Integrable systems approach to the Schottky problem and related questions

This paper provides an informal introduction to the integrable systems approach to the Schottky problem, detailing how Jacobian theta functions yield solutions to the KP equation and culminating in Krichever's proof of Welters' trisecant conjecture for the most degenerate flex line case.

The Gist

Imagine you are trying to solve a massive, impossible puzzle. On one side of the table, you have a set of complex, moving machines (differential equations) that describe how things change over time, like waves in the ocean or particles in a quantum field. On the other side, you have static, beautiful geometric shapes (algebraic curves and their associated "Jacobian" shapes).

For a long time, mathematicians thought these two sides of the table were completely unrelated. One was about motion and change; the other was about shape and structure.

This paper, written by Samuel Grushevsky and Yuancheng Xie, is a guidebook to a revolutionary discovery: These two sides are actually the same thing. It explains how a specific type of geometric shape can be used to solve the most difficult motion problems, and conversely, how solving a motion problem can tell you exactly what a geometric shape looks like.

Here is the story broken down into simple analogies:

1. The Magic Substitution (The "Magic Trick")

Imagine you are trying to calculate the area of a weird, curved shape. Usually, this is a nightmare. But sometimes, if you find the right "magic substitution" (like changing your perspective), the messy curve suddenly turns into a simple circle or a straight line that is easy to measure.

In the 19th century, mathematicians found that for certain complex curves (called elliptic curves), there was a "magic substitution" that turned difficult integrals into simple trigonometric functions. This paper asks: Can we find a magic substitution for any complex curve?

The answer is yes, but the "magic" isn't a simple formula. It's a deep connection between the curve and a set of special functions called Theta Functions. These functions act like a universal translator, turning the language of geometry into the language of physics equations.

2. The "Jacobian" (The Shape's Shadow)

Every curve has a "shadow" or a "soul" called its Jacobian.

• Think of a curve as a piece of string.
• The Jacobian is a multi-dimensional doughnut (a torus) that captures all the possible ways you can wrap that string around itself.

The Schottky Problem is the central mystery of this paper. It asks: "If I give you a multi-dimensional doughnut, how can you tell if it is the 'shadow' of a real string (a curve), or if it's just a random, fake doughnut?"

For a long time, no one knew how to distinguish a "real" Jacobian from a "fake" one.

3. The "Trisecant" (The Straight Line Test)

The paper focuses on a brilliant solution proposed by the late mathematician Igor Krichever. He found a way to test if a doughnut is "real" by looking for a very specific geometric trick: The Trisecant Line.

• The Analogy: Imagine you have a 3D sculpture (the doughnut). You take a long, straight stick (a line).
  • If you poke the sculpture, the stick might hit it once.
  • If you are lucky, it might hit it twice (a secant).
  • The Miracle: For a "fake" doughnut, it is statistically impossible for a straight stick to hit the surface at three distinct points at the same time.
  • The Real Deal: However, if the doughnut is the "shadow" of a real string (a Jacobian), there is a special stick that always hits it at three points.

Krichever proved that if you can find even one of these "three-point sticks" (specifically, a "flex line" where the stick is tangent to the surface in a very degenerate way), then the doughnut must be the shadow of a real curve.

4. The "KP Equation" (The Wave Machine)

On the other side of the table, we have the KP Equation. This is a famous, complex equation used to describe waves (like tsunamis or solitons).

• Usually, solving this equation is incredibly hard.
• But Krichever showed that if you build your solution using the "Theta Functions" from a real curve, the equation solves itself perfectly.

The paper explains that the KP Equation and the Trisecant Line are two sides of the same coin.

• If a shape has a "Trisecant Line" (Geometry), it automatically satisfies the KP Equation (Physics).
• If a shape satisfies the KP Equation, it must have a "Trisecant Line."

5. The "Baker-Akhiezer" Function (The Universal Translator)

To make all this work, the authors use a special tool called the Baker-Akhiezer function.

