超周期与超弦测度在超曲线模空间边界附近的行为
Superperiods and superstring measure near the boundary of the moduli space of supercurves
AI总结：
本文研究超曲线模空间边界附近超周期映射的行为，证明其与经典周期相似，并应用于亏格2模空间，刻画其投影性质，结合Mumford同构展开超弦测度，同时给出Deligne正则性证明。
摘要：
我们研究了稳定超曲线模空间边界附近超周期映射的行为，并证明了它与经典曲线周期的行为相似。我们考虑将该结果应用于亏格为 $2$ 的模空间（记为 $\bar{\mathcal S}_2$）的几何。首先，我们根据其在边界附近的行为刻画了 $\bar{\mathcal S}_2$ 的典范投影，特别证明了 $\bar{\mathcal S}_2$ 不是投影的。其次，我们将超周期的信息与 Witten 给出的亏格 $2$ Mumford 同构的显式计算相结合，研究亏格 $2$ 超弦测度在边界附近的展开。我们还给出了 Deligne 关于任意亏格 $g$ 的超弦测度在 $\bar{\mathcal S}_g$ 上正则性的证明。
英文摘要：
We study the behavior of the superperiod map near the boundary of the moduli space of stable supercurves and prove that it is similar to the behavior of periods of classical curves. We consider two applications to the geometry of this moduli space in genus $2$, denoted as $\bar{\mathcal S}_2$. First, we characterize the canonical projection of $\bar{\mathcal S}_2$ in terms of its behavior near the boundary, proving in particular that $\bar{\mathcal S}_2$ is not projected. Secondly, we combine the information on superperiods with the explicit calculation of genus $2$ Mumford isomorphism, due to Witten, to study the expansion of the superstring measure for genus $2$ near the boundary. We also present the proof, due to Deligne, of regularity of the superstring measure on $\bar{\mathcal S}_g$ for any genus.
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