For the dualising sheaf $\omega_{X/B}$ of the symplectic Lefschetz fibration $\pi: X \to B$, we show that $c_1^2(\omega_{X/B}) \geq 0$ for relatively minimal symplectic 4-manifolds. Some consequences related to the Hodge bundle are obtained.

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9v2 [math.SG] 24 Jan 2002ONTHERELATIVEDUALISINGSHEAFOFASYMPLECTICLEFSCHETZFIBRATIONWEIPINGLIANDPETERVERMEIREAbstract.ForthedualisingsheafωX/BofarelativelyminimalsymplecticLefschetz brationπ:X→B,weshowthatc21(ωX/B)≥0.SomeconsequencesrelatedtotheHodgebundleareobtained.1.IntroductionALefschetz brationπ:X→Bisananalogueofastableholomorphic brationinthesmoothcategory.AsymplecticLefschetz brationisaversionofanalmost-complexholo-morphic bration.SinceDonaldson’sseminalwork[6,onthetopologicaldescriptionofclosedsymplecticmanifolds,thesymplecticLefschetz brationhasreceivedmoreattentioninRecallthefollowingresults:Theorem1.1.LetXbeasmooth,compactsymplecticfourmanifold.Then →X,thereexistsasymplectic1.[7]AfterblowingupXa nitenumberoftimesf:X →CP1.Lefschetz brationπ:X22.Ifb+2(X)>1,thenc1(X)≥0.OurworkismotivatedbyTheorem1ofwhichstatesthatforarelativelyminimal1symplecticLefschetz brationπ:X→B(notrationalorruled)c21(X)≥

Date:February1,2008.

1991MathematicsSubjectClassi cation.14C15,14C20,14H10,57M25,57N10.

Keywordsandphrases.CanonicalClass,SymplecticLefschetzFibration,DualisingSheaf.

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