A Darboux theorem for derived schemes with shifted symplectic structure

By Christopher Brav, Vittoria Bussi, D. Joyce. Journal of The American Mathematical Society, 2013

78 citations recorded by the source corpus.

We prove a Darboux theorem for derived schemes with symplectic forms of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, in the sense of Pantev, Toën, Vaquié, and Vezzosi. More precisely, we show that a derived scheme <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic upper X"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold-italic">X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbfit {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with symplectic form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega overTilde"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>ω</mml:mi> <mml:mo stretchy="false">~</mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\tilde {\omega }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is locally equivalent to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper S p e c upper A comma omega right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Spec</mml:mi> <mml:mo>⁡</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ω</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\operatorname {Spec} A,\omega )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S p e c upper A"> <mml:semantics> <mml:mrow> <mml:mi>Spec</mml:mi> <mml:mo>⁡</mml:mo> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Spec} A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> an affine derived scheme in which the cdga <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has Darboux-like coordinates with respect to which the symplectic form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega"> <mml:semantics> <mml:mi>ω</mml:mi> <mml:annotation encoding="application/x-tex">\omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is standard, and in which the differential in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is given by a Poisson bracket with a Hamiltonian function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi"> <mml:semantics> <mml:mi mathvariant="normal">Φ</mml:mi> <mml:annotation encoding="application/x-tex">\Phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k plus 1"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k+1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. When <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k equals negative 1"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k=-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, this implies that a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="negative 1"> <mml:semantics> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-shifted symplectic derived scheme <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis bold-italic upper X comma omega overTilde right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold-italic">X</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>ω</mml:mi> <mml:mo stretchy="false">~</mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\mathbfit {X}, \tilde {\omega })</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is Zariski locally equivalent to the derived critical locus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C r i t left-parenthesis normal upper Phi right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Crit</mml:mi> <mml:mo>⁡</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Φ</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Crit}(\Phi )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a regular function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi colon upper U right-arrow double-struck upper A Superscript 1"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">Φ</mml:mi> <mml:mo>:</mml:mo> <mml:mi>U</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">A</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\Phi :U\rightarrow \mathbb {A}^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on a smooth scheme <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U"> <mml:semantics> <mml:mi>U</mml:mi> <mml:annotation encoding="application/x-tex">U</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We use this to show that the classical scheme <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X equals t 0 left-parenthesis bold-italic upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>t</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold-italic">X</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">X=t_0(\mathbfit {X})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has the structure of an <italic>algebraic d-critical locus</italic>, in the sense of Joyce. In a series of works, the authors and their collaborators extend these results to (derived) Artin stacks, and discuss a Lagrangian neighbourhood theorem for shifted symplectic derived schemes, and applications to categorified and motivic Donaldson–Thomas theory of Calabi–Yau 3-folds, and to defining new Donaldson–Thomas type invariants of Calabi–Yau 4-folds, and to defining Fukaya categories of Lagrangians in algebraic symplectic manifolds using perverse sheaves.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e356dfd490d7000c80e70f462bac9d1bcd346115