抛物算子与微分算子
Parabolic opers and differential operators
摘要：
在[BDP]中，在除子上具有抛物结构的曲线上的向量丛的集合中，定义并研究了抛物SL（r，C）-算子.本文引入并研究了曲线上抛物型向量丛之间的全纯微分算子。本文考虑黎曼曲面X上的抛物型SL（r，C）-算子，其奇异因子S是给定的，抛物权固定，且满足S中任意点的抛物权$x_i$都是$\frac{1}{2N_i+1}$的整数倍，其中$N_i > 1$是固定整数.我们证明了这个算子空间是X上两个自然抛物线丛之间的全纯r阶微分算子的仿射空间（只依赖于因子S和权$N_i$），满足以下条件：微分算子的主符号是常数函数1，次主象征同为零.子主象征的消失确保秩r丛上的对数联络实际上是对数SL（r，C）-联络。
英文摘要：
Parabolic SL(r,C)-opers were defined and investigated in [BDP] in the set-up of vector bundles on curves with a parabolic structure over a divisor. Here we introduce and study holomorphic differential operators between parabolic vector bundles over curves. We consider the parabolic SL(r,C)-opers on a Riemann surface X with given singular divisor S and with fixed parabolic weights satisfying the condition that all parabolic weights at any point $x_i$ in S are integral multiples of $\frac{1}{2N_i+1}$, where $N_i > 1$ are fixed integers. We prove that this space of opers is canonically identified with the affine space of holomorphic differential operators of order r between two natural parabolic line bundles on X (depending only on the divisor S and the weights $N_i$) satisfying the conditions that the principal symbol of the differential operators is the constant function 1 and the sub-principal symbol vanishes identically. The vanishing of the sub-principal symbol ensures that the logarithmic connection on the rank r bundle is actually a logarithmic SL(r, C)-connection.
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