• Think of this as a universal translator or a Rosetta Stone.
• It takes the static data of a curve (the geometry) and translates it into a dynamic function that solves the moving wave equations (the physics).
• It's like having a machine that takes a picture of a mountain and instantly writes the script for a movie about how the wind blows over that mountain.

The Big Picture: Why Does This Matter?

This paper is a celebration of the work of Igor Krichever, who passed away recently. It shows that the universe is deeply interconnected.

• Geometry isn't just static shapes; it contains the secrets of how things move and change.
• Physics isn't just chaotic motion; it is governed by hidden, beautiful geometric structures.

By proving that a single "flex line" (a specific geometric alignment) is enough to identify a Jacobian, the authors solved a 150-year-old mystery. They showed that you don't need to check the whole shape; you just need to find that one special line where three points align, and you instantly know the entire structure of the universe behind it.

In short: The paper tells us that if you look closely enough at the geometry of a shape, you can hear the music of the waves it creates. And if you listen to the waves, you can reconstruct the shape. They are one and the same.

Technical Summary

Based on the provided lecture notes by Samuel Grushevsky and Yuancheng Xie, here is a detailed technical summary of the paper "Integrable Systems Approach to the Schottky Problem and Related Questions."

1. Problem Statement: The Schottky Problem

The central problem addressed is the Schottky problem, which asks to characterize the locus of Jacobians of algebraic curves within the moduli space of principally polarized abelian varieties (ppav).

• Context: Let $M_g$ be the moduli space of genus $g$ curves and $A_g$ be the moduli space of ppavs. The Torelli map $J: M_g \to A_g$ sends a curve $C$ to its Jacobian $Jac(C)$.
• The Challenge: For $g \ge 4$, $\dim M_g = 3g-3$ while $\dim A_g = g(g+1)/2$. Thus, the image $J(M_g)$ is a proper subvariety of $A_g$. The problem is to find intrinsic geometric or analytic conditions that distinguish Jacobians from general ppavs.
• Specific Focus: The paper focuses on Welters' Trisecant Conjecture and its most degenerate case: the existence of a flex line (a line tangent to the Kummer variety with multiplicity 3) on the Kummer variety of a ppav. The goal is to prove that the existence of even a single such flex line characterizes Jacobians among indecomposable ppavs.

2. Methodology: Integrable Systems and Algebraic Geometry

The authors employ a synthesis of classical algebraic geometry and the theory of integrable systems, specifically the Kadomtsev-Petviashvili (KP) hierarchy. The methodology proceeds in two main directions that are eventually unified:

A. From Geometry to Differential Equations (The "Forward" Direction)

1. Commuting Differential Operators: The theory begins with the study of commuting differential operators $L_1, L_2$ in one variable. The Burchnall-Chaundy theorem states that such operators satisfy a polynomial relation $Q(L_1, L_2) = 0$, defining an algebraic spectral curve.
2. Baker-Akhiezer Functions: Given a curve $C$, a marked point $p_\infty$, and a local coordinate, one constructs the Baker-Akhiezer function $\psi(x, p)$. This is a meromorphic function on $C$ with an essential singularity at $p_\infty$ and prescribed poles.
3. KP Hierarchy Solutions: These functions serve as common eigenfunctions for a hierarchy of commuting differential operators. The compatibility conditions of these operators yield the KP equation (and the full KP hierarchy).
4. Theta Functions: The coefficients of the KP equation are expressed explicitly in terms of the Riemann theta function associated with the Jacobian of $C$. Specifically, the potential $u(x,y,t)$ in the KP equation is given by $u = 2\partial_x^2 \ln \theta(Ux + Vy + Z)$.

B. From Differential Equations to Geometry (The "Reverse" Direction)

1. Trisecants and Flex Lines: The paper analyzes the geometry of the Kummer variety $Kum(A) \subset \mathbb{P}^{2g-1}$, the image of the abelian variety under the map defined by second-order theta functions.
2. Fay-Gunning Identity: For Jacobians, the Kummer images of specific linear combinations of points on the curve are collinear, forming trisecant lines.
3. Degeneration to Flex Lines: By taking limits where the points on the curve coalesce, the trisecant condition degenerates into a differential equation satisfied by the theta function. This differential equation corresponds to the KP equation (specifically, the condition that the theta function satisfies the KP equation).
4. Welters' Conjecture: The conjecture posits that if a ppav $A$ has a flex line on its Kummer variety, then $A$ must be a Jacobian.

