I. Krichever
We prove that an indecomposable principally polarized abelian variety $X$ is the Jacobain of a curve if and only if there exist vectors $U\neq 0,V$ such that the roots $x_i(y)$ of the theta-functional equation $θ(Ux+Vy+Z)=0$ satisfy the equations of motion of the {\it formal infinite-dimensional Calogero-Moser system}
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Integrable systems approach to the Schottky problem and related questions
本文通过可积系统方法探讨施托克西问题，利用雅可比θ函数解决KP方程，并证明了世界者三secant猜想的最退化情况。
We present a computational approach to the classical Schottky problem based on Fay's trisecant identity for genus $g\geq 4$. For a given Riemann matrix $\mathbb{B}\in\mathbb{H}^{g}$, the Fay identity establishes linear dependence of secants in the Kummer variety if and only if the Riemann matrix corresponds to a Jacobian variety as shown by Krichever. The theta functions in terms of which these secants are expressed depend on the Abel maps of four arbitrary points on a Riemann surface. However, there is no concept of an Abel map for general $\mathbb{B} \in \mathbb{H}^{g}$. To establish linear …
Translating auxiliary symmetries between Schottky uniformization and Jacobi parametrization
本文研究了在Schottky统一化和Jacobi参数化之间翻译辅助对称性，以实现函数关系的双向转换，从而提升特殊函数的数值计算效率。
We introduce and study (strict) Schottky G-bundles over a compact Riemann surface X, where G is a connected reductive algebraic group. Strict Schottky representations are shown to be related to branes in the moduli space of G-Higgs bundles over X, and we prove that all Schottky $G$-bundles have trivial topological type. Generalizing the Schottky moduli map introduced in Florentino to the setting of principal bundles, we prove its local surjectivity at the good and unitary locus. Finally, we prove that the Schottky map is surjective onto the space of flat bundles for two special classes: when G…
We study a natural map from representations of a free (resp. free abelian) group of rank g in GL_r(C), to holomorphic vector bundles of degree zero over a compact Riemann surface X of genus g (resp. complex torus X of dimension g). This map defines what is called a Schottky functor. Our main result is that this functor induces an equivalence between the category of unipotent representations of Schottky groups and the category of unipotent vector bundles on X. We also show that, over a complex torus, any vector or principal bundle with a flat holomorphic connection is Schottky.
Novikov's conjecture on the Riemann-Schottky problem: {\it the Jacobians of smooth algebraic curves are precisely those indecomposable principally polarized abelian varieties (ppavs) whose theta-functions provide solutions to the Kadomtsev-Petviashvili (KP) equation}, was the first evidence of nowadays well-established fact: connections between the algebraic geometry and the modern theory of integrable systems is beneficial for both sides. The purpose of this paper is twofold. Our first goal is to present a proof of the strongest known characterization of a Jacobian variety in this direction: …
An Ansatz for the Poincaré metric on compact Riemann surfaces is proposed. This implies that the Liouville equation reduces to an equation resembling a non chiral analogous of the higher genus relationships (KP equation) arising in the framework of Schottky's problem solution. This approach connects uniformization (Fuchsian groups) and moduli space theories with KP hierarchy. Besides its mathematical interest, the Ansatz has some applications in the framework of quantum Riemann surfaces arising in 2D gravity.
We give an explicit weak solution to the Schottky problem, in the spirit of Riemann and Schottky. For any genus $g$, we write down a collection of polynomials in genus $g$ theta constants, such that their common zero locus contains the locus of Jacobians of genus $g$ curves as an irreducible component. These polynomials arise by applying a specific Schottky-Jung proportionality to an explicit collection of quartic identities for theta constants in genus $g-1$, which are suitable linear combinations of Riemann's quartic relations.
KP solitons and the Schottky uniformization
本文针对TNN格拉斯曼流形的每个元素构造Schottky群以一致化相应黎曼曲面，并通过退化有限间隙解得到KP孤子。
In this survey of works on a characterization of Jacobians and Prym varieties among indecomposable principally polarized abelian varieties via the soliton theory we focus on a certain circle of ideas and methods which show that the characterization of Jacobians as ppav whose Kummer variety admits a trisecant line and the Pryms as ppav whose Kummer variety admits a pair of symmetric quadrisecants can be seen as an abelian version of pole systems arising in the theory of elliptic solutions to the basic soliton hierarchies. We present also recent results in this direction on the characterization …
Crossing the transcendental divide: from Schottky groups to algebraic curves
本文构造亏格 $g\geq 3$ 且带对称性与 $M$-curve 实结构的非超椭圆 Schottky Riemann 面，并数值计算其对应代数曲线与 Riemann 矩阵，以沟通解析与代数表示。
Using the arithmetic Schottky uniformization theory, we show the arithmeticity of $PSL_{2}({\mathbb C})$ Chern-Simons invariant. In terms of this invariant, we give an explicit formula of the Riemann-Roch isomorphism as Zograf-Mcintyre-Takhtajan's infinite product for families of algebraic curves. By this formula, we determine the unknown constant which appears in the holomorphic factorization formula of determinant of Laplacians on Riemann surfaces via the classical Liouville action. As an application, we show the rationality of Ruelle zeta values for Schottky uniformized $3$-manifolds.
