Topology of holomorphic lefschetz pencils on the four-torus

Noriyuki Hamada, Kenta Hayano

Research output: Contribution to journalArticle

Abstract

We discuss topological properties of holomorphic Lefschetz pencils on the four-torus. Relying on the theory of moduli spaces of polarized abelian surfaces, we first prove that, under some mild assumptions, the (smooth) isomorphism class of a holomorphic Lefschetz pencil on the four-torus is uniquely determined by its genus and divisibility. We then explicitly give a system of vanishing cycles of the genus-3 holomorphic Lefschetz pencil on the four-torus due to Smith, and obtain those of holomorphic pencils with higher genera by taking finite unbranched coverings. One can also obtain the monodromy factorization associated with Smith’s pencil in a combinatorial way. This construction allows us to generalize Smith’s pencil to higher genera, which is a good source of pencils on the (topological) four-torus. As another application of the combinatorial construction, for any torus bundle over the torus with a section we construct a genus-3 Lefschetz pencil whose total space is homeomorphic to that of the given bundle.

Original languageEnglish
Pages (from-to)1515-1572
Number of pages58
JournalAlgebraic and Geometric Topology
Volume18
Issue number3
DOIs
Publication statusPublished - 2018 Apr 9

Keywords

  • Lefschetz pencil
  • Mapping class groups
  • Monodromy factorizations
  • Polarized abelian surfaces
  • Symplectic calabi–yau four-manifolds

ASJC Scopus subject areas

  • Geometry and Topology

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