Symplectic four-manifolds and projective surfaces

Fedor Bogomolov, L. Katzarkov

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)79-109
Number of pages31
JournalTopology and its Applications
Volume88
Issue number1-2
StatePublished - 1998

Fingerprint

Four-manifolds
Fundamental Group
Infinite Groups
Monodromy
Riemann Surface
Self-intersection
Symplectic Structure
Obstruction
Fiber
Analogue
Invariant
Series
Arbitrary
Theorem

Keywords

  • Fundamental groups
  • Monodromy
  • Projective surfaces
  • Symplectic fourfolds

ASJC Scopus subject areas

  • Geometry and Topology

Cite this

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

In: Topology and its Applications, Vol. 88, No. 1-2, 1998, p. 79-109.

Research output: Contribution to journalArticle

Bogomolov, Fedor ; Katzarkov, L. / Symplectic four-manifolds and projective surfaces. In: Topology and its Applications. 1998 ; Vol. 88, No. 1-2. pp. 79-109.
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