There is the famous Serre intersection formula in algebraic geometry using the Tor functor (see for example here). I would like to know if there is such a formula in analytic (i.e. complex) geometry. Thanks.

In somewhat more detail: given two subvarieties of complementary codimensions in a smooth projective variety, the intersection number between them can be computed in terms of an alternating sum involving the Tor function, no matter if their intersection is transversal (and more generally, proper) or not. My question is then how far we can do so for compact complex manifolds. Is there any subtlety of using sheaf theory and so on in this general case? I would expect there is no such, the generalization should go through.

But I see no references on this topic (either on the Serre intersection formula in the compact complex manifolds setting or the more general topic of subtlety in using sheaf theory in this general setting), so I would like to know.

  • $\begingroup$ I do not understand what the OP is asking. I would like the OP to clarify the question. $\endgroup$ Dec 18 '15 at 19:29
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    $\begingroup$ I assume the spirit of the question is "Serre's formula in algebraic geometry is nice as it gives an algebraic way (Tor's) to compute intersection numbers. What if I do the same with analytic spaces (say working in the local ring at the origin in affine space), does a sensible theory exist?". $\endgroup$
    – pro
    Dec 18 '15 at 22:40
  • $\begingroup$ Note that Serre's formula is for proper intersections so there is actually no sheaf theory involved, only a computation in the local rings of the points of intersection. It should be easy to check that it does hold in the complex analytic setting, but I do not know any explicit reference. $\endgroup$
    – naf
    Dec 20 '15 at 7:08
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    $\begingroup$ @ulrich: Serre's formula is also for improper intersection also. I am interested here in the global intersection number. $\endgroup$
    – A.N.Ony
    Dec 20 '15 at 23:14
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    $\begingroup$ I don' t think so. Proper is not the same as transversal; the link that you give assumes that the intersection is proper. $\endgroup$
    – naf
    Dec 21 '15 at 4:49

Serre's formula works in the analytic category as well. If X is a smooth complex manifold, there is a ring structure on the Grothendieck group $K(X)$ of coherent sheaves given by the usual formula $$ [\mathcal{F}] . [\mathcal{G}]=\sum_{i \geq 0} (-1)^i [\mathrm{Tor}^i_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})].$$ The Chern character $ch \colon K(X) \rightarrow H^*(X, \mathbb{Q})$ is a ring morphism. If $Z$ and $W$ are two complex subvarieties of $X$ of codimensions $p$ and $q$, write $$ ch(\mathcal{O}_Z). ch(\mathcal{O}_W)=\sum_{i \geq 0} (-1)^i ch\,(\mathrm{Tor}^i_{\mathcal{O}_X}(\mathcal{O}_Z, \mathcal{O}_W)).$$ Now $ch(\mathcal{O}_Z)=[Z] + \,\mathrm{classes\,in}\, H^k(X, \mathbb{Q})$ for $k>2p$ and similarly for $W$. Hence the term with the smallest cohomological degree appearing in $ch(\mathcal{O}_Z). ch(\mathcal{O}_W)$ is exactly the homological intersection of $Z$ and $W$ in $H^{2(p+q)}(X, \mathbb{Q})$. This proves that $$[Z]. [W]=\sum_{i \geq 0} (-1)^i ch_{p+q}(\mathrm{Tor}^i_{\mathcal{O}_X}(\mathcal{O}_Z, \mathcal{O}_W)).$$ Next recall that if $\mathcal{F}$ is a coherent sheaf whose support consists of irreducible components $D_i$ of codimension $\geq d$, then $$ch_d(\mathcal{F})=\sum_{codim(D_i)=d} \ell_i(\mathcal{F})\, [D_i^{red}]$$ where $\ell_i(\mathcal{F})$ is the length of $\mathcal{F}$ at the generic point of $D_i$. This gives Serre's formula (at least over $\mathbb{Q}$).

  • $\begingroup$ Thanks for the answer. Do you have any reference for this? $\endgroup$
    – A.N.Ony
    Dec 23 '15 at 12:58
  • $\begingroup$ Also, I would like to know if there is any restriction on the manifold X: compact, Kahler, ... in the result you claimed? $\endgroup$
    – A.N.Ony
    Dec 23 '15 at 22:08
  • $\begingroup$ Well, I don't know a good reference in the analytic case. You have to put several things together. The fact that the alternate sum of the Tor sheaves defines a ring structure on K(X) comes from the fact that $[F]. [G]$ is the image of the derived tensor product of F with G in the derived category $D^b_{coh}(X)$ by the natural map $\chi \colon D^b_{coh}(X) \rightarrow K(X)$ defined by $\chi(\mathcal{K})=\sum_i (-1)^i \mathcal{H}^i(\mathcal{K})$. Then you must use a standard spectral sequence argument. $\endgroup$ Dec 23 '15 at 23:30
  • $\begingroup$ To prove that the Chern character is multiplicative, you factor it by the topological K-theory $K^{top}(X)$: the map $K(X) \rightarrow K^{top}(X)$ is given by associating to $\mathcal{F}$ the alternate sum of smooth complex vector bundles resolving $\mathcal{F} \otimes_{\mathcal{O}_X} \mathscr{C}^{\infty}_X$. This map is a ring morphism. Then you compose it with the usual topological Chern character, which is also multiplicative. $\endgroup$ Dec 23 '15 at 23:51
  • $\begingroup$ For the formula of the Chern character of a torsion sheaf, this is more painful: you have to use devissage, resolutions of singularities and Grothendieck-Riemann-Roch for smooth immersions. I think this works on arbitrary paracompact complex manifolds. $\endgroup$ Dec 23 '15 at 23:52

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