A quantum splitting principle and an application
Annales de l'Institut Fourier, Volume 69 (2019) no. 5, pp. 2067-2088.

We propose an analogy of splitting principle in genus-0 Gromov–Witten theory. More precisely, we show how the Gromov–Witten theory of a variety X can be embedded into the theory of the projectivization of a vector bundle over X. An application is also given.

Nous proposons un analogue du principe de décomposition en théorie de Gromov–Witten de genre zéro. Plus précisément, nous montrons comment réaliser la théorie de Gromov–Witten d’une variété X dans la théorie de la projectivisation d’un fibré vectoriel sur X. Nous donnons également une application.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3289
Classification: 14N35
Keywords: Gromov–Witten theory, splitting principle, projective bundle
Mots-clés : théorie de Gromov–Witten, principe de décomposition, fibré projectif

Fan, Honglu 1

1 ETH Zürich Department of Mathematics Rämistrasse 101 Zürich, 8092 (Switzerland)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{AIF_2019__69_5_2067_0,
     author = {Fan, Honglu},
     title = {A quantum splitting principle and an application},
     journal = {Annales de l'Institut Fourier},
     pages = {2067--2088},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {69},
     number = {5},
     year = {2019},
     doi = {10.5802/aif.3289},
     zbl = {07034550},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3289/}
}
TY  - JOUR
AU  - Fan, Honglu
TI  - A quantum splitting principle and an application
JO  - Annales de l'Institut Fourier
PY  - 2019
SP  - 2067
EP  - 2088
VL  - 69
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3289/
DO  - 10.5802/aif.3289
LA  - en
ID  - AIF_2019__69_5_2067_0
ER  - 
%0 Journal Article
%A Fan, Honglu
%T A quantum splitting principle and an application
%J Annales de l'Institut Fourier
%D 2019
%P 2067-2088
%V 69
%N 5
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3289/
%R 10.5802/aif.3289
%G en
%F AIF_2019__69_5_2067_0
Fan, Honglu. A quantum splitting principle and an application. Annales de l'Institut Fourier, Volume 69 (2019) no. 5, pp. 2067-2088. doi : 10.5802/aif.3289. https://aif.centre-mersenne.org/articles/10.5802/aif.3289/

[1] Bernardara, Marcello A semiorthogonal decomposition for Brauer Severi schemes, Math. Nachr., Volume 282 (2009) no. 10, pp. 1406-1413 | DOI | MR | Zbl

[2] Brown, Jeffrey Gromov-Witten Invariants of Toric Fibrations (2009) (https://arxiv.org/abs/0901.1290)

[3] Coates, Tom; Givental, Alexander Quantum Riemann–Roch, Lefschetz and Serre, Ann. Math., Volume 165 (2007) no. 1, pp. 15-53 | MR | Zbl

[4] Fan, Honglu Chern classes and Gromov–Witten theory of projective bundles (2017) (https://arxiv.org/abs/1705.07421) | MR

[5] Fan, Honglu; Lee, Yuan-Pin On Gromov–Witten theory of projective bundles (2016) (https://arxiv.org/abs/1607.00740)

[6] Graber, Tom; Pandharipande, Rahul Localization of virtual classes, Invent. Math., Volume 135 (1999) no. 2, pp. 487-518 | DOI | MR | Zbl

[7] Grothendieck, Alexander Le groupe de Brauer I, II, III, Dix Exposés sur la Cohomologie des Schémas (Advanced Studies in Pure Mathematics), Volume 3, North-Holland, 1968, pp. 46-188 | MR

[8] Kim, Bumsig; Kresch, Andrew; Pantev, Tony Functoriality in intersection theory and a conjecture of Cox, Katz, and Lee, J. Pure Appl. Algebra, Volume 179 (2003) no. 1, pp. 127-136 | MR | Zbl

[9] Kresch, Andrew Cycle groups for Artin stacks, Invent. Math., Volume 138 (1999) no. 3, pp. 495-536 | DOI | MR | Zbl

[10] Lai, Hsin-Hong Gromov–Witten invariants of blow-ups along submanifolds with convex normal bundles, Geom. Topol., Volume 13 (2007) no. 1, pp. 1-48 | MR | Zbl

[11] Lee, Yuan-Pin; Lin, Hui-Wen; Qu, Feng; Wang, Chin-Lung Invariance of quantum rings under ordinary flops III: A quantum splitting principle, Camb. J. Math., Volume 4 (2014) no. 3, pp. 333-401 | DOI | MR | Zbl

[12] Liu, Chiu-Chu M. Localization in Gromov–Witten theory and orbifold Gromov-Witten theory, Handbook of moduli. Volume II (Advanced Lectures in Mathematics (ALM)), Volume 25, International Press., 2013, pp. 353-425 | MR | Zbl

[13] Manolache, Cristina Virtual pull-backs, J. Algebr. Geom., Volume 21 (2012) no. 2, pp. 201-245 | MR | Zbl

[14] Mustata, Anca; Mustata, Andrei Gromov–Witten invariants for varieties with C* action (2015) (https://arxiv.org/abs/1505.01471)

[15] Ruan, Yongbin Surgery, quantum cohomology and birational geometry, Northern California symplectic geometry seminar (Advances in the Mathematical Sciences), Volume 196, American Mathematical Society, 1999, pp. 183-198 | MR | Zbl

Cited by Sources: