Duality in the smooth projective case for motivic spectra

Duality in the smooth projective case for (generalized) motivic spectra

Workgroup on motivic homotopy theory
13–17 July 2026, precise schedule to be fixed
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Schedule: dates and speakers to be fixed, within the week 13–17 July 2026
Location: TBA

Presentation: This workgroup will be devoted to the proof of the duality results of Annala–Hoyois–Iwasa in the non- \(\mathbb A^1\)-invariant framework of their motivic spectra. The aim is to understand the ambidexterity theorem for smooth projective morphisms and the resulting Atiyah duality statement, while keeping close to the geometry of the proof.

A central point will be the construction and use of Gysin maps in this setting, following Tang. We will attach particular attention to the arguments which allow one to circumvent the classical purity arguments: in the category \(MS_S\), the usual purity map attached to a closed immersion is no longer an isomorphism in general, but one still has enough Gysin functoriality, normalization and compatibility with composition to prove duality in the smooth projective case.

The emphasis will be on the geometric route: deformation to the normal cone, compactified and multiple deformation spaces, normalization and composition of Gysin maps, the geometric evaluation map, the induction for projective spaces, and the final passage from projective spaces to smooth projective morphisms. Applications, such as \(\mathbb A^1\)-colocalization and logarithmic cohomology theories, will be surveyed at the end if time allows.

Program of the workgroup:

TBA	The classical case and overview of the proof	Frédéric Déglise	Quick introduction: the classical picture of purity (Gysin triangles and morphisms, the formal route from Gysin morphisms to duality, enhancement/root of the six-functor formalism). We will review, in a comparative way, the general strategy of the proof of the theorem of Annala–Hoyois–Iwasa, as well as Tang's approach and the main new ideas. Ref: Gysin II, Six functors, AHI2.
TBA	Generalized motivic spectra	Frédéric Déglise	Reminders on derived algebraic geometry. Construction of the category \(MS_S\) of generalized motivic spectra: blowup excision, \(P^1\)-inversion, Thom spaces and the \(J\)-homomorphism. The goal is to isolate the formalism needed for Thom spectra \(\operatorname{Th}_X(\xi)\). Ref: Annala–Iwasa, §§1–3; Annala–Hoyois–Iwasa, Conner–Floyd, §§2–4; AHI2, §2.
TBA	Gysin maps: construction	Davit Harutyunyan	Construction of the Gysin map for a closed immersion between smooth schemes in \(MS_S\). Compactified deformation to the normal cone, derived blowups, excess intersection squares, functoriality, and the factorization through the open complement. Comparison with the classical construction of the deformation space of Fulton and Rost. Compatibility with the monoidal structure. Ref: Tang, §§2–3 and Appendix A; see also AHI2, §2.
TBA	Gysin maps: normalization, composition	Davit Harutyunyan	Proof of the normalization theorem for the Gysin map of the zero section and for \(\mathbb P_Y(\mathcal O)\subset \mathbb P_Y(E\oplus\mathcal O)\). Compatibility of Gysin maps with composition. Ref: Tang, Theorem 3.10 and Corollaries 3.11–3.12; Theorem 3.17.
TBA	Gysin transformation, geometric evaluation map	Leonardo Colombo	Basic functoriality of \(MS\), premotivic structure: \(f^*\), \(p_\sharp\), correspondences. Gysin transformations and their properties. Comparison with the classical construction in \(SH\) via \(\mathbb A^1\)-localization. Definition of the comparison map and its properties. Ref: AHI2, §§2–3 (no need to state Theorem 2.3).
TBA	Projective spaces and geometric duality	Leonardo Colombo	Gysin null-sequences and Gysin quadruples. Duality for \(\mathbb P^n\). Possibly: Euler class in \(MS\), \(SL\)- and \(Sp\)-orientations à la Panin–Walter, associated projective bundle theorems. Ref: AHI2, §§3–4; complementary references: Panin–Walter, Panin–Walter.
TBA	Ambidexterity, Atiyah duality and applications	Ayman Toufik	New form of the Gysin transformation, properties. Proof of ambidexterity for smooth projective morphisms, and deduction of Atiyah duality for Thom spectra. Applications (to be chosen if needed): \(\mathbb A^1\)-colocalization, logarithmic cohomology theories, Landweber exactness and operations in algebraic \(K\)-theory. Ref: AHI2, §5. Applications: §§6–9.
Date	Title	Speaker	Description	Notes

References:

Tang: Tang, Longke. The \(P^1\)-motivic Gysin map. arXiv:2604.24888v1, 2026. arXiv.
Annala–Hoyois–Iwasa (Atiyah duality): Annala, Toni; Hoyois, Marc; and Iwasa, Ryomei. Atiyah duality for motivic spectra. arXiv:2403.01561v1, 2024. arXiv.
Annala–Hoyois–Iwasa (Conner–Floyd): Annala, Toni; Hoyois, Marc; and Iwasa, Ryomei. Algebraic cobordism and a Conner–Floyd isomorphism for algebraic \(K\)-theory. arXiv:2303.02051v2, 2024. arXiv.
Annala–Iwasa: Annala, Toni and Iwasa, Ryomei. Motivic spectra and universality of \(K\)-theory. arXiv:2204.03434v3, 2025. arXiv.
Gysin II: D. Around the Gysin triangle II. Documenta Mathematica 13 (2008), 613–675. arXiv.
Motivic Six functors: Cisinski, Denis-Charles and D. Triangulated categories of mixed motives. Springer Monographs in Mathematics, Springer, 2019. arXiv.
Fulton: Fulton, William. Intersection Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3rd series, vol. 2. Second edition, Springer, 1998. doi.
Rost: Rost, Markus. Chow groups with coefficients. Documenta Mathematica 1 (1996), 319–393. EUDML.
Ayoub: Ayoub, Joseph. Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. Astérisque 314–315, Société Mathématique de France, 2007. SMF.
Röndigs: Röndigs, Oliver. Functoriality in motivic homotopy theory. Unpublished preprint, 2005.
Panin–Walter (MSL and MSp): Panin, Ivan and Walter, Charles. On the algebraic cobordism spectra MSL and MSp. St. Petersburg Math. J. 34 (2023), no. 1, 144–187. arXiv, doi.
Panin–Walter (quaternionic projective bundle): Panin, Ivan and Walter, Charles. Quaternionic Grassmannians and Pontryagin classes in algebraic geometry. Preprint, 2010. arXiv.