Study of moduli spaces, and K3 moonshine

1. 我们在不使用Grothendieck’s quote格式的情况下重构了几何不变理论，构造了向量束的模空间。两者都大大简化了向量束的模理论。我们期待在此基础上取得新的发展。例如，用我们的描述2来研究雅可比矩阵的退化是很有趣的。此外，还简化了具有附加结构的向量束的模空间的构造，如抛物线结构或稳定对。由此，著名的Verlinde公式现在被认为是某不变环的Cayley-Sylvester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，将成为现代不变理论中的定理。在刺破的黎曼球或等点投影线上，存在二阶抛物矢量束模的主空间。它的坐标环是多项式环上二维加性群的某平方零线性作用的不变性环。特别地，不变环是有限生成的。与下面提到的结果一起，我们解决了多维加性群的平方自由作用的（原来的）Hilbert第十四问题。我们构造了三维加性群的希尔伯特第十四问题的一个反例。这个环与5维射影空间在九个点上的放大的总坐标环同构。我们也给出了这个同构的简化证明。在两个K3曲面积上的一类Hodge环的代数性上，给出了Shafarevich猜想的一个新的证明。我们定义了一种阿贝尔变体的双能级结构，并研究了配备这种结构的阿贝尔曲面的模量。当偏振类型为（1,d）和d【小于等于】5时，模空间非常简单，且具有很多几何特征。将迹公式应用于这一模问题并确定自同构环的乘结构是非常有趣的。少

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