# Which nonlinear PDEs are of interest to algebraic geometers and why?

Motivation

I have recently started thinking about the interrelations among algebraic geometry and nonlinear PDEs. It is well known that the methods and ideas of algebraic geometry have lead to a number of important achievements in the study of PDEs, suffice it to mention the construction of finite-gap solutions to integrable PDEs (see e.g. this book) and the geometric approach to PDEs developed by A.M. Vinogradov et al. which revolves around the concept of diffiety (the word itself was merged from "differential" and "variety") and which the authors themselves consider, at least to some extent, as a "translation" of ideas from algebraic geometry and commutative algebra to the realm of PDEs, see e.g. these two books.

On the other hand, it appears (as far as my googling skills allow me to tell) that the other way around, i.e., in the applications of nonlinear PDEs in algebraic geometry, the interaction is at least somewhat less intense. I was able to come up with basically just two things: the Novikov conjecture (proved by Shiota) on the relation of the KP equation to the Schottky problem and the applications of the Monge-Ampere equations in Kahler geometry.

Question

Which are the other applications of the nonlinear PDEs in (broadly understood) algebraic geometry? In other words, which nonlinear PDEs are of interest to algebraic geometers and why?

EDIT: It should be obvious, but to play it safe I would like to spell this out loud and clear: please feel free to share not only the already known cases where PDEs have helped the algebraic geometers but also the more open-problem-type cases where, say, there is a PDE that could be of use in algebraic geometry but some crucial bits of information about this PDE (for instance, about the existence of solution(s) with the desired properties) are still missing.

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this must be community wiki –  Denis Serre May 26 '11 at 14:57
@Dennis: I believe that this question is not that trivial and easy to answer, even by experts, so I'd like to keep it unwikified to give people an extra incentive (in the form of gaining rep) to answer. Also, I am not convinced that having a sorted list of answers to this particular question is a right thing to do. –  mathphysicist May 26 '11 at 17:37
Denis with one N, please... –  Denis Serre May 27 '11 at 8:08
@Denis: Sorry, just a typo. Mille pardons! –  mathphysicist May 27 '11 at 10:36

Strictly speaking, this is not meant as an answer to the question---it's more like a suggestion that you might find it interesting to also ask a related, but different, question.

I would say the most interesting feature of many interactions between nonlinear (especially integrable) PDE and algebraic geometry, from the point of view of an algebraic geometer, is that they show us new structures that nobody knew existed classically.

One instance of this is the story that you're referring to relating the conformal field theory of free fermions, KP, and the geometry of moduli of bundles, mediated (geometrically) by the Sato Grassmannian. This is what leads to Shiota's characterization of the Schottky locus. That's well before my time, but I wouldn't have guessed that most classical algebraic geometers thought of this as a very big advance on the Schottky problem at the time: if you'd like equations for the Schottky locus, or even some nice algebro-geometric description of it, you probably aren't satisfied with what Shiota tells you (I'm a huge fan of this story, so my remark isn't meant to convey my own opinion of this work, rather to guess at what some others might have thought). Algebraic geometers were interested in moduli of curves, bundles, etc. long before this story appeared, but a whole new world emerged from it.

In a different direction, the discovery of integrability in topological string theory (as formulated, for example, in Witten's conjecture, later Kontsevich's theorem) shows us striking new structure, again governed by integrable PDE (KdV, n-KdV) in the intersection theory on moduli spaces of curves (and later 2D Toda in the Gromov-Witten theory of curves). Again, I would argue (as someone who, admittedly, is incredibly far from expert in the subject) that the most interesting part of the story is the amazingly rich structure (involving integrable PDE, matrix models, and intersection theory on moduli spaces) that was revealed by these discoveries.

EDIT: What I wrote initially takes a slightly misleading tone about applications back to classical AG: for example, Krichever's stunning work on the trisecant conjecture, which grows directly out of integrable PDE, is surely something of classical interest!

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One important use of PDEs in algebraic geometry is in so-called "Hitchin-Kobayashi correspondences". The original example of this is the following theorem.

Theorem (Donaldson, Uhlenbeck-Yau) Let $L \to X$ be an ample line bundle over a compact complex manifold and $\omega$ a Kähler metric representing $c_1(L)$. A holomorphic vector bundle $E \to X$ is slope polystable with respect to $L$ if and only if it admits a Hermitian-Einstein metric with respect to $\omega$.

To spell this out in detail: the slope of a bundle $\mu(E)$ is its degree divided by its rank, where degree is $\langle c_1(E) \cup c_1(L)^{n-1}, [X] \rangle$. A bundle is slope stable if all proper coherent subsheaves have strictly smaller slope. A bundle is called slope polystable if it is the sum of stable bundles of equal slope. Meanwhile, a Hermitian metric in $E$ is called Hermitian-Einstein if its curvature $F$ satisfies the equation $(F,\omega) = c \cdot \mathrm{Id}$, for a constant $c$. (Here we are taking the innerproduct on 2-forms, the result being an endomorphism of $E$.) This is a non-linear PDE on the Hermitian metric.

