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I'd like to have a definition of intersection on non compact complex surfaces because all i have found so far is only about projective surfaces. for example how can i define the self intersection of curves on a complex surfaces which is not projective. and are there any references which clarify this.

thanks for your help.

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Specifically for curves on surfaces, you can look at Barth-Hulek-Peters-Van de Ven Chapter II, Section 10 ("Intersection numbers") for a quick summary. The idea is as follows. For noncompact spaces, you can still define a nondegenerate pairing between $H^k(X)$ and $H^{dim(X)-k}_c(X)$ where the subscript c means compactly supported cohomology (and dim means real dimension). This is because when you wedge an arbitrary form with something compactly supported, you get something compactly supported which you can integrate. Since you can integrate an arbitrary differential form over a compact subvariety, or a compactly-supported form over an arbitrary subvariety, this allows you to define an intersection theory between compact and noncompact cycles. The only thing you can't do this way is define a well-behaved intersection number between two noncompact subvarieties (which you wouldn't expect to because they can "intersect at infinity").

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  • $\begingroup$ I still don't understand even though i tried to read the book you suggested, i want to do intersecting between two divisor ( such that one of them is compactly supported ) but i don't the link with the intersection between forms is there somme sort of bijection between compactly suuported divisor on a complex surfaces $X$ and the de rahm cohomology group $H^{2}_c(X)$ ? $\endgroup$
    – singularity
    Commented May 18, 2021 at 19:11
  • $\begingroup$ A (compact respectively non-compact) subvariety defines a Poincaré dual class in (ordinary respectively compactly supported) cohomology. You can think of it using whatever model of cohomology you prefer. Intersection then corresponds to cup product of cohomology classes. Of course the answer is just a (compactly supported) cohomology class, but if the subvarieties have complementary dimension then you get a number by integrating over the ambient variety. $\endgroup$
    – Jonny Evans
    Commented May 19, 2021 at 6:58
  • $\begingroup$ thanks for your answer, but i don't see how a subvariety defines a class in the cohomology group i will try to study more about the poincaré duality, i a also would like to know if that intersection correcponds to the intersection multiplicity. for example let's take to subvariety $A$ et $B$ which meet at a point $x$ and suppose their local equation are f=0 and g=0 ( f and g are holomorphic function) can i say that $(A,B)_x = dis /frac{/mathbb{C}{x,y}}{(f,g)}$ ? $\endgroup$
    – singularity
    Commented May 19, 2021 at 8:15

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