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I heard from a recent talk that the "number of $n$-degree curves" in a Calabi-Yau manifold is an invariant of the space. But what does that mean? (Specifically I would like to ask the following.)

  1. Does the sequence of "number of $n$-degree curves in a CY-manifold has a name? (i.e. the sequence for which Physics wins over Mathematics in calculating $n_3$.)

  2. So does it make sense to ask, for example, 'What's the number of quadratic rational curves on a torus?'

  3. Is this sequence related to something more familiar, like homology groups, Betti number, torsion, Euler characteristic (not its weighted version), etc.?

Thank you very much for your attention.

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    $\begingroup$ Usually one called $n_d$ the number of degree $d$ curves in $\mathbb P^2$ passing through $3d-1$ points. That can be calculated using Gromov Witten invariants. $\endgroup$ – user99914 Mar 10 '14 at 7:12
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    $\begingroup$ But $\mathbb P^2$ is not CY. $\endgroup$ – Brenin Mar 10 '14 at 11:21
  • $\begingroup$ The question has a number of issues. First, the title asks why a false statement is true. Second, the title doesn't match the questions in the body. Third, there is no such thing as "the sequence of degree-n curves in a CY manifold"; you have to choose the genus of the curves you are talking about in an appropriate way, because usually there will be infinitely many curves of a given degree. $\endgroup$ – user64687 Mar 10 '14 at 12:05
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The theory of curve counting in CY manifolds is well established in dimension $3$, and it goes under the name of Donaldson-Thomas theory.

Example. The general quintic threefold (which is a CY) is expected to contain a finite number $N_d$ of degree $d$ rational curves, for any $d$. The history of this sequence $\{N_d\}$ is fascinating. The number of lines, $2875$, was known in the $19$th century. The number of conics, $609250$, was found by Katz, while the number of twisted cubics, $317206375$, was found by Ellingsrud and Strømme (and did agree with a conjectured formula for the $N_d$'s coming from the world of String Theory; such a formula turned out to be correct later in the $1990$'s). I do not know, however, whether the above sequence has a name.

Remark. The number $2875$ (say) is not an invariant of the quintic threefold $Q\subset \mathbb P^4$. Indeed, any quintic with the "correct" number of lines deforms to the Fermat quintic $$x_0^5+x_1^5+x_2^5+x_3^5+x_4^5=0,$$ which contains $50$ one-dimensional families of lines (this was proved Albano-Katz). There are other examples of quintic threefolds with infinitely many lines.

The idea of DT theory is to count curves "virtually", so that the count will always be a finite number, even when an actual count in a special situation would give the answer $\infty$.

For a projective CY$3$ $X$, one can look at one-dimensional subschemes $Z\subset X$ such that $\chi(\mathscr O_Z)=n$ (view this as "fixing the genus") and the homology class of the pure $Z$ (i.e. the class of $Z$ with embedded and isolated points removed) is some $\beta\in H_2(X,\mathbb Z)$. These $(n,\beta)$ are the discrete invariants one has to fix in order to get a "moduli space of degree $\beta$ curves in $X$". A standard notation for such moduli space is $I_n(X,\beta)$.

The virtual dimension of this moduli space is $-K_X\cdot\beta=0$, thanks to the CY condition, and the virtual count of its points, which is the Donaldson-Thomas invariant of $I_n(X,\beta)$, is an integer $$N_{n,\beta}=DT(I_n(X,\beta))\in\mathbb Z\,\,\,\,\,\,\,\,\,\,\,\,\,(\star)$$ which is invariant under deformations of the CY$3$. This is the main reason why such integer deserves to be called an "invariant".

Again, I do not know whether $\{N_{n,\beta}\}$ has a name, but I guess an equivalent datum, namely the generating series $$Z_{DT}(X;q,u)=\sum_{n,\beta}N_{n,\beta}q^nu^\beta$$ over all $n,\beta$ does have a name: it is called the Donaldson-Thomas partition function of $X$.

As for question $(2)$, an Abelian variety $A$ contains no rational curve, as every map $\mathbb P^1\to A$ is constant.

The only thing that comes to my mind for your question $(3)$ is the one you want to avoid, namely weighted Euler characteristics, which compute the numbers $(\star)$.

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  • $\begingroup$ Nice! +1 Do you have any good reference for learning the basics of DT theory? :) $\endgroup$ – Alex Youcis Mar 10 '14 at 12:58
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    $\begingroup$ @AlexYoucis: Thanks :) I like Maulik's notes very much: www-fourier.ujf-grenoble.fr/sites/ifmaquette.ujf-grenoble.fr/…. $\endgroup$ – Brenin Mar 10 '14 at 15:24
  • $\begingroup$ By 'rational curves', do you mean curves where there points with rational coordinates lying on them? $\endgroup$ – wilsonw Oct 5 '14 at 8:50
  • $\begingroup$ I mean a smooth curve of genus $0$, i.e. a curve isomorphic to $\mathbb P^1$. $\endgroup$ – Brenin Oct 5 '14 at 9:08
  • $\begingroup$ I think I was asking for a name for the sequence A076912 on OEIS which says the "number of degree-$n$ rational curves on a general quintic". $\endgroup$ – wilsonw Apr 16 at 13:56

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