---
OA_place: repository
OA_type: green
_id: '20622'
abstract:
- lang: eng
  text: We first recall Solomon’s relations for Welschinger invariants counting real
    curves in real symplectic fourfolds and the Witten–Dijkgraaf–Verlinde–Verlinde
    (WDVV)-style relations for Welschinger invariants counting real curves in real
    symplectic sixfolds with some symmetry. We then explicitly demonstrate that, in
    some important cases (projective spaces with standard conjugations, real blowups
    of the projective plane, and two- and threefold products of the one-dimensional
    projective space with two involutions each), these relations provide complete
    recursions determining all Welschinger invariants from basic input. We include
    extensive tables of Welschinger invariants in low degrees obtained from these
    recursions with Mathematica. These invariants provide lower bounds for counts
    of real rational curves, including with curve insertions in smooth algebraic threefolds.
article_processing_charge: No
arxiv: 1
author:
- first_name: Xujia
  full_name: Chen, Xujia
  id: 968ad14a-fd86-11ee-a420-ea29715511a3
  last_name: Chen
- first_name: Aleksey
  full_name: Zinger, Aleksey
  last_name: Zinger
citation:
  ama: 'Chen X, Zinger A. WDVV-type relations for Welschinger’s invariants: Applications.
    <i>Kyoto Journal of Mathematics</i>. 61(2):339-376. doi:<a href="https://doi.org/10.1215/21562261-2021-0005">10.1215/21562261-2021-0005</a>'
  apa: 'Chen, X., &#38; Zinger, A. (n.d.). WDVV-type relations for Welschinger’s invariants:
    Applications. <i>Kyoto Journal of Mathematics</i>. Duke University Press. <a href="https://doi.org/10.1215/21562261-2021-0005">https://doi.org/10.1215/21562261-2021-0005</a>'
  chicago: 'Chen, Xujia, and Aleksey Zinger. “WDVV-Type Relations for Welschinger’s
    Invariants: Applications.” <i>Kyoto Journal of Mathematics</i>. Duke University
    Press, n.d. <a href="https://doi.org/10.1215/21562261-2021-0005">https://doi.org/10.1215/21562261-2021-0005</a>.'
  ieee: 'X. Chen and A. Zinger, “WDVV-type relations for Welschinger’s invariants:
    Applications,” <i>Kyoto Journal of Mathematics</i>, vol. 61, no. 2. Duke University
    Press, pp. 339–376.'
  ista: 'Chen X, Zinger A. WDVV-type relations for Welschinger’s invariants: Applications.
    Kyoto Journal of Mathematics. 61(2), 339–376.'
  mla: 'Chen, Xujia, and Aleksey Zinger. “WDVV-Type Relations for Welschinger’s Invariants:
    Applications.” <i>Kyoto Journal of Mathematics</i>, vol. 61, no. 2, Duke University
    Press, pp. 339–76, doi:<a href="https://doi.org/10.1215/21562261-2021-0005">10.1215/21562261-2021-0005</a>.'
  short: X. Chen, A. Zinger, Kyoto Journal of Mathematics 61 (n.d.) 339–376.
date_created: 2025-11-10T08:45:12Z
date_published: 2021-06-01T00:00:00Z
date_updated: 2025-11-10T15:13:58Z
day: '01'
doi: 10.1215/21562261-2021-0005
extern: '1'
external_id:
  arxiv:
  - '1809.08938'
intvolume: '        61'
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.1809.08938
month: '06'
oa: 1
oa_version: Preprint
page: 339-376
publication: Kyoto Journal of Mathematics
publication_identifier:
  eissn:
  - 2154-3321
publication_status: submitted
publisher: Duke University Press
quality_controlled: '1'
status: public
title: 'WDVV-type relations for Welschinger''s invariants: Applications'
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 61
year: '2021'
...