no. 1
The Intrinsic Normal Cone for Artin Stacks
[Le cône normal intrinsèque pour les champs d’Artin]
Aranha, Dhyan1 ; Pstrągowski, Piotr2
1 Universität Duisburg-Essen Essen (Germany)
2 Institute for Advanced Study Princeton (USA)
Annales de l'Institut Fourier, Tome 74 (2024) no. 1, pp. 71-120.

Nous étendons la construction du cône normal d’une immersion fermée de schémas à tout morphisme localement de type fini de champs d’Artin supérieurs et montrons que dans le cas de Deligne-Mumford notre construction retrouve le cône normal intrinsèque relatif de Behrend et Fantechi. Nous caractérisons notre extension comme l’unique satisfaisant une courte liste d’axiomes, et l’utilisons pour construire la déformation du cône normal. Comme application de nos méthodes, nous associons à tout morphisme de champs d’Artin muni d’un choix d’une théorie d’obstruction parfaite globale une classe fondamentale virtuelle relative dans le groupe de Chow de Kresch.

We extend the construction of the normal cone of a closed embedding of schemes to any locally morphism of finite type of higher Artin stacks and show that in the Deligne-Mumford case our construction recovers the relative intrinsic normal cone of Behrend and Fantechi. We characterize our extension as the unique one satisfying a short list of axioms, and use it to construct the deformation to the normal cone. As an application of our methods, we associate to any morphism of Artin stacks equipped with a choice of a global perfect obstruction theory a relative virtual fundamental class in the Chow group of Kresch.

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DOI : 10.5802/aif.3583
Classification : 14C15
Keywords: Moduli space, normal cone, deformation theory, Artin stack, Chow group
Mot clés : Espace des modules, cône normal, théorie des déformations, champs d’Artin, groupe de Chow

Aranha, Dhyan 1 ; Pstrągowski, Piotr 2

1 Universität Duisburg-Essen Essen (Germany)
2 Institute for Advanced Study Princeton (USA)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Aranha, Dhyan; Pstrągowski, Piotr. The Intrinsic Normal Cone for Artin Stacks. Annales de l'Institut Fourier, Tome 74 (2024) no. 1, pp. 71-120. doi : 10.5802/aif.3583. https://aif.centre-mersenne.org/articles/10.5802/aif.3583/

[1] André, Michel Homologie des algèbres commutatives, Grundlehren der Mathematischen Wissenschaften, 206, Springer, 1974, xv+341 pages | DOI | MR

[2] Antieau, Benjamin; Gepner, David Brauer groups and étale cohomology in derived algebraic geometry, Geom. Topol., Volume 18 (2014) no. 2, pp. 1149-1244 | DOI | MR | Zbl

[3] Battistella, Luca; Carocci, Francesca; Manolache, Cristina Virtual classes for the working mathematician (2018) (https://arxiv.org/abs/1804.06048)

[4] Behrend, Kai Gromov–Witten invariants in algebraic geometry, Invent. Math., Volume 127 (1997) no. 3, pp. 601-617 | DOI | MR | Zbl

[5] Behrend, Kai; Fantechi, Barbara The intrinsic normal cone, Invent. Math., Volume 128 (1997) no. 1, pp. 45-88 | DOI | MR | Zbl

[6] Bhatt, Bhargav Completions and derived de Rham cohomology (2012) (https://arxiv.org/abs/1207.6193)

[7] Donaldson, Simon K.; Thomas, Richard P. Gauge theory in higher dimensions, The geometric universe (Oxford, 1996), Oxford University Press, 1998, pp. 31-47 | DOI | MR | Zbl

[8] Fulton, William Intersection theory, 2, Springer, 2013

[9] Gaitsgory, Dennis; Rozenblyum, Nick A study in derived algebraic geometry: Volume I: correspondences and duality, Mathematical Surveys and Monographs, 221, American Mathematical Society, 2017

[10] Grothendieck, Alexander Catégories cofibrées additives et complexe cotangent relatif, Lecture Notes in Mathematics, 79, Springer, 1968, ii+167 pages | MR

