01043nas a2200109 4500008004300000245005700043210005400100520069600154100001600850700003100866856003600897 2006 en_Ud 00aSemistability vs. nefness for (Higgs) vector bundles0 aSemistability vs nefness for Higgs vector bundles3 aAccording to Miyaoka, a vector bundle E on a smooth projective curve is semistable if and only if a certain numerical class in the projectivized bundle PE is nef. We establish a similar criterion for the semistability of Higgs bundles: namely, such a bundle is semistable if and only if for every integer s between 0 and the rank of E, a suitable numerical class in the scheme parametrizing the rank s locally-free Higgs quotients of E is nef. We also extend this result to higher-dimensional complex projective varieties by showing that the nefness of the above mentioned classes is equivalent to the semistability of the Higgs bundle E together with the vanishing of the discriminant of E.1 aBruzzo, Ugo1 aHernandez Ruiperez, Daniel uhttp://hdl.handle.net/1963/223700851nas a2200145 4500008004300000245005500043210005500098260001000153520041100163100002200574700001600596700003100612700002600643856003600669 2002 en_Ud 00aRelatively stable bundles over elliptic fibrations0 aRelatively stable bundles over elliptic fibrations bWiley3 aWe consider a relative Fourier-Mukai transform defined on elliptic fibrations over an arbitrary normal base scheme. This is used to construct relative Atiyah sheaves and generalize Atiyah\\\'s and Tu\\\'s results about semistable sheaves over elliptic curves to the case of elliptic fibrations. Moreover we show that this transform preserves relative (semi)stability of sheaves of positive relative degree.1 aBartocci, Claudio1 aBruzzo, Ugo1 aHernandez Ruiperez, Daniel1 aMunoz Porras, Jose M. uhttp://hdl.handle.net/1963/3132