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15 November 2017 Enumeration of real curves in CP2n1 and a Witten–Dijkgraaf–Verlinde–Verlinde relation for real Gromov–Witten invariants
Penka Georgieva, Aleksey Zinger
Duke Math. J. 166(17): 3291-3347 (15 November 2017). DOI: 10.1215/00127094-2017-0023


We establish a homology relation for the Deligne–Mumford moduli spaces of real curves which lifts to a Witten–Dijkgraaf–Verlinde–Verlinde (WDVV)-type relation for a class of real Gromov–Witten invariants of real symplectic manifolds; we also obtain a vanishing theorem for these invariants. For many real symplectic manifolds, these results reduce all genus 0 real invariants with conjugate pairs of constraints to genus 0 invariants with a single conjugate pair of constraints. In particular, we give a complete recursion for counts of real rational curves in odd-dimensional projective spaces with conjugate pairs of constraints and specify all cases when they are nonzero and thus provide nontrivial lower bounds in high-dimensional real algebraic geometry. We also show that the real invariants of the 3-dimensional projective space with conjugate point constraints are congruent to their complex analogues modulo 4.


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Penka Georgieva. Aleksey Zinger. "Enumeration of real curves in CP2n1 and a Witten–Dijkgraaf–Verlinde–Verlinde relation for real Gromov–Witten invariants." Duke Math. J. 166 (17) 3291 - 3347, 15 November 2017.


Received: 26 January 2016; Revised: 20 February 2017; Published: 15 November 2017
First available in Project Euclid: 11 October 2017

zbMATH: 1384.53070
MathSciNet: MR3724219
Digital Object Identifier: 10.1215/00127094-2017-0023

Primary: 53D45
Secondary: 14N35

Keywords: enumeration of real curves , real Gromov–Witten invariants recursion

Rights: Copyright © 2017 Duke University Press


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Vol.166 • No. 17 • 15 November 2017
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