3. Key Contributions and Proof Strategy

The core contribution is the exposition and detailed proof of Krichever's Theorem (2010), which solves the flex line case of Welters' conjecture. The proof strategy is non-trivial because, unlike previous results that assumed the existence of a curve or a jet of a curve, this proof starts with only the existence of a single flex line.

Step-by-Step Proof Outline:

1. Formal Solutions: Assume the existence of a flex line. This implies the existence of vectors $U, V$ and a function $\tau(x,y) = \theta(Ux + Vy + Z)$ satisfying a specific differential equation (equivalent to the KP equation restricted to a specific direction).
2. Construction of Wave Solutions: The authors construct formal power series solutions (wave solutions) $\psi$ to the linear equation $(\partial_x^2 + u)\psi = \partial_y \psi$. A critical technical hurdle is ensuring these solutions have only simple poles (and not higher-order poles) along the theta divisor. This relies on the indecomposability of the abelian variety.
3. $\lambda$-Periodicity: To globalize the solution, the authors impose $\lambda$-periodicity conditions on the coefficients of the wave solution with respect to the lattice of the abelian variety. This ensures the solution is well-defined on the variety.
4. Construction of the Spectral Curve: Using the unique $\lambda$-periodic wave solution, they construct a pseudo-differential operator $L$ for which $\psi$ is an eigenfunction. From the commutativity of the hierarchy, they derive a commutative ring of ordinary differential operators.
5. Recovering the Curve: By the correspondence between commutative rings of differential operators and algebraic curves (Burchnall-Chaundy theory), this ring defines a spectral curve $\hat{C}$.
6. Isomorphism: The final and most difficult step is proving that the original abelian variety $A$ is isomorphic to the Jacobian of this spectral curve $\hat{C}$.
   • The authors show that the KP flows on the space of operators are linearized on the Jacobian of $\hat{C}$.
   • They demonstrate that these flows are also induced by abelian functions on $A$.
   • This establishes a holomorphic embedding $A \hookrightarrow Jac(\hat{C})$ and vice versa, proving $A \cong Jac(\hat{C})$.

4. Key Results

• Theorem 4.20 (Krichever 2010): An indecomposable principally polarized abelian variety $(A, \Theta)$ is the Jacobian of an algebraic curve if and only if its Kummer variety admits a flex line (a line tangent with multiplicity 3).
• Connection to KP Hierarchy: The existence of a flex line is analytically equivalent to the theta function of the abelian variety satisfying the KP equation.
• Global Existence: The proof establishes the global existence of Baker-Akhiezer functions and the associated spectral curve starting from purely local differential data (the flex line condition), without assuming the existence of a curve a priori.

5. Significance

• Resolution of a Major Conjecture: This work provides a complete solution to the "strong" Schottky problem (characterizing Jacobians without assuming a curve exists inside the variety) for the flex line case, a long-standing open problem in algebraic geometry.
• Unification of Fields: It beautifully unifies the theory of integrable systems (KP hierarchy, Baker-Akhiezer functions, spectral curves) with classical algebraic geometry (Jacobian varieties, theta divisors, Kummer varieties).
• Methodological Innovation: The proof introduces powerful techniques for handling singularities and global existence of solutions on abelian varieties, specifically dealing with the "bad locus" $\Sigma$ where the theta function and its derivatives vanish.
• Historical Tribute: The paper serves as a comprehensive guide to the work of Igor Krichever, whose insights bridged these two fields for decades. It clarifies the logical flow from the degenerate geometry of the Kummer variety to the algebraic structure of the Jacobian.

In summary, the paper demonstrates that the geometric property of having a "flex line" on a Kummer variety is a sufficient condition to force an abelian variety to be the Jacobian of a curve, by reconstructing the curve via the machinery of integrable systems.