Mathematics – Algebraic Geometry
Scientific paper
2012-01-30
Mathematics
Algebraic Geometry
44 pages, 6 figures
Scientific paper
The moduli space of stable quotients introduced by Marian-Oprea-Pandharipande provides a natural compactification of the space of morphisms from nonsingular curves to a nonsingular variety. When the latter is a Grassmannian, the moduli space of stable quotients carries a canonical virtual class. We show that the analogue of Givental's J-function for the resulting twisted projective invariants is described by the same mirror hypergeometric series as the corresponding Gromov-Witten invariants (which arise from the moduli space of stable maps), but without the mirror transform (in the Calabi-Yau case). This implies that the stable quotients and Gromov-Witten twisted invariants agree if there is enough "positivity", but not in all cases. As a corollary of the proof, we show that certain twisted Hurwitz numbers arising in the stable quotients theory are also described by a fundamental object associated with this hypergeometric series. We thus completely answer some of the questions posed by Marian-Oprea-Pandharipande concerning their invariants. Our results suggest a deep connection between the stable quotients invariants of complete intersections and the geometry of the mirror families.
Cooper Yaim
Zinger Aleksey
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