Symplectic surgeries along certain singularities and new Lefschetz fibrations

Anar Akhmedov; Ludmil Katzarkov

doi:10.1016/j.aim.2019.106920

Symplectic surgeries along certain singularities and new Lefschetz fibrations

Anar Akhmedov, Ludmil Katzarkov

Mathematics

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2 Scopus citations

Abstract

We define a new 4-dimensional symplectic cut and paste operations arising from the generalized star relations (t_a₀t_a₁t_a₂⋯t_{a_2g+1})^2g+1=t_b₁t_b₂ ^gt_b₃, also known as the trident relations, in the mapping class group Γ_g,3 of an orientable surface of genus g≥1 with 3 boundary components. We also construct new families of Lefschetz fibrations by applying the (generalized) star relations and the chain relations to the families of words (t_c₁t_c₂⋯t_{c_2g−1}t_{c_2g}t_{c_2g+1} ²t_{c_2g}t_{c_2g−1}⋯t_c₂t_c₁)²ⁿ=1, (t_c₁t_c₂⋯t_{c_2g}t_{c_2g+1})^(2g+2)n=1 and (t_c₁t_c₂⋯t_{c_2g−1}t_{c_2g})^2(2g+1)n=1 in the mapping class group Γ_g of the closed orientable surface of genus g≥1 and n≥1. Furthermore, we show that the total spaces of some of these Lefschetz fibrations are irreducible exotic symplectic 4-manifolds. Using the degenerate cases of the generalized star relations, we also realize all elliptic Lefschetz fibrations and genus two Lefschetz fibrations over S² with non-separating vanishing cycles.

Original language	English (US)
Article number	106920
Journal	Advances in Mathematics
Volume	360
DOIs	https://doi.org/10.1016/j.aim.2019.106920
State	Published - Jan 22 2020

Bibliographical note

Funding Information:
We are very grateful to D. Auroux for some helpful discussions we have had back in September 2018 regarding some of the material presented here. We also thank H. Endo and N. Monden, with whom the first author had an email exchange with at the early stages of this project. We also thank C. Karakurt and J. Dorfmeister for their interest in our work and many comments which led to the present improved version of the article. We also thank the referee for the valuable comments and suggestions that helped us further to improve the presentation of our paper. Both authors would like to thank Simons Foundation for supporting our projects and the Department of Mathematics at Harvard University, where part of this work was done, for its hospitality. A. Akhmedov was supported by Simons Research Fellowship and Collaboration Grants for Mathematicians by Simons Foundation . L. Katzarkov was supported by Simons research grant, NSF DMS 150908 , ERC Gemis, DMS-1265230 , DMS-1201475 OISE-1242272 PASI. Simons collaborative Grant - HMS, Simons investigator grant - HMS. HSE - grant, HMS and automorphic forms. L. Katzarkov was also partially supported by Laboratory of Mirror Symmetry NRU HSE, RF Government grant, ag. No 14.641.31.0001 .

Funding Information:
We are very grateful to D. Auroux for some helpful discussions we have had back in September 2018 regarding some of the material presented here. We also thank H. Endo and N. Monden, with whom the first author had an email exchange with at the early stages of this project. We also thank C. Karakurt and J. Dorfmeister for their interest in our work and many comments which led to the present improved version of the article. We also thank the referee for the valuable comments and suggestions that helped us further to improve the presentation of our paper. Both authors would like to thank Simons Foundation for supporting our projects and the Department of Mathematics at Harvard University, where part of this work was done, for its hospitality. A. Akhmedov was supported by Simons Research Fellowship and Collaboration Grants for Mathematicians by Simons Foundation. L. Katzarkov was supported by Simons research grant, NSF DMS 150908, ERC Gemis, DMS-1265230, DMS-1201475 OISE-1242272 PASI. Simons collaborative Grant - HMS, Simons investigator grant - HMS. HSE - grant, HMS and automorphic forms. L. Katzarkov was also partially supported by Laboratory of Mirror Symmetry NRU HSE, RF Government grant, ag. No 14.641.31.0001.

Publisher Copyright:
© 2019 Elsevier Inc.

Keywords

4-manifold
Lefschetz fibration
Mapping class group
Symplectic surgery
Trident relation

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title = "Symplectic surgeries along certain singularities and new Lefschetz fibrations",

abstract = "We define a new 4-dimensional symplectic cut and paste operations arising from the generalized star relations (ta0 ta1 ta2 ⋯ta2g+1 )2g+1=tb1 tb2 gtb3 , also known as the trident relations, in the mapping class group Γg,3 of an orientable surface of genus g≥1 with 3 boundary components. We also construct new families of Lefschetz fibrations by applying the (generalized) star relations and the chain relations to the families of words (tc1 tc2 ⋯tc2g−1 tc2g tc2g+1 2tc2g tc2g−1 ⋯tc2 tc1 )2n=1, (tc1 tc2 ⋯tc2g tc2g+1 )(2g+2)n=1 and (tc1 tc2 ⋯tc2g−1 tc2g )2(2g+1)n=1 in the mapping class group Γg of the closed orientable surface of genus g≥1 and n≥1. Furthermore, we show that the total spaces of some of these Lefschetz fibrations are irreducible exotic symplectic 4-manifolds. Using the degenerate cases of the generalized star relations, we also realize all elliptic Lefschetz fibrations and genus two Lefschetz fibrations over S2 with non-separating vanishing cycles.",

