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Medientyp: E-Artikel Titel: The McKay correspondence as an equivalence of derived categories Beteiligte: Bridgeland, Tom; King, Alastair; Reid, Miles Erschienen: American Mathematical Society (AMS), 2001 Erschienen in: Journal of the American Mathematical Society Sprache: Englisch DOI: 10.1090/s0894-0347-01-00368-x ISSN: 0894-0347; 1088-6834 Schlagwörter: Applied Mathematics ; General Mathematics Entstehung: Anmerkungen: Beschreibung: <p>Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a finite group of automorphisms of a nonsingular three-dimensional complex variety <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, whose canonical bundle <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega Subscript upper M"> <mml:semantics> <mml:msub> <mml:mi>ω<!-- ω --></mml:mi> <mml:mi>M</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\omega _M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is locally trivial as a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-sheaf. We prove that the Hilbert scheme <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y equals upper G"> <mml:semantics> <mml:mrow> <mml:mi>Y</mml:mi> <mml:mo>=</mml:mo> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">Y = G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H i l b upper M"> <mml:semantics> <mml:mrow> <mml:mi>Hilb</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Hilb}M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> parametrising <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-clusters in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a crepant resolution of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X equals upper M slash upper G"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>=</mml:mo> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">X=M/G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and that there is a derived equivalence (Fourier–Mukai transform) between coherent sheaves on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and coherent 𝐺-sheaves</p> <p> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This identifies the K theory of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with the equivariant K theory of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and thus generalises the classical McKay correspondence. Some higher-dimensional extensions are possible.</p> Zugangsstatus: Freier Zugang