# Picard-Fuchs equations

If I have the periods $$\pi_1(\lambda)=\int_0^1\frac{dx}{\sqrt{x(x-1)(x-\lambda)}}$$ and $\pi_2(\lambda)$ similarly defined of the cubic curve $$y^2z=z(x-z)(x-\lambda z)$$ Such functions will be holomorphic on $\lambda \in \mathbb{C}-\{0,1\}$. Then the sum $\pi(\lambda)=\pi_1(\lambda)+\pi_2(\lambda)$, as well as each $\pi_i$, satisfy the differential equation called Picard-Fuchs equation $$0=\tfrac{1}{4}\pi+(2\lambda-1)\pi'+\lambda(\lambda-1)\pi''$$ derivation with respect to $\lambda$.

My questions are the following. Where does such a differential equation come from? why is it important for? has it something to do with Gromov-Witten Theory?

I'll appreciate any kind of comment related with such an equation.

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Here is a very rough outline:

Take your family $E$ of elliptic curves over $B := \mathbb{C} - \{0,1\}$. Then take the associated "(co)homology bundle" over $B$, whose fibre over $\lambda$ is the (singular) (co)homology of the elliptic curve $E_\lambda$. To be rigorous, the $i$-th cohomology bundle is $R^i \pi_\ast\mathbb{C}$, where $\pi$ is the map $E \to B$ and $\mathbb{C}$ is the constant sheaf (let us work in the analytic topology). Actually to be precise I should say that $R^i\pi_\ast\mathbb{C}$ is a (locally constant) sheaf of $\mathbb{C}$ vector spaces, and the corresponding vector bundle is $R^i\pi_\ast\mathbb{C} \otimes_\mathbb{C} \mathcal{O}\_B$. It is a fact that these cohomology bundles come with flat (Gauss-Manin) connections $\nabla$. One way to see that the vector bundles are flat is to observe that there are integral lattices $R^i\pi_\ast \mathbb{Z} \subset R^i\pi_\ast\mathbb{C}$.

Let $\omega$ be a 1-form on the family $E$. Note that the 1st cohomology of an elliptic curve is rank 2, so the cohomology bundle $R^1\pi_\ast \mathbb{C}$ is rank 2, thus if we have 3 sections, then they will be (fiber-wise) linearly dependent. So here are 3 sections: $\omega, \nabla_{d/d\lambda}\omega, (\nabla_{d/d\lambda})^2 \omega$. The Picard-Fuchs equation is essentially just the equation which expresses that these sections are linearly dependent. Your equation involving "$\pi$" (not the same as what I am calling "$\pi$") and its derivatives comes from taking this linear dependence equation and "plugging in" (i.e. integrating along) homology classes extended by parallel transport.

The story that I've described above generalizes to arbitrary families of smooth compact varieties.

Thomas Riepe's answer explains some of the more classical reasons why we might be interested in period integrals and Picard-Fuchs equations, so let me say a few words about the relation to Gromov-Witten theory.

The relation to GW theory arises from mirror symmetry, which is a duality between type IIA and type IIB string theories. One of the reasons why mathematicians first became interested in mirror symmetry was because of the prediction of the physicists Candelas-de la Ossa-Green-Parkes in the early 90s that the genus 0 GW invariants of a quintic threefold (type IIA theory) could be computed via an analysis of period integrals and Picard-Fuchs equations coming from a "mirror" family of Calabi-Yau manifolds (type IIB theory). This is a general principle of mirror symmetry: that GW invariants of certain manifolds can be computed via completely different methods on the "mirror manifold". Usually, studying the mirror manifold is "easier" than trying to study the GW theory of the original manifold directly; although by now our knowledge of GW theory has grown considerably, so this is less true than it used to be.

A very nice introductory paper on this material is "Picard-Fuchs equations and mirror maps for hypersurfaces" by David Morrison: http://arxiv.org/abs/alg-geom/9202026

If you're interested in reading further, you should check out the book "Mirror symmetry and algebraic geometry" by Cox-Katz, which covers all of this material in detail and explains the proofs (due to Givental and Lian-Liu-Yau) of the Candelas-et. al. prediction.

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Two small points. If you "extend $\gamma$ by parallel transport" the local derivatives are zero. Secondly, as you say, the section will "obviously" satisfy a DE; however, the key point, which does not follow from what you observe, is that DE is of algebraic origin. The point is that you can construct the linear dependence in the (algebraic) de Rham cohomology. – moonface Nov 16 '09 at 15:04
Perhaps I am missing something, but I don't quite see what is your connection above?. – Csar Lozano Huerta Nov 16 '09 at 16:14
@moonface: Thanks, I've edited my answer, hopefully it is now correct. As for your second comment, I don't know if it is "the" key point, but it is indeed an important point. I believe that everything that I discussed works equally well in the algebraic category if you replace $R^i \pi_\ast \mathbb{C}$ with the hypercohomology $\mathbb{R}^i \pi_\ast \Omega^\bullet_{E/B}$. Then to show that the Picard-Fuchs equation is algebraic, it suffices to show that the Gauss-Manin connection is algebraic, which comes from Riemann-Hilbert: locally constant sheaves $\leftrightarrow$ flat vector bundles. – Kevin H. Lin Nov 16 '09 at 17:31
Anyway, I think that what I wrote describes the essential idea. I don't know how to do everything purely algebraically without mentioning fancy things like hypercohomology, relative differentials, and Riemann-Hilbert correspondence. I had wanted to avoid using too much fancy algebraic geometry. – Kevin H. Lin Nov 16 '09 at 18:02
Kevin -- In fact, I also don't know how to do it in a "concrete" way in general. In the case of curves, though, you can get by in classical language because you can express the de Rham cohomology in terms of meromorphic forms. In concrete terms: You can write down an meromorphic $f$ on the curve $E_{\lambda}$, depending on $\lambda$, so that, if $\omega = dx/\sqrt{\dots}$ is the pertinent differential above, then a linear combination of $\omega, \omega', \omega''$ equals $df$. So this linear combination is zero in cohomology. – moonface Nov 16 '09 at 18:14

There exist many beautifull texts on that, on the basics about and the startling arithmetic application of the Picard-Fuch eq., I recommend to look into Clemens' "Scrap book of complex curve theory" and Katz' articles, esp. "On the differential equations satisfied by period matrices" (probably in numdam). General infos are in Brieskorn's "plane algebraic curves" and in Griffiths-Harris. I don't know about a connection to Gromov-Witten theory. The curiosity in Periods of algebraic varieties comes probably from that they are special numbers like pi or e coming from a natural process, satisfying some set of relations and having transcentality properties. Then, Manin showed that the diff.-eq. they satisfy encodes arithmetic infos too (via "Gaus-Manin connection"). Further one wonders if one can produce with periods like with e and pi interesting algebraic numbers. There is a very beautifull article by Zagier "Periods" on the web somewhere.

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Thanks! I just realized because of your reference, that Clemens was a professor at my university! – Csar Lozano Huerta Nov 21 '09 at 5:45