MODULI OF SEMISTABLE LOGARITHMIC CONNECTIONS
MODULI OF SEMISTABLE LOGARITHMIC CONNECTIONS
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半稳态对数连接模
DOI：
10.1090/s0894-0347-1993-1182671-2
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发表时间：
1993
影响因子：
3.9
通讯作者：
N. Nitsure
中科院分区：
文献类型：
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作者：
N. Nitsure
在范畴等价的Riemann Hilbert对应下，非奇异复射影簇X上的局部系统对应于对E =（s，V），其中'是X上的局部自由层，V：8 Q1 @是F上的可积联络。Deligne（见[D]）将这种对应推广到非完全拟投射簇，证明了对应于这种簇Y上的局部系统的正确代数几何对象是Y上的所谓正则联络。Y上代数联络的正则性条件可以表示如下。确定Y的Hironaka完备化X，即包含Y作为开子簇的非奇异投射簇X，使得补S = XY是具有正规交叉的光滑因子。X上的一个对数联络F =（Y，V）在S上具有奇异性，根据定义，它是X上的一个无挠相干层Y和一个映射V：9 -* 21[log S]？9，其中i14[logS]是X上的有理1-形式层，其对数极点在S上.假设连接V的曲率为零。最后，Y上的代数联络E =（F，V）是正则的当且仅当存在X上的对数联络F，其在S上具有奇点，且扩展E。如果存在这样一个扩张F，那么下面的层F可以被选择为局部自由的，并且将被称为E的对数格。一个给定的正则联络的对数格当然不是唯一的，但是有一个选择它的规范方法叫做德利涅构造，它只依赖于映射exp：C -的集合论部分的选择。C*。由于这样的截面甚至不是连续的，德利涅格在正则连接族中表现不好。Simpson在[S]中给出了射影簇X上（非奇异）联络的模概型的构造。然而，一个简单的例子表明，期望在拟投射簇Y上正则联络的模概型的普遍存在是不合理的。为此，设Y为仿射线减去原点，坐标为x。带有参数t的微分方程dy/dx tylx = 0定义了层Y上的一族正则联络，由坐标为t的仿射线T参数化。我们可以看到t1和t2参数化同构连接当且仅当t1 t2
Under the Riemann Hilbert correspondence, which is an equivalence of categories, local systems on a nonsingular complex projective variety X correspond to pairs E = (s, V) where ' is a locally free sheaf on X, and V: 8 Q1 @ is an integrable connection on F. Generalizing this correspondence to noncomplete quasi-projective varieties, Deligne (see [D]) showed that the correct algebro-geometric objects which correspond to local systems on such a variety Y are the so-called regular connections on Y. The condition of regularity on algebraic connections on Y can be expressed as follows. Fix a Hironaka completion X of Y, that is, nonsingular projective variety X which contains Y as an open subvariety, such that the complement S = X Y is a smooth divisor with normal crossings. A logarithmic connection F = (Y, V) on X with singularities over S is by definition a torsion-free coherent sheaf Y on X together with a map V: 9 -* 21[log S]? 9 which satisfies C-linearity and the Leibniz rule, where i14[logS] is the sheaf of rational 1-forms on X with logarithmic poles on S. The curvature of the connection V is assumed to be zero. Finally, an algebraic connection E = (F, V) on Y is regular if and only if there exists a logarithmic connection F on X with singularities over S which extends E. If such an extension F exists, the underlying sheaf F can be chosen to be locally free, and will be called a logarithmic lattice for E. A logarithmic lattice for a given regular connection is of course not unique, but there is a canonical way of choosing it called Deligne's construction, which only depends on the choice of a set-theoretic section of the map exp: C -. C* . As such a section is not even continuous, the Deligne lattice does not behave well in families of regular connections. Simpson has given in [S] a construction of a moduli scheme for (nonsingular) connections on a projective variety X. However, a simple example shows that it would be unreasonable to expect in general the existence of a moduli scheme for regular connections on a quasi-projective variety Y. For this, let Y be the affine line minus the origin, with coordinate x. The differential equation dy/dx tylx = 0 with parameter t defines a family of regular connections on the sheaf Y, parametrized by the affine line T with coordinate t. We can see that t1 and t2 parametrize isomorphic connections if and only if t1 t2