Generating the Fukaya categories of Hamiltonian 𝐺-manifolds

By J. Evans, YankΔ± Lekili. Journal of The American Mathematical Society, 2015

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<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 compact Lie group, and let <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> be a field of characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than-or-equal-to 0"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>β‰₯<!-- β‰₯ --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p \geq 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript asterisk Baseline left-parenthesis upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mo>βˆ—<!-- βˆ— --></mml:mo> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H^*(G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has no <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-torsion if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that a free Lagrangian orbit of a Hamiltonian <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>-action on a compact, monotone, symplectic manifold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> split-generates an idempotent summand of the monotone Fukaya category <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper F left-parenthesis upper X semicolon k right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">F</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>;</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {F}(X; k)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if and only if it represents a nonzero object of that summand (slightly more general results are also provided). Our result is based on an explicit understanding of the wrapped Fukaya category <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper W left-parenthesis upper T Superscript asterisk Baseline upper G semicolon k right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">W</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>T</mml:mi> <mml:mo>βˆ—<!-- βˆ— --></mml:mo> </mml:msup> <mml:mi>G</mml:mi> <mml:mo>;</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {W}(T^*G; k)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> through Koszul twisted complexes involving the zero-section and a cotangent fibre and on a functor <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D Superscript b Baseline script upper W left-parenthesis upper T Superscript asterisk Baseline upper G semicolon k right-parenthesis right-arrow upper D Superscript b Baseline script upper F left-parenthesis upper X Superscript minus Baseline times upper X semicolon k right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>D</mml:mi> <mml:mi>b</mml:mi> </mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">W</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>T</mml:mi> <mml:mo>βˆ—<!-- βˆ— --></mml:mo> </mml:msup> <mml:mi>G</mml:mi> <mml:mo>;</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">β†’<!-- β†’ --></mml:mo> <mml:msup> <mml:mi>D</mml:mi> <mml:mi>b</mml:mi> </mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">F</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>βˆ’<!-- βˆ’ --></mml:mo> </mml:mrow> </mml:msup> <mml:mo>Γ—<!-- Γ— --></mml:mo> <mml:mi>X</mml:mi> <mml:mo>;</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">D^b \mathcal {W}(T^*G; k) \to D^b\mathcal {F}(X^{-} \times X; k)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> canonically associated to the Hamiltonian <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>-action on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We explore several examples which can be studied in a uniform manner, including toric Fano varieties and certain Grassmannians.</p>

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/79c38982144f9cb82a9d3fd2434b4c43acb58ddc