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In Wendl's book J-holomorphic curve, he claim ...embedded 2- sphere with self-intersection $−1$. As far as I know, the definition of self-intersection of a J-holomorphic curve $u$ is defined by

$\frac{1}{2}\#\{(z_1,z_2)|u(z_1)=u(z_2)\}$,

how could it be a negative number?

Ted Shifrin
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Grey
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    If you can perturb your curve to another such that the two intersect transversely, then you can regard the self-intersection number as a signed count of intersection points. In general though, you can't guarantee the existence of this second curve; this is the case for the exceptional divisor of a blowup of a complex surface. You may be interested in this question. – Michael Albanese May 31 '22 at 02:51
  • Could you please say something about how to regard the the self-intersection number as a signed count of intersection points? – Grey May 31 '22 at 03:08
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    Consider reading a differential topology textbook, such as Guillemin and Pollack: It is expected that you know differential topology when you read about J-holomorphic curves. – Moishe Kohan May 31 '22 at 05:29
  • Thanks, I will read it. – Grey May 31 '22 at 07:38

1 Answers1

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The following might be a good (challenging) exercise for you to work out. I wrote this for the graduate students in my Guillemin & Pollack course about 15 years ago.

  1. Prove that any complex vector space has a canonical orientation. Deduce that intersection numbers are always nonnegative for (transverse) complex submanifolds of a complex manifold.

  2. Let $H=\{[z_0,z_1,z_2]: z_2=0\}\subset\Bbb CP^2$. Calculate $I(H,H)$ explicitly.

  3. Let $\xi$ denote the tautological line bundle on $\Bbb CP^1$. This is the line bundle consisting of points $\{(\ell,z)\in \Bbb CP^1\times\Bbb C^2: z\in\ell\}$. Show that there is a smooth section $\sigma$ of $\xi$ that intersects the zero-section $Z$ precisely once (say at $[1,0]$) with intersection number $-1$. Deduce that $I(Z,Z)=-1$. Conclude that the only global holomorphic (complex analytic) section of $\xi$ is the zero section. Is there an obvious global meromorphic section?

For example, $\sigma([z_0,z_1]) = \dfrac{\bar z_1}{|z_0|^2+|z_1|^2}(z_0,z_1)$.

  1. Let $\widetilde{\Bbb C^2}$ denote the blow-up of $\Bbb C^2$ at the origin, and let $E$ denote the “exceptional fiber," i.e., the $\Bbb CP^1$ lying over the origin. Observe first that $\widetilde{\Bbb C^2}\to \Bbb CP^1$ is the tautological bundle $\xi$. Use your answer to part 3 and the tubular neighborhood theorem to prove that $I(E,E)=-1$. Why does this not contradict part 1?
Ted Shifrin
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