Derived supergeometry and perfect obstruction theories on algebraic superstacks
Abstract
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 ℳst−SUSYg,n(Y) of stable supermaps of fixed genus, homology class and numbers of Neveu--Schwarz and Ramond--Ramond punctures, with target a smooth projective superscheme Y. We exhibit a natural derived enhancement ℝ ℳst−SUSYg,n(Y) and compute its cotangent supercomplex; we prove that the induced obstruction theory on ℳst−SUSYg,n(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.
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BibTeX
@misc{bruzzo2026derived,
title = {{Derived supergeometry and perfect obstruction theories on algebraic superstacks}},
author = {Ugo Bruzzo and Daniel Hernández Ruipérez and Emanuele Pavia and Andrea T. Ricolfi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08608},
eprint = {2610.08608},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08608},
}
RIS
TY - UNPB
TI - Derived supergeometry and perfect obstruction theories on algebraic superstacks
AU - Ugo Bruzzo
AU - Daniel Hernández Ruipérez
AU - Emanuele Pavia
AU - Andrea T. Ricolfi
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08608
UR - https://arxiv.org/abs/2610.08608
AB - 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 ℳ^st−SUSY_g,n(Y) of stable supermaps of fixed genus, homology class and numbers of Neveu--Schwarz and Ramond--Ramond punctures, with target a smooth projective superscheme Y. We exhibit a natural derived enhancement ℝ ℳ^st−SUSY_g,n(Y) and compute its cotangent supercomplex; we prove that the induced obstruction theory on ℳ^st−SUSY_g,n(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.
ER -