Relative quasimaps and mirror formulae
Abstract
We construct and study the theory of relative quasimaps in genus zero, in the spirit of Gathmann. When $X$ is a smooth toric variety and $Y$ is a smooth very ample hypersurface in $X$, we produce a virtual class on the moduli space of relative quasimaps to $(X,Y)$, which we use to define relative quasimap invariants. We obtain a recursion formula which expresses each relative invariant in terms of invariants of lower tangency, and apply this formula to derive a quantum Lefschetz theorem for quasimaps, expressing the restricted quasimap invariants of $Y$ in terms of those of $X$. Finally, we show that the relative $I$-function of Fan-Tseng-You coincides with a natural generating function for relative quasimap invariants, providing mirror-symmetric motivation for the theory.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2017
- DOI:
- 10.48550/arXiv.1710.11158
- arXiv:
- arXiv:1710.11158
- Bibcode:
- 2017arXiv171011158B
- Keywords:
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- Mathematics - Algebraic Geometry;
- 14N35
- E-Print:
- 32 pages, 1 figure