On the monodromy of moduli spaces of sheaves on K3 surfaces

Abstract

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Baseline 3"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">K3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> surface and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A u t upper D left-parenthesis upper S right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Aut</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>S</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Aut}D(S)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the group of auto-equivalences of the derived category of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We construct a natural representation of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A u t upper D left-parenthesis upper S right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Aut</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>S</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Aut}D(S)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on the cohomology of all moduli spaces of stable sheaves (with primitive Mukai vectors) on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The main result of this paper is the precise relation of this action with the monodromy of the Hilbert schemes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Superscript left-bracket n right-bracket"> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">[</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">S^{[n]}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of points on the surface. A formula is provided for the monodromy representation in terms of the Chern character of the universal sheaf. Isometries of the second cohomology of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Superscript left-bracket n right-bracket"> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">[</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">S^{[n]}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are lifted, via this formula, to monodromy operators of the whole cohomology ring of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Superscript left-bracket n right-bracket"> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">[</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">S^{[n]}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .