Weighted Euler characteristic of the moduli space of higher rank Joyce–Song pairs

Research output: Contribution to journalArticlepeer-review


The invariants of rank 2 Joyce–Song semistable pairs over a Calabi–Yau threefold were computed in Sheshmani (Illinois J Math 59(1):55–83, 2016), using the wall-crossing formula of Joyce and Song (A Theory of Generalized Donaldson–Thomas Invariants. Memoirs of American Mathematical Society, American Mathematical Society, Providence, 2012), and Kontsevich and Soibelman (Stability structures, motivic Donaldson–Thomas invariants and cluster transformations, arXiv:0811.2435, 2008). Such wallcrossing computations often depend on the combinatorial properties of certain elements of a Hall-algebra [these are the stack functions defined by Joyce (Adv Math 210(2):635–706, 2007)]. These combinatorial computations become immediately complicated and hard to carry out, when studying higher rank stable pairs with rank > 2. The main purpose of this article is to introduce an independent approach to computation of rank 2 stable pair invariants, without applying the wallcrossing formula and rather by stratifying their corresponding moduli space and directly computing the weighted Euler characteristic of the strata. This approach may similarly be used to avoid complex combinatorial wallcrossing calculations in rank > 2 cases.

Original languageEnglish (US)
Pages (from-to)661-715
Number of pages55
JournalEuropean Journal of Mathematics
Issue number3
StatePublished - Sep 1 2016
Externally publishedYes


  • Generalized Donaldson–Thomas invariants
  • Joyce–Song stable pairs
  • Ringel–Hall algebra
  • Stack functions
  • Weighted Euler characteristic

ASJC Scopus subject areas

  • Mathematics(all)


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