In this survey we discuss some of the classical and modern methods in studying the (Riemann-)Schottky problem, the problem of characterizing Jacobians of curves among principally polarized abelian varieties. We present many of the recent results in this subject, and describe some directions of current research. This paper is based on the talk given at the "Classical algebraic geometry today" workshop at MSRI in January 2009.
In this paper, an explicit hierarchy of differential equations for the $τ$-functions defining the moduli space of curves with automorphisms as a subscheme of the Sato Grassmannian is obtained. The Schottky problem for Riemann surfaces with automorphisms consists of characterizing those p.p.a.v. that are Jacobian varieties of a curve with a non-trivial automorphism. A characterization in terms of hierarchies of p.d.e. for theta functions is also given.
We prove the following converse of Riemann's Theorem: let (A,Θ) be an indecomposable principally polarized abelian variety whose theta divisor can be written as a sum of a curve and a codimension two subvariety Θ=C+Y. Then C is smooth, A is the Jacobian of C, and Y is a translate of W_{g-2}(C). As applications, we determine all theta divisors that are dominated by a product of curves and characterize Jacobians by the existence of a d-dimensional subvariety with curve summand whose twisted ideal sheaf is a generic vanishing sheaf.
We extend the spectral theory of generalized Laplacians to integrable metrics on compact Riemann surfaces. As a consequence, we attach in a direct way, a holomorphic analytic torsion to any integrable metrics. We also provide a different approach to define the holomorphic analytic torsion. We prove that both approaches agree.
We obtain variational formulas for holomorphic objects on Riemann surfaces with respect to arbitrary local coordinates on the moduli space of complex structures. These formulas are written in terms of a canonical object on the moduli space which corresponds to the pairing between the space of quadratic differentials and the tangent space to the moduli space. This canonical object satisfies certain commutation relations which appear to be the same as the ones that emerged in the integrability theory of Whitham type hierarchies. Driven by this observation, we develop the theory of Whitham type h…
We continue the study of the spectral theory associated to integrable metrics, started in our previous paper arXiv:1301.1793 [math.SP]. We introduce the notion of 1-integrable metric on line-bundles on a compact Riemann surface. We extend the spectral theory of generalized Laplacians to line-bundles equipped with 1-integrable metrics. As an application, we recover the following identity: [ζ'_{Δ_{\bar{\mathcal{O}(m)}_\infty}}(0)=T_g\bigl((\p^1,ω_\infty); \bar{\mathcal{O}(m)}_\infty \bigr),] obtained using direct computations in arXiv:1301.1792 [math.NT].
On the universality of integrable deformations of solutions of degenerate Riemann-Hilbert-Birkhoff problems
本文研究退化 RHB 问题解的可积形变分类，证明 Sabbah 形变虽非严格普适，却具有相对普适性，并刻画了可局部全纯 Jordan 化的矩阵值映射。
We present the correspondence between Lax integrable systems with spectral parameter on a Riemann surface, and Conformal Field Theories, in quite general set-up suggested earlier by the author. This correspondence turns out to give a prequantization of the integrable systems in question.
Physically meaningful periodic solutions to certain integrable partial differential equations are given in terms of multi-dimensional theta functions associated to real Riemann surfaces. Typical analytical problems in the numerical evaluation of these solutions are studied. In the case of hyperelliptic surfaces efficient algorithms exist even for almost degenerate surfaces. This allows the numerical study of solitonic limits. For general real Riemann surfaces, the choice of a homology basis adapted to the anti-holomorphic involution is important for a convenient formulation of the solutions an…
The Information-Theoretic Schottky Problem treats the period matrix of a compact Riemann Surface as a compressible signal. In this case, the period matrix of a smooth plane curve is characterized by only 4 of its columns, a significant compression.
General tropical convergence of harmonic amoebas
本文通过施托克西统一化理论研究了和谐阿莫埃博到非简单热带曲线的收敛性，并扩展了兰格的结果以应用于更广泛的情况。
This paper is a survey on relations between secant identities and soliton equations and applications of soliton equations to problems of algebraic geometry, i.e., the Riemann-Schottky problem and its analogues. A short introduction into the analytic theory of theta functions is also given.
We give a geometric derivation of Schottky's equation in genus four for the period matrices of Riemann surfaces among all period matrices. The equation arises naturally from the singularity theory of the Gauss map on the theta divisor, and thus generalizes for any genus $g \ge 4$ to a certain ideal of Siegel modular forms vanishing on period matrices of Riemann surfaces. This ideal is generated by modular forms associated to the invariants of cubic forms in $g - 1$ variables which vanish on the Fermat cubic.
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