Stability and the Fourier-Mukai transform I

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<jats:title>Abstract</jats:title><jats:p>We consider the problem of preservation of stability under the Fourier–Mukai transform ℱ<jats:sub>ℰ</jats:sub>:<jats:italic>D</jats:italic>(<jats:italic>X</jats:italic>)→<jats:italic>D</jats:italic>(<jats:italic>Y</jats:italic>) on an abelian surface and a<jats:italic>K</jats:italic>3 surface. If<jats:italic>Y</jats:italic>is the moduli space of<jats:italic>μ</jats:italic>-stable sheaves on<jats:italic>X</jats:italic>with respect to a polarization<jats:italic>H</jats:italic>, we have a canonical polarization<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X08003758_inline1"><jats:alt-text>$\widehat {H}$</jats:alt-text></jats:inline-graphic>on<jats:italic>Y</jats:italic>and we have a correspondence between (<jats:italic>X</jats:italic>,<jats:italic>H</jats:italic>) and<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X08003758_inline2"><jats:alt-text>$(Y,\widehat {H})$</jats:alt-text></jats:inline-graphic>. We show that the stability with respect to these polarizations is preserved under ℱ<jats:sub>ℰ</jats:sub>, if the degree of stable sheaves on<jats:italic>X</jats:italic>is sufficiently large.</jats:p>

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