Notice that the slope polystability is purely algebraic - it makes no mention whatsoever of the metric $\omega$. What is remarkable is that this is equivalent to the existence of a solution of a non-linear PDE. The story behind this theorem is quite a long one. It can be seen as an infinite dimensional example of the equivalence between quotients via GIT and symplectic quotients. The PDE plays the role of the moment map.

Since this result was proved there have been many other versions involving bundles with additional data, say, e.g., a Higgs field, which appears both in the definition of stability and the corresponding PDE. The corresponding "Hitchin-Kobayashi correspondences" play important roles in the study of the moduli of the algebraic objects. For example the fact that one can always solve the relevant PDE leads directly to an interesting Kähler metric on the moduli space of stable objects. In the case of Higgs fields, this is a hyperkähler metric. This metric is one of the starting points of the approach to geometric Langlands proposed by Kapustin and Witten. There are also other applications of the PDE point of view here leading to, amongst other things, strong restrictions on the fundamental groups of Kähler manifolds. This subject sometimes goes by the name "Non-abelian Hodge theory". (I should stress that this is a long way from my expertise!)

In a different vein, one version of the Hitchin-Kobayashi correspondence (which is still conjectural) concerns not metrics on bundles but rather metrics on the manifold itself. Many people in Kähler geometry are currently working on understanding both the conjecture and its ramifications. The idea (originally due to Yau, later refined by Donaldson and Tian) is that given an ample line bundle $L \to X$, one should be able to find a so-called "extremal" Kähler metric in $c_1(L)$ if and only if the polarised variety is "stable". Here a metric is "extremal" if it's a critical point of the $L^2$-norm of the curvature tensor, restricted to metrics in $c_1(L)$. This turns out to be equivalent to the gradient of the scalar curvature being holomorphic, a sixth-order fully non-linear PDE. "Simple" examples are Kähler-Einstein metrics (when $L$ is a multiple of the canonical bundle). The correct definition of stability, known as K-polystability, is a little too involved to give neatly here, but it is important to mention that, just as for slope polystabilty of a vector bundle, it is a purely algebraic concept. This whole subject is vast, and I could write about it for pages and pages, but I've probably already said too much for one answer!

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Thanks a lot! Could you recommend some references where the PDEs in question are written down explicitly so that the PDE people could have a first look at them without having to dig through all of the algebro-geometric machinery? –  mathphysicist Jun 6 '11 at 17:15
@mathphysicist: Sorry to take so long to get back to you. Good places to read about this are the two articles of Donaldson: "A new proof of a theorem of Narasimhan and Seshadri" in the Journal of Differential Geometry and "Anti-self-dual connections over complex algebraic surfaces and stable vector bundles" in Proceedings of the London Mahtematical Society. Another great article is "The Yang-Mills equations over Riemann surfaces" by Atiyah and Bott, in, I think, Transactions of the Royal Society. –  Joel Fine Jun 10 '11 at 13:46

It's a bit embarrassing to be giving this answer when there are many others far better qualified, but: in connection with the minimal model program, there have been efforts to understand canonical models by producing canonical singular Kahler-Einstein metrics via Kahler-Ricci flow. See for example http://arxiv.org/pdf/math.AG/0603064.pdf.

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The complex Monge-Ampère equation gives rise to remarkable metrics on complex line bundles. In that respect, the theorem of Calabi-Yau has very important applications in algebraic variety, particularly for understanding the geometry of complex algebraic varieties with trivial canonical class.

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This is a bit borderline. But the notion of hyperbolicity, which is associated to the well-posedness of the Cauchy problem, led Garding to introduce the concept of hyperbolic polynomials. The theory of HPs is an important part of so-called real algebraic geometry. In particular, the study of lacunas of the fundamental solution of a hyperbolic operator is intimately related to the algebraic topology of real algebraic surfaces.

Edit. A remarkable result about HPs (Helton, Vinnikov, CPAM 2009, conjectured by P. Lax in 1972): let $P(T,X,Y)$ be a homogeneous polynomial of degree $n$, hyperbolic in the $T$-direction. Then there exist Hermitian matrices $H,K$ such that $P(T,X,Y)=c\det(TI_n+XH+YK)$. This is obviously false for polynomials in more than $3$ variables, for instance for $T^2-X^2-Y^2-Z^2$.

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Thanks, Denis! Could you please suggest a good introductory book or survey article on the subject? –  mathphysicist May 27 '11 at 15:16
Maybe this Séminaire Bourbaki article of Atiyah: numdam.org/item?id=SB_1966-1968__10__87_0 ? –  Michael Bächtold May 28 '11 at 11:39
Thanks for the references once more! –  mathphysicist May 28 '11 at 14:00