Study of moduli spaces, and K3 moonshine

模空间和 K3 月光的研究

• 批准号: 10304001
• 负责人: MUKAI Shigeru
• 金额: $ 7.48万
• 依托单位: KYOTO UNIVERCITY (2001)Nagoya University (1998-2000)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (A)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10304001/
• 关键词: moduli; vector bundle; invariant; Verlinde formula; conformal block; Hilbert's 14th problem; fundamental domain; K3 surface; Hodeg予想; モジュライ空間; 不変式論; Brill-Noether理論; trisecant line公式; 有界対称領域; ラータ関数; アーベル曲面; 佐武コンパクト化; テータ関数; フーベル曲面; 正多面体

1. We reconstructed the geometric invariant theory and constructed the moduli space of vector bundles without using the Grothendieck's Quot-scheme. Both simplified the moduli theory of vector bundles a lot. We expect new development will be followed on this foundation. For example, it is interesting to study the degeneration of Jacobian using our description.2. The construction of moduli spaces of vector bundles with additional structure, say parabolic structure or stable pair, were also simplified. By virtue of this, the celebrated Verlinde formula is now regarded as the Cayley-Sylvester type explicit formula for a certain invariant ring. We hope that many mathematics around the formula, including the affine Lie algebra, Hecke algebra and quantum group, will become theorems in a modern invariant theory.3. The master space of the moduli of rank two parabolic vector bundles over punctured Riemann sphere, or equivalently pointed projective line, exists. Its coordinate ring is the invaria … More nt ring of a certain square zero linear action of the 2-dimensional additive group on a polynomial ring. In particular, the invariant ring is finitely generated. Together with the results mentioned below, we have solved the (original) Hilbert fourteenth problem for the square free action of multi-dimensional additive groups.4. We constructed a counterexample of Hilbert's fourteenth problem for the 3-dimensional additive group. This ring is isomorphic to the total coordinate ring of the blow-up of the 5-dimensional projective space at nine points. We also gave a simplified proof of this isomorphism.5. We found a new proof of the Shafarevich conjecture on the algebraicity of a certain class of Hodge cycles on the product of two K3 surfaces.6. We defined a bi-level structure of an abelian variety and studied the moduli of abelian surfaces equipped with this structures. The moduli spacce is very simple and has a lot of geometry when the polarization type is (1,d) and d 【less than or equal】 5. It is very interesting to apply the trace formula to this moduli problem and determine the multiplicative structure of the ring of automorphic forms. Less

1.本文在不使用Grothendieck公式的情况下，重建了几何不变理论，构造了向量丛的模空间。两者都大大简化了向量丛的模理论。我们期待在此基础上有新的发展。例如，用我们的方法研究雅可比矩阵的退化是很有趣的.对具有附加结构的向量丛的模空间的构造也作了简化，如抛物结构或稳定对。因此，著名的Verlinde公式现在被认为是某个不变环的凯莱-西尔维斯特型显式公式。我们希望围绕这个公式的许多数学，包括仿射李代数、Hecke代数和量子群，都能成为现代不变量理论中的定理.证明了穿孔黎曼球面上二秩抛物向量丛的模的主空间存在。它的坐标环是不变量 ...更多信息 多项式环上二维加法群的某个平方零线性作用的环。特别地，不变环是双生成的。结合下面提到的结果，我们解决了多维加法群的平方自由作用的（原）Hilbert第十四问题.构造了三维可加群的Hilbert第十四问题的一个反例.这个环同构于5维射影空间在9个点上爆破的全坐标环。我们也给出了这种同构的一个简化证明.我们发现了Shafarevich猜想关于两个K3曲面乘积上的一类Hodge圈的代数性的一个新证明.定义了阿贝尔簇的双层结构，并研究了具有双层结构的阿贝尔曲面的模。当极化类型为（1，d）且d [小于或等于] 5时，模空间非常简单，并有许多几何形状。将迹公式应用于这个模问题并确定自守形式环的乘法结构是非常有趣的。少

项目成果

MUKAI,Shigeru: "Duality of polarized K3 surfaces" proc.Euroconference on Algebraic Geometry. 107-122 (1998)

MUKAI，Shigeru：“偏振 K3 表面的对偶性”proc.欧洲代数几何会议。

UMEMURA,Hiroshi: "On the transformation group of the second Paninleve equation" Nagoya Math.J.to appear.

UMEMURA、Hiroshi：“论第二个 Paninleve 方程的变换群”Nagoya Math.J. 出现。

MUKAI, Shigeru: "Duality of polarized K3 surfaces"Proc. Euroconference on Alg. Geom.. 107-122 (1998)

MUKAI，Shigeru：“偏振 K3 表面的对偶性”Proc。

向井 茂: "モジュライ理論1, 2"岩波書店. 455 (2000)

向井茂：“Modurai理论1、2”岩波书店455（2000）。

SAITO, Masa-Hiko, UMEMURA, Hiroshi: "Painleve equations and deformations of rational surfaces with rational double points"Physics and combinatorics. 320-365 (1999)

SAITO、Masa-Hiko、UMEMURA、Hiroshi：“Painleve 方程和有理双点有理曲面的变形”物理学和组合数学。