cancel in pairs. To be able to calculate the index as a whole number you need an orientation and so you consider complex vector fields instead.

Thus $S^2$ regarded as the complex projective line $\mathbf{CP}^1$ can be embedded in complex two-dimensional space in such a way that it admits a normal complex vector field with index -1. This is the kind of embedding you get when you blow up a point.

Note that a complex vector field with index -1 is not holomorphic. If you were to try convert such a vector field into a holomorphic one, you would need to replace the -1-zero with a pole. Algebraically this means that the expression for the -1-vector field necessarily contains conjugations ($\bar{z}$) which gives the orientation-reversion characteristic of negative index. The closest way to get orientation-reversion in a holomorphic setting is with inversion ($1/z$) which would introduce a singularity. (And that's why algebraic geometers say that an exceptional divisor cannot move.)

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I think you mean the following: suppose you have a complex surface (this is a space with two complex dimensions, ie. 4 real dimensions). For example, $\mathbb{P}^2$, the complex projective space. This space is important because it is the "right" setting to study polynomial equations in two variables. Now, the zero set of this polynomial equations may not be a smooth set; that is, it may have singularities. For example, the zero set of $$xy = 0$$ is not smooth at the point $(0,0)$, it looks like an $X$ (two lines crossing at a point) there. You smooth out this singularities via the blow up. This is how you get a sphere with $-1$ self intersection. Please look up the precise definition of blow up in say, wikipedia, here I will give a somewhat wrong overall picture of it. The idea is to remove a ball around $(0,0)$ and glue the antipodal points (this is what you would be doing if you were working in real manifold and not carrying about the smooth structure), doing this you remove the singularity. As you go up the $x$-axis at some point you "teleport" from the negative numbers to the positive numbers - you never see the $y$-axis. It turns out that in complex dimension, when you blow up a point in $\mathbb{P}^2$ you get a sphere inside your space. This sphere has $-1$ self intersection (that is, if you try to move if off its position you will get two spheres that intersect at a point but oriented in such a way that the intersection is negative). In particular, this new sphere (the moved off one) is not a complex submanifold since transverse complex submanifolds always intersect positively.

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  • $\begingroup$ Sorry, what do you mean by "is not a complex submanifold"? It is for sure an analytic subspace, as it is locally given as the zero set of holomorphic functions, then it is also smooth ... $\endgroup$ – wisefool Oct 13 '14 at 10:08
  • $\begingroup$ The moved off sphere is not the zero set of holomorphic functions You can prove this claim as stated - if it were it would have negative intersection with the $\mathbb{P}^1$ you got after the blowup. $\endgroup$ – Esp F Oct 13 '14 at 16:26
  • $\begingroup$ Oh, sorry, I misread … I hadn't got that "this sphere" in the two consecutive sentences was referred once to the exceptional divisor and once to an alleged "shift" of that... $\endgroup$ – wisefool Oct 13 '14 at 17:23

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