Good formal structures for flat meromorphic connections, I: Surfaces
Good formal structures for flat meromorphic connections, I: Surfaces
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平坦亚晶连接的良好形式结构，I：表面
DOI：
10.1215/00127094-2010-041
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发表时间：
2008
影响因子：
2.5
通讯作者：
K. Kedlaya
中科院分区：
文献类型：
--
作者：
K. Kedlaya
我们给出了一个标准，在该标准下，人们可以在任意维度的复杂解析或代数簇上获得形式平坦连接的良好分解（在马尔格朗日意义上）。该准则根据微分算子的谱行为来表述，并推广了一维情况下 Robba 的 Hukuhara-Levelt-Turrittin 分解构造。作为一个应用，我们证明了在适当的爆炸后，表面上平坦亚纯连接存在良好的形式结构；这验证了 Sabbah 的猜想，并扩展了 Mochizuki 的代数联系的结果。我们的证明在与表面上的点相关的价值树上使用有限性论证，以验证数值标准。
We give a criterion under which one can obtain a good decomposition (in the sense of Malgrange) of a formal flat connection on a complex analytic or algebraic variety of arbitrary dimension. The criterion is stated in terms of the spectral behavior of differential operators, and generalizes Robba's construction of the Hukuhara-Levelt-Turrittin decomposition in the one-dimensional case. As an application, we prove the existence of good formal structures for flat meromorphic connections on surfaces after suitable blowing up; this verifies a conjecture of Sabbah, and extends a result of Mochizuki for algebraic connections. Our proof uses a finiteness argument on the valuative tree associated to a point on a surface, in order to verify the numerical criterion.