TY - JOUR

T1 - Weighted Euler characteristic of the moduli space of higher rank Joyce–Song pairs

AU - Sheshmani, Artan

N1 - Funding Information:
I would like to thank Sheldon Katz for suggesting the computation in Sect. . Thanks to Yukinobu Toda for many valuable discussions and for explaining to me his paper on rank 2 Donaldson–Thomas invariants. Thanks to Arend Bayer for kindly answering some of my questions related to this work in spring 2011. I am grateful to Richard Thomas for his help and support and for providing me the opportunity for being a member at the Isaac Newton Institute for Mathematical Sciences during 2010–2011. I also thank the Newton Institute for hospitality. I acknowledge partial support from NSF grants DMS 0244412, DMS 0555678 and DMS 08–38434 EMSW21–MCTP (R.E.G.S) during the time that the first drafts of project were being completed. I would also like to thank Kavli IPMU and MIT for their kind and wonderful hospitality. My work at IPMU was supported by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.
Publisher Copyright:
© 2016, Springer International Publishing AG.

PY - 2016/9/1

Y1 - 2016/9/1

N2 - 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.

AB - 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.

KW - Generalized Donaldson–Thomas invariants

KW - Joyce–Song stable pairs

KW - Ringel–Hall algebra

KW - Stack functions

KW - Weighted Euler characteristic

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U2 - 10.1007/s40879-016-0104-6

DO - 10.1007/s40879-016-0104-6

M3 - Article

AN - SCOPUS:84981713445

VL - 2

SP - 661

EP - 715

JO - European Journal of Mathematics

JF - European Journal of Mathematics

SN - 2199-675X

IS - 3

ER -