keywords = "4-manifold, Lefschetz fibration, Mapping class group, Symplectic surgery, Trident relation",

author = "Anar Akhmedov and Ludmil Katzarkov",

note = "Funding Information: We are very grateful to D. Auroux for some helpful discussions we have had back in September 2018 regarding some of the material presented here. We also thank H. Endo and N. Monden, with whom the first author had an email exchange with at the early stages of this project. We also thank C. Karakurt and J. Dorfmeister for their interest in our work and many comments which led to the present improved version of the article. We also thank the referee for the valuable comments and suggestions that helped us further to improve the presentation of our paper. Both authors would like to thank Simons Foundation for supporting our projects and the Department of Mathematics at Harvard University, where part of this work was done, for its hospitality. A. Akhmedov was supported by Simons Research Fellowship and Collaboration Grants for Mathematicians by Simons Foundation . L. Katzarkov was supported by Simons research grant, NSF DMS 150908 , ERC Gemis, DMS-1265230 , DMS-1201475 OISE-1242272 PASI. Simons collaborative Grant - HMS, Simons investigator grant - HMS. HSE - grant, HMS and automorphic forms. L. Katzarkov was also partially supported by Laboratory of Mirror Symmetry NRU HSE, RF Government grant, ag. No 14.641.31.0001 . Funding Information: We are very grateful to D. Auroux for some helpful discussions we have had back in September 2018 regarding some of the material presented here. We also thank H. Endo and N. Monden, with whom the first author had an email exchange with at the early stages of this project. We also thank C. Karakurt and J. Dorfmeister for their interest in our work and many comments which led to the present improved version of the article. We also thank the referee for the valuable comments and suggestions that helped us further to improve the presentation of our paper. Both authors would like to thank Simons Foundation for supporting our projects and the Department of Mathematics at Harvard University, where part of this work was done, for its hospitality. A. Akhmedov was supported by Simons Research Fellowship and Collaboration Grants for Mathematicians by Simons Foundation. L. Katzarkov was supported by Simons research grant, NSF DMS 150908, ERC Gemis, DMS-1265230, DMS-1201475 OISE-1242272 PASI. Simons collaborative Grant - HMS, Simons investigator grant - HMS. HSE - grant, HMS and automorphic forms. L. Katzarkov was also partially supported by Laboratory of Mirror Symmetry NRU HSE, RF Government grant, ag. No 14.641.31.0001. Publisher Copyright: {\textcopyright} 2019 Elsevier Inc.",

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doi = "10.1016/j.aim.2019.106920",

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N2 - We define a new 4-dimensional symplectic cut and paste operations arising from the generalized star relations (ta0 ta1 ta2 ⋯ta2g+1 )2g+1=tb1 tb2 gtb3 , also known as the trident relations, in the mapping class group Γg,3 of an orientable surface of genus g≥1 with 3 boundary components. We also construct new families of Lefschetz fibrations by applying the (generalized) star relations and the chain relations to the families of words (tc1 tc2 ⋯tc2g−1 tc2g tc2g+1 2tc2g tc2g−1 ⋯tc2 tc1 )2n=1, (tc1 tc2 ⋯tc2g tc2g+1 )(2g+2)n=1 and (tc1 tc2 ⋯tc2g−1 tc2g )2(2g+1)n=1 in the mapping class group Γg of the closed orientable surface of genus g≥1 and n≥1. Furthermore, we show that the total spaces of some of these Lefschetz fibrations are irreducible exotic symplectic 4-manifolds. Using the degenerate cases of the generalized star relations, we also realize all elliptic Lefschetz fibrations and genus two Lefschetz fibrations over S2 with non-separating vanishing cycles.

AB - We define a new 4-dimensional symplectic cut and paste operations arising from the generalized star relations (ta0 ta1 ta2 ⋯ta2g+1 )2g+1=tb1 tb2 gtb3 , also known as the trident relations, in the mapping class group Γg,3 of an orientable surface of genus g≥1 with 3 boundary components. We also construct new families of Lefschetz fibrations by applying the (generalized) star relations and the chain relations to the families of words (tc1 tc2 ⋯tc2g−1 tc2g tc2g+1 2tc2g tc2g−1 ⋯tc2 tc1 )2n=1, (tc1 tc2 ⋯tc2g tc2g+1 )(2g+2)n=1 and (tc1 tc2 ⋯tc2g−1 tc2g )2(2g+1)n=1 in the mapping class group Γg of the closed orientable surface of genus g≥1 and n≥1. Furthermore, we show that the total spaces of some of these Lefschetz fibrations are irreducible exotic symplectic 4-manifolds. Using the degenerate cases of the generalized star relations, we also realize all elliptic Lefschetz fibrations and genus two Lefschetz fibrations over S2 with non-separating vanishing cycles.

KW - 4-manifold

KW - Lefschetz fibration

KW - Mapping class group

KW - Symplectic surgery

KW - Trident relation

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Symplectic surgeries along certain singularities and new Lefschetz fibrations

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