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Topology of holomorphic
Lefschetz pencils on the four-torus
Noriyuki Hamada and Kenta Hayano |
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Algebraic & Geometric Topology 18 (2018) 1515–1572
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Abstract |
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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- 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- Lefschetz pencil whose total space is homeomorphic to that of the given bundle. |
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Keywords
Lefschetz pencil, polarized abelian surfaces, symplectic
Calabi–Yau four-manifolds, monodromy factorizations,
mapping class groups
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Mathematical Subject Classification 2010
Primary: 57R35
Secondary: 14D05, 20F38, 32Q55, 57R17
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References |
Publication
Received: 23 January 2017
Revised: 4 September 2017
Accepted: 1 October 2017
Published: 3 April 2018
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Authors |
| Noriyuki Hamada | |
| Department of Mathematics and
Statistics University of Massachusetts Amherst, MA United States |
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| Kenta Hayano | |
| Department of Mathematics Keio University Yagami Campus Yokohama Japan |
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| http://www.math.keio.ac.jp/~k-hayano/index_en.html | |
