### Abstract

In this paper we consider symplectic versions of some results and constructions from the theory of complex projective surfaces with infinite fundamental groups. We introduce series of simple examples of symplectic fourfolds which are not Kähler. All of them have infinite fundamental groups which are fundamental groups of complex projective surfaces and contain symplectically embedded Riemann surfaces with positive self-intersection and a small image of their fundamental groups inside the fundamental group of the ambient symplectic fourfold. We have shown that there are no analogues of Zariski-Nori theorems for symplectic fourfolds. Our main results concern symplectic pencils of symplectically embedded Riemann surfaces. We give a universal construction of such pencils with rather arbitrary properties (any fundamental group in particular). We also give an obstruction for a symplectic Lefschetz pencil to be Kähler. Our construction suggests that the embedding of the local monodromy of the fiber of the above pencils in their global monodromy is an invariant of the symplectic structure.

Original language | English (US) |
---|---|

Pages (from-to) | 79-109 |

Number of pages | 31 |

Journal | Topology and its Applications |

Volume | 88 |

Issue number | 1-2 |

State | Published - 1998 |

### Fingerprint

### Keywords

- Fundamental groups
- Monodromy
- Projective surfaces
- Symplectic fourfolds

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Topology and its Applications*,

*88*(1-2), 79-109.

**Symplectic four-manifolds and projective surfaces.** / Bogomolov, Fedor; Katzarkov, L.

Research output: Contribution to journal › Article

*Topology and its Applications*, vol. 88, no. 1-2, pp. 79-109.

}

TY - JOUR

T1 - Symplectic four-manifolds and projective surfaces

AU - Bogomolov, Fedor

AU - Katzarkov, L.

PY - 1998

Y1 - 1998

N2 - In this paper we consider symplectic versions of some results and constructions from the theory of complex projective surfaces with infinite fundamental groups. We introduce series of simple examples of symplectic fourfolds which are not Kähler. All of them have infinite fundamental groups which are fundamental groups of complex projective surfaces and contain symplectically embedded Riemann surfaces with positive self-intersection and a small image of their fundamental groups inside the fundamental group of the ambient symplectic fourfold. We have shown that there are no analogues of Zariski-Nori theorems for symplectic fourfolds. Our main results concern symplectic pencils of symplectically embedded Riemann surfaces. We give a universal construction of such pencils with rather arbitrary properties (any fundamental group in particular). We also give an obstruction for a symplectic Lefschetz pencil to be Kähler. Our construction suggests that the embedding of the local monodromy of the fiber of the above pencils in their global monodromy is an invariant of the symplectic structure.

AB - In this paper we consider symplectic versions of some results and constructions from the theory of complex projective surfaces with infinite fundamental groups. We introduce series of simple examples of symplectic fourfolds which are not Kähler. All of them have infinite fundamental groups which are fundamental groups of complex projective surfaces and contain symplectically embedded Riemann surfaces with positive self-intersection and a small image of their fundamental groups inside the fundamental group of the ambient symplectic fourfold. We have shown that there are no analogues of Zariski-Nori theorems for symplectic fourfolds. Our main results concern symplectic pencils of symplectically embedded Riemann surfaces. We give a universal construction of such pencils with rather arbitrary properties (any fundamental group in particular). We also give an obstruction for a symplectic Lefschetz pencil to be Kähler. Our construction suggests that the embedding of the local monodromy of the fiber of the above pencils in their global monodromy is an invariant of the symplectic structure.

KW - Fundamental groups

KW - Monodromy

KW - Projective surfaces

KW - Symplectic fourfolds

UR - http://www.scopus.com/inward/record.url?scp=12244302289&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=12244302289&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:12244302289

VL - 88

SP - 79

EP - 109

JO - General Topology and its Applications

JF - General Topology and its Applications

SN - 0016-660X

IS - 1-2

ER -