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• W2100024891 abstract "We study homological mirror symmetry conjecture of symplectic and complex torus. We will associate a mirror torus<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper T Superscript 2 n Baseline comma omega plus StartRoot negative 1 EndRoot upper B right-parenthesis Superscript logical-and><mml:semantics><mml:mrow><mml:mo stretchy=false>(</mml:mo><mml:msup><mml:mi>T</mml:mi><mml:mrow class=MJX-TeXAtom-ORD><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mi>ω<!-- ω --></mml:mi><mml:mo>+</mml:mo><mml:msqrt><mml:mo>−<!-- − --></mml:mo><mml:mn>1</mml:mn></mml:msqrt><mml:mi>B</mml:mi><mml:msup><mml:mo stretchy=false>)</mml:mo><mml:mrow class=MJX-TeXAtom-ORD><mml:mo>∧<!-- ∧ --></mml:mo></mml:mrow></mml:msup></mml:mrow><mml:annotation encoding=application/x-tex>(T^{2n},omega +sqrt {-1}B)^{wedge }</mml:annotation></mml:semantics></mml:math></inline-formula>to each symplectic torus<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper T Superscript 2 n Baseline comma omega right-parenthesis><mml:semantics><mml:mrow><mml:mo stretchy=false>(</mml:mo><mml:msup><mml:mi>T</mml:mi><mml:mrow class=MJX-TeXAtom-ORD><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mi>ω<!-- ω --></mml:mi><mml:mo stretchy=false>)</mml:mo></mml:mrow><mml:annotation encoding=application/x-tex>(T^{2n},omega )</mml:annotation></mml:semantics></mml:math></inline-formula>together with a closed 2 form<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper B><mml:semantics><mml:mi>B</mml:mi><mml:annotation encoding=application/x-tex>B</mml:annotation></mml:semantics></mml:math></inline-formula>which we call a<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper B><mml:semantics><mml:mi>B</mml:mi><mml:annotation encoding=application/x-tex>B</mml:annotation></mml:semantics></mml:math></inline-formula>-field. We will associate a coherent sheaf<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper E left-parenthesis upper L comma script upper L right-parenthesis><mml:semantics><mml:mrow><mml:mrow class=MJX-TeXAtom-ORD><mml:mrow class=MJX-TeXAtom-ORD><mml:mi class=MJX-tex-caligraphic mathvariant=script>E</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy=false>(</mml:mo><mml:mi>L</mml:mi><mml:mo>,</mml:mo><mml:mrow class=MJX-TeXAtom-ORD><mml:mrow class=MJX-TeXAtom-ORD><mml:mi class=MJX-tex-caligraphic mathvariant=script>L</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy=false>)</mml:mo></mml:mrow><mml:annotation encoding=application/x-tex>{mathcal E}(L,{mathcal L})</mml:annotation></mml:semantics></mml:math></inline-formula>on<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper T Superscript 2 n Baseline comma omega plus StartRoot negative 1 EndRoot upper B right-parenthesis Superscript logical-and><mml:semantics><mml:mrow><mml:mo stretchy=false>(</mml:mo><mml:msup><mml:mi>T</mml:mi><mml:mrow class=MJX-TeXAtom-ORD><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mi>ω<!-- ω --></mml:mi><mml:mo>+</mml:mo><mml:msqrt><mml:mo>−<!-- − --></mml:mo><mml:mn>1</mml:mn></mml:msqrt><mml:mi>B</mml:mi><mml:msup><mml:mo stretchy=false>)</mml:mo><mml:mrow class=MJX-TeXAtom-ORD><mml:mo>∧<!-- ∧ --></mml:mo></mml:mrow></mml:msup></mml:mrow><mml:annotation encoding=application/x-tex>(T^{2n},omega +sqrt {-1}B)^{wedge }</mml:annotation></mml:semantics></mml:math></inline-formula>to each pair<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper L comma script upper L right-parenthesis><mml:semantics><mml:mrow><mml:mo stretchy=false>(</mml:mo><mml:mi>L</mml:mi><mml:mo>,</mml:mo><mml:mrow class=MJX-TeXAtom-ORD><mml:mrow class=MJX-TeXAtom-ORD><mml:mi class=MJX-tex-caligraphic mathvariant=script>L</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy=false>)</mml:mo></mml:mrow><mml:annotation encoding=application/x-tex>(L,{mathcal L})</mml:annotation></mml:semantics></mml:math></inline-formula>of affine Lagrangian submanifolds<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L><mml:semantics><mml:mi>L</mml:mi><mml:annotation encoding=application/x-tex>L</mml:annotation></mml:semantics></mml:math></inline-formula>and a flat complex line bundle<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper L><mml:semantics><mml:mrow class=MJX-TeXAtom-ORD><mml:mrow class=MJX-TeXAtom-ORD><mml:mi class=MJX-tex-caligraphic mathvariant=script>L</mml:mi></mml:mrow></mml:mrow><mml:annotation encoding=application/x-tex>{mathcal L}</mml:annotation></mml:semantics></mml:math></inline-formula>on<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L><mml:semantics><mml:mi>L</mml:mi><mml:annotation encoding=application/x-tex>L</mml:annotation></mml:semantics></mml:math></inline-formula>. In the case of affine Lagrangian submanifolds, we show that the Floer homology of Langrangian submanifolds is isomorphic to the extension of the mirror sheaf<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper E left-parenthesis upper L comma script upper L right-parenthesis><mml:semantics><mml:mrow><mml:mrow class=MJX-TeXAtom-ORD><mml:mrow class=MJX-TeXAