Real orientations, real Gromov-Witten theory, and real enumerative geometry

Penka Georgieva; Aleksey Zinger

doi:10.3934/era.2017.24.010

Article Contents

2017, Volume 24: 87-99. Doi: 10.3934/era.2017.24.010

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Real orientations, real Gromov-Witten theory, and real enumerative geometry

Penka Georgieva^1, and
Aleksey Zinger^2,

1.
Institut de Mathématiques de Jussieu – Paris Rive Gauche, Université Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris Cedex 5, France
2.
Department of Mathematics, Stony Brook University, Stony Brook, NY, 11794, USA

Supported by ERC grant STEIN-259118.
Partially supported by NSF grant DMS 1500875 and MPIM.

Received: April 03, 2017

Published: August 2017

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Abstract

The present note overviews our recent construction of real Gromov-Witten theory in arbitrary genera for many real symplectic manifolds, including the odd-dimensional projective spaces and the renowned quintic threefold, its properties, and its connections with real enumerative geometry. Our construction introduces the principle of orienting the determinant of a differential operator relative to a suitable base operator and a real setting analogue of the (relative) spin structure of open Gromov-Witten theory. Orienting the relative determinant, which in the now-standard cases is canonically equivalent to orienting the usual determinant, is naturally related to the topology of vector bundles in the relevant category. This principle and its applications allow us to endow the uncompactified moduli spaces of real maps from symmetric surfaces of all topological types with natural orientations and to verify that they extend across the codimension-one boundaries of these spaces, thus implementing a far-reaching proposal from C.-C. Liu's thesis.

Keywords:

Real Gromov-Witten invariants.

Mathematics Subject Classification: 14N35(Primary), 53D45(Secondary).

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Table . The extendability of the canonical orientations factoring into (13) and of the parity of the number of components of $\sum ^{\sigma }$ across the codimension 1 strata: $+$ extends, $-$ flips

	(E)/(H1)	(H2)/(H3)
orientation on (11) with $(V, \varphi ) = u^*(TX, \text{d}\phi)$	$+$	$+$
orientation induced by KS isomorphism	$-$	$-$
orientation induced by SD isomorphism	$+$	$+$
orientation on (11) with $(V, \varphi ) = (T^\sum , (\text{d}\sigma )^)^{\otimes2}$	$+$	$-$
parity of $\|\pi_0(\sum ^{\sigma })\|$	$-$	$+$

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