Hilbert schemes and stable pairs: GIT and derived category wall crossings
By J. Stoppa, Richard P. Thomas. 2009
60 citations recorded by the source corpus.
We show that the Hilbert scheme of curves and Le Potier's moduli space of stable pairs with one dimensional support have a common GIT construction. The two spaces correspond to chambers on either side of a wall in the space of GIT linearisations. We explain why this is not enough to prove the "DT/PT wall crossing" conjecture relating the invariants derived from these moduli spaces when the underlying variety is a 3-fold. We then give a gentle introduction to a small part of Joyce's theory for such wall crossings, and use it to give a short proof of an identity relating the Euler characteristics of these moduli spaces. When the 3-fold is Calabi-Yau the identity is the Euler-characteristic analogue of the DT/PT wall crossing conjecture, but for general 3-folds it is something different, as we discuss.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/6e2f96272ed3be0c3f13b3bdf70b1df6770a379a
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In Geometry and Topology
- Yukinobu Toda
- D. Joyce
- T. Bridgeland
- Yinan Song
- Emanuele Macrì
- Arend Bayer
- P. Stellari
- D. Huybrechts
In Mathematical Physics
Citations in the corpus
- Hall algebras and curve-counting invariants (2010, methodology): “ involving unweighted Euler characteristic invariants defined without a Behrend function. This was first proved by Toda [26, 27]. Our argument gives a second proof of this result, and Stoppa and Thomas [24] have given a third (and simpler) proof, which even extends to the non-CalabiYau case. The appro”
- A theory of generalized Donaldson–Thomas invariants (2008, methodology): “Recent papers by Toda [101] and Stoppa and Thomas [98] prove a version of this for invariants without Behrend functions as weights, and using our methods to include Behrend functions should yield the proof.”
- Curve counting theories via stable objects I. DT/PT correspondence (2009, methodology): “ Therefore we should now be able to give a complete proof of Conjecture 1.1. In the Appendix, we will give a proof of Conjecture 1.1 using the result of [4]. Finally we comment that Stoppa and Thomas [27] investigate DT/PT correspondence via the wall-crossing of GIT stability. Then they show the sam”