Intrinsic Moduli Theory Seminar

This seminar has two parts. The first develops the theory of good moduli spaces and culminates in the existence theorem of Alper, Halpern-Leistner, and Heinloth (AHLH), which characterizes when an algebraic stack admits a good moduli space.

The second introduces Halpern-Leistner’s intrinsic theory of stability and GIT (also known as beyond GIT, or BGIT) using filtrations, numerical invariants, and Θ-stratifications. Combined with the first part, this culminates in the main theorem of intrinsic moduli theory.

This is a joint seminar bringing together algebraic geometers from the University of Melbourne (UniMelb) and the Australian National University (ANU).

When: Mondays, 4pm - 5pm (17 August to 2 November 2026) Where: Zoom (details by email)

Part I

Good Moduli Spaces: Existence and Local Structure

  1. Good Moduli Spaces

    Stacks and Moduli §7.5, Good moduli spaces for Artin stacks

    Adam MonteleoneUniMelb

  2. Luna’s Fundamental Lemma

    Stacks and Moduli §7.5, The étale local structure of algebraic stacks §3, Slices étales §II.2

    Tianqi FengUniMelb

  3. Coherent Completeness and Tannaka Duality

    Stacks and Moduli §7.6, A Luna étale slice theorem for algebraic stacks

    Oliver LiUniMelb

  4. Local Structure Theorems

    Stacks and Moduli §7.7, A Luna étale slice theorem for algebraic stacks

    Fei PengUniMelb

  5. Θ- and S-completeness

    Stacks and Moduli §7.9, Existence of moduli spaces for algebraic stacks

    Gufang ZhaoUniMelb

  6. Reductivity and Unpunctured Inertia

    Stacks and Moduli §7.10, Existence of moduli spaces for algebraic stacks

    Nicolas VilchesANU

  7. Existence of Good Moduli Spaces

    Stacks and Moduli §7.10, Existence of moduli spaces for algebraic stacks

    Anand DeopurkarANU

Part II

The Intrinsic Approach to Stability and GIT

  1. The Stacks Filt and Grad, and Θ-Stratifications

    Moduli theory §§16.2, 17.1, On the structure of instability in moduli theory

    Sveta MakarovaANU

  2. Degenerations and Numerical Invariants

    Moduli theory §17.2, On the structure of instability in moduli theory

    Adam MonteleoneUniMelb

  3. Harder-Narasimhan Filtrations, Specialization, and Boundedness

    Moduli theory §17.3, On the structure of instability in moduli theory

    Sveta MakarovaANU

  4. The Intrinsic GIT Theorem

    Moduli theory §18.1, The intrinsic approach to moduli theory

    Adam MonteleoneUniMelb

  5. Applications

    Moduli theory §§18.2, 18.3, The intrinsic approach to moduli theory

    Sveta MakarovaANU

References

  1. [1] J. Alper, Stacks and Moduli, draft of 5 January 2026, §§7.5–7.10.
  2. [2] J. Alper, Good moduli spaces for Artin stacks, Ann. Inst. Fourier (Grenoble) 63 (2013), no. 6, 2349–2402. doi:10.5802/aif.2833.
  3. [3] J. Alper, J. Hall, and D. Rydh, A Luna étale slice theorem for algebraic stacks, Ann. of Math. (2) 191 (2020), no. 3, 675–738. doi:10.4007/annals.2020.191.3.1.
  4. [4] J. Alper, J. Hall, and D. Rydh, The étale local structure of algebraic stacks, Ann. Sci. Éc. Norm. Supér. (4) 59 (2026), no. 1, 125–198. doi:10.24033/asens.2637.
  5. [5] J. Alper and D. Halpern-Leistner, The intrinsic approach to moduli theory, preprint, arXiv:2603.21412 (2026), ICM 2026 proceedings contribution.
  6. [6] J. Alper, D. Halpern-Leistner, and J. Heinloth, Existence of moduli spaces for algebraic stacks, Invent. Math. 234 (2023), no. 3, 949–1038. doi:10.1007/s00222-023-01214-4.
  7. [7] D. Halpern-Leistner, On the structure of instability in moduli theory, preprint, arXiv:1411.0627 (2014), revised 4 February 2022.
  8. [8] D. Halpern-Leistner, Θ-stratifications, Θ-reductive stacks, and applications, in Algebraic Geometry: Salt Lake City 2015, Proc. Sympos. Pure Math., vol. 97.1, Amer. Math. Soc., Providence, RI, 2018, pp. 349–379. doi:10.1090/pspum/097.1/01678.
  9. [9] D. Halpern-Leistner, Moduli theory, lecture notes, Cornell University, version of 4 January 2021, Chapters 14–18.
  10. [10] D. Luna, Slices étales, in Sur les groupes algébriques, Bull. Soc. Math. France, Mém. 33 (1973), 81–105. doi:10.24033/msmf.110.