导出超几何与代数超叠上的完美障碍理论
Derived supergeometry and perfect obstruction theories on algebraic superstacks
浏览论文内容
中文总结 AI 辅助
本文发展代数超叠的形变理论，构造导出超叠与余切超复形，证明拟光滑导出提升给出完美障碍理论，并应用于稳定超映射模空间，计算其虚拟维数。
中文摘要 AI 辅助
我们为代数超叠发展了一个形变理论框架，并将其应用于稳定超映射模空间上完美障碍理论的构造。为实现这一目标，我们采用了导出代数几何中的工具与概念，并将其推广到超几何情形。我们的主要技术贡献包括：(i) 导出超叠的概念，(ii) 其余切超复形，以及 (iii) 代数超叠的完美障碍理论。我们证明了代数超叠的拟光滑导出提升总能产生一个完美障碍理论：这推广了经典形变理论中一个众所周知的原则。随后我们转向 Deligne--Mumford 超叠 $\boldsymbol{\mathcal{M}}^{\mathsf{st-}\mathrm{SUSY}}_{g,\mathfrak{n}}(\boldsymbol{Y})$ 的情形，该超叠参数化固定亏格、同调类以及 Neveu--Schwarz 和 Ramond--Ramond 孔数目的稳定超映射，其目标为光滑射影超概形 $\boldsymbol{Y}$。我们构造了一个自然的导出提升 $\mathbb{R} \boldsymbol{\mathcal{M}}^{\mathsf{st-}\mathrm{SUSY}}_{g,\mathfrak{n}}(\boldsymbol{Y})$ 并计算了其余切超复形；我们证明了在 $\boldsymbol{\mathcal{M}}^{\mathsf{st-}\mathrm{SUSY}}_{g,\mathfrak{n}}(\boldsymbol{Y})$ 上诱导的障碍理论是完美的。最后，我们通过超 Grothendieck--Riemann--Roch 计算了其虚拟维数。所得公式特别给出了先前对稳定超映射所 conjectured 的虚拟维数公式的一个形变理论证明，并退化为经典稳定映射模栈的通常虚拟维数。
英文摘要
We develop a deformation-theoretic framework for algebraic superstacks and apply it to the construction of perfect obstruction theories on moduli spaces of stable supermaps. In order to achieve this we employ the tools and concepts from derived algebraic geometry, extending them to the supergeometric setting. Our main technical contributions are (i) the notions of derived superstacks, (ii) their cotangent supercomplexes and (iii) perfect obstruction theories for algebraic superstacks. We prove that a quasi-smooth derived enhancement of an algebraic superstack always yields a perfect obstruction theory: this generalizes a well-known principle from classical deformation theory. We then turn to the case of the Deligne--Mumford superstack $\boldsymbol{\mathcal{M}}^{\mathsf{st-}\mathrm{SUSY}}_{g,\mathfrak{n}}(\boldsymbol{Y})$ of stable supermaps of fixed genus, homology class and numbers of Neveu--Schwarz and Ramond--Ramond punctures, with target a smooth projective superscheme $\boldsymbol{Y}$. We exhibit a natural derived enhancement $\mathbb{R} \boldsymbol{\mathcal{M}}^{\mathsf{st-}\mathrm{SUSY}}_{g,\mathfrak{n}}(\boldsymbol{Y})$ and compute its cotangent supercomplex; we prove that the induced obstruction theory on $\boldsymbol{\mathcal{M}}^{\mathsf{st-}\mathrm{SUSY}}_{g,\mathfrak{n}}(\boldsymbol{Y})$ is perfect. Finally, we compute its virtual dimension by super Grothendieck--Riemann--Roch. The resulting formula gives, in particular, a deformation-theoretic proof of the virtual dimension formula previously conjectured for stable supermaps and specializes to the usual virtual dimension of the classical moduli stack of stable maps.
