Fridays 14-16, M 103

The Donaldson–Thomas invariants of a Calabi–Yau threefold X are certain virtual curve counts produced using the moduli stack of stable sheaves on X. The MNOP conjecture, recently proven for most Calabi–Yau threefolds by Pandharipande and Pixton, predicts a tight relationship between Donaldson–Thomas and Gromov–Witten invariants. In this course, we will give an introduction to Donaldson–Thomas theory and discuss a refinement that produces invariants of motivic nature. To achieve this, we will study and apply the following tools: the local structure of quasi-smooth derived stacks with (-1)–shifted symplectic structure, the formalism of motivic vanishing cycles, and microlocal geometry.

*Prerequisites*: Algebraic geometry. Familiarity with homotopical algebra / infinity-categories, and mixed motives or ℓ-adic sheaves, is recommended.

- [Be] K. Behrend,
*Donaldson-Thomas type invariants via microlocal geometry*, Ann. of Math. (2) 170 (2009), no. 3, 1307--1338, DOI. - [KL] Y.-H. Kiem, J. Li,
*Categorification of Donaldson-Thomas invariants via perverse sheaves*, arXiv:1212.6444. - [PTVV] T. Pantev, B. Toën, M. Vaquié, G. Vezzosi,
*Shifted symplectic structures*, Publ. Math. Inst. Hautes Études Sci. 117 (2013), 271--328, DOI. - [BJM] V. Bussi, D. Joyce, S. Meinhardt,
*On motivic vanishing cycles of critical loci*, J. Algebraic Geom. 28 (2019), no. 3, 405--438, DOI.