Dhyan Aranha
Since arriving in September of 2020 in the research group of Prof. Dr. Marc Levine I have started to undertake 2 research projects both of which are in their infancy.
The first one, which is joint with Jochen Heinloth, is to understand the decomposition theorem of Beilinson, Bernstein, Deligne and Gabber [BBD82] for morphisms of algebraic stacks to their good moduli spaces. What has been done so far has been to reinterpret and simplify various results and constructions in [BL94] about \(G\)-equivariant perverse sheaves via more modern language involving $\infty$-categories and stacks. We are currently thinking about how to use Kirwan’s desingularization procedure [Kir85] to reduce ourselves to the results that have been established in [BL94], however this is still very much work in progress.
The second project is about understanding how to do microlocal sheaf theory in derived algebraic geometry. This project is joint with Adeel Khan, Tasuki Kinjo, and Alexei Latynetsev.
Here the main goal is to study the categorification of Donaldson-Thomas invariants and in particular attack a conjecture of Toda’s [Tod21] that roughly says given a a \((-1)\)-shifted symplectic derived scheme \(M\) there should exist an \(\infty\)-category \(DT(M)\) which contains "all" of the information about the Donaldson-Thomas theory of \(M\). For example, taking the periodic cyclic homology of \(DT(M)\) should give the \(\mathbb{Z}/2\mathbb{Z}\)-graded cohomological Donaldson-Thomas invariant of [BBD+15]. Roughly the problem is one of gluing \(\infty\)-categories and it was a suggestion of Adeel Khan’s that perhaps some of the methods and constructions in microlocal sheaf theory might be relevant. This project is joint with Adeel Khan, Tasuki Kinjo, and Alexei Latynetsev.
References
[BBD82] Alexander A. Beilinson, Joseph Bernstein, and Pierre Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981), Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5– 171.
[BBD+15] C. Brav, V. Bussi, D. Dupont, D. Joyce, and B. Szendröi, Symmetries and stabilization for sheaves of vanishing cycles, J. Singul. 11 (2015), 85–151, With an appendix by Jörg Schürmann. MR 3353002
[BL94] Joseph Bernstein and Valery Lunts, Equivariant sheaves and functors, Lecture Notes in Mathematics, vol. 1578, Springer-Verlag, Berlin, 1994. MR 1299527
[Kir85] Frances Clare Kirwan, Partial desingularisations of quotients of nonsingular varieties and their Betti numbers, Ann. of Math. (2) 122 (1985), no. 1, 41–85. MR 799252
[Tod21] Yukinobu Toda, Categorical donaldson-thomas theory for local suraces, 2021.
Project related publications
Project related preprints
[AP] D. Aranha, P. Pstrągowski, The Intrinsic Normal Cone For Artin Stacks. https://arxiv.org/abs/1909.07478
Works in progress
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