Counting curves on surfaces in Calabi–Yau 3-folds

@article{Gholampour2014CountingCO,
  title={Counting curves on surfaces in Calabi–Yau 3-folds},
  author={Amin Gholampour and Artan Sheshmani and Richard P. Thomas},
  journal={Mathematische Annalen},
  year={2014},
  volume={360},
  pages={67-78}
}
Motivated by S-duality modularity conjectures in string theory, we define new invariants counting a restricted class of two-dimensional torsion sheaves, enumerating pairs $$Z\subset H$$Z⊂H in a Calabi–Yau threefold $$X$$X. Here $$H$$H is a member of a sufficiently positive linear system and $$Z$$Z is a one-dimensional subscheme of it. The associated sheaf is the ideal sheaf of $$Z\subset H$$Z⊂H, pushed forward to $$X$$X and considered as a certain Joyce–Song pair in the derived category of $$X… 
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