[11] Illusie, Luc Complexe cotangent et déformations. I, Lecture Notes in Mathematics, 239, Springer, 1971, xv+355 pages | DOI | MR

[12] Khan, Adeel A. Virtual fundamental classes of derived stacks I (2019) (https://arxiv.org/abs/1909.01332)

[13] Khan, Adeel A. Algebraic K-theory of quasi-smooth blow-ups and cdh descent, Ann. Henri Lebesgue, Volume 3 (2020), pp. 1091-1116 | DOI | Numdam | MR | Zbl

[14] Kontsevich, Maxim Enumeration of rational curves via torus actions, The moduli space of curves (Texel Island, 1994) (Progress in Mathematics), Volume 129, Birkhäuser, 1995, pp. 335-368 | DOI | MR | Zbl

[15] Kontsevich, Maxim; Manin, Yurii Gromov–Witten classes, quantum cohomology, and enumerative geometry, Commun. Math. Phys., Volume 164 (1994) no. 3, pp. 525-562 http://projecteuclid.org/euclid.cmp/1104270948 | DOI | MR | Zbl

[16] Kresch, Andrew Cycle groups for Artin stacks, Invent. Math., Volume 138 (1999), pp. 495-536 | DOI | MR | Zbl

[17] Lee, Yuan-Pin Quantum K-theory, I: Foundations, Duke Math. J., Volume 121 (2004) no. 3, pp. 389-424 | MR | Zbl

[18] Lurie, Jacob Derived Algebraic Geometry, Ph. D. Thesis, Massachusetts Institute of Technology, Dept. of Mathematics (USA) (2004) includes bibliographical references (p. 191-193).

[19] Lurie, Jacob Higher Topos Theory, Annals of Mathematics Studies, 170, Princeton University Press, 2009, xviii+925 pages | DOI | MR

[20] Lurie, Jacob Higher Algebra (2017) (http://www.math.harvard.edu/~lurie/papers/HA.pdf)

[21] Lurie, Jacob Spectral Algebraic Geometry (2018) (http://www.math.harvard.edu/~lurie/papers/SAG-rootfile.pdf)

[22] Pandharipande, Rahul; Thomas, Richard P. Curve counting via stable pairs in the derived category, Invent. Math., Volume 178 (2009) no. 2, pp. 407-447 | DOI | MR | Zbl

[23] Quillen, Daniel On the (co-) homology of commutative rings, Applications of Categorical Algebra (New York, 1968) (Proceedings of Symposia in Pure Mathematics), Volume 17, American Mathematical Society (1970), pp. 65-87 | DOI | MR | Zbl

[24] Raynaud, Michele; Grothendieck, Alexander Revêtements étales et groupe fondamental: Séminaire de Géométrie Algébrique du Bois Marie 1960/61 (SGA 1), Lecture Notes in Mathematics, 224, Springer, 2006

[25] Simpson, Carlos Algebraic (geometric) n-stacks (1996) (https://arxiv.org/abs/alg-geom/9609014)

[26] Tanaka, Yuuji; Thomas, Richard P. Vafa–Witten invariants for projective surfaces I: stable case, J. Algebr. Geom., Volume 29 (2020) no. 4, pp. 603-668 | DOI | MR | Zbl

[27] Toën, Bertrand Higher and derived stacks: a global overview, Algebraic geometry—Seattle 2005. Part 1 (Proceedings of Symposia in Pure Mathematics), Volume 80, American Mathematical Society, 2009, pp. 435-487 | DOI | MR | Zbl

[28] Toën, Bertrand; Vezzosi, Gabriele Homotopical algebraic geometry I: Topos theory, Adv. Math., Volume 193 (2005) no. 2, pp. 257-372 | DOI | MR | Zbl

[29] Toen, Bertrand; Vezzosi, Gabriele Homotopical algebraic geometry. II. Geometric stacks and applications., Memoirs of the American Mathematical Society, 902, American Mathematical Society, 2008, x+224 pages

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