论文标题
关于不变喷气差异的有限生成纤维环
On finite generation of fiber ring of invariant jet differentials
论文作者
论文摘要
本文的主要结果是涉及不变$ k $ jet的总环，作为猜想的概括，该猜想是在\ cite {dmr}中首次提到的，后来在\ cite {d1，d2，me，bk}中进行了研究。 \ cite {dmr}中的猜想指出，在常规位点有限地生成了射击歧管的不变喷射差速器空间的光纤环，即不变的$ k $ jets的归一化环有限地生成。我们还独立证明了这个猜想。 Berczi-Kirwan在\ cite {bk}（2012）中进行了部分尝试，但猜想仍然开放。我们的方法是不同的，并且对这个问题很新。在投影量$ x $上的常规$ k $ jets的分析自动形态组是通用线性组$ gl_k \ mathbb {c} $的非还原子组。在这种情况下，纤维环中不变多项式的Chevalley定理的证明陷入困境。需要不同的方法来证明不变的环有限生成。我们采用了一些代数谎言组的技术（不一定是还原），以及Berczi和Kirwan（2012）获得的主要结果，以证明不变喷射环的有限生成。此外，在上面，我们对非还原谎言组$ g_k $及其Lie代数的结构理论进行了系统的研究。我们考虑了$ g_k $的两个动作，这是对其谎言代数的第一个动作，第二个动作是$ k $ -Jets的上述操作。我们证明了这两个动作中的这两个动作都有通用稳定器，并在副商图上获得了各种结果。最后，我们介绍了$ g_k $的表示环的结构。
The main result of this paper concerns the finite generation of the total ring of invariant $k$-jets as a generalization of a conjecture which was first mentioned in \cite{DMR} and later studied in \cite{D1, D2, Me, BK}. The conjecture in \cite{DMR} states that the fiber ring of the space of invariant jet differentials of a projective manifold is finitely generated on the regular locus, i.e. the normalized ring of invariant $k$-jets is finitely generated. We also prove this conjecture independently. Berczi-Kirwan has a partial attempt toward the question in \cite{BK} (2012), but the conjecture remains open. Our method is different and quite new to this question. The analytic automorphism group of regular $k$-jets of holomorphic curves on a projective variety $X$ is a non-reductive subgroup of the general linear group $GL_k \mathbb{C}$. In this case, the proof of the Chevalley theorem on the invariant polynomials in the fiber rings falls into difficulties. Different methods are required for the proof of the finite generation of the ring of invariants. We employ some techniques of algebraic Lie groups (not necessarily reductive) together with primary results obtained by Berczi and Kirwan (2012) to prove the finite generation of the ring of invariant jets. Besides, to above we present a systematic study of the structure theory of the non-reductive Lie group $G_k$ and its Lie algebra. We consider two actions of $G_k$, the first its adjoint action on its Lie algebra and the second the above action on the space of $k$-jets. We prove both of these two actions have generic stabilizers and obtain various results on the associate quotient maps. Finally, we present a result on the structure of the representation ring of $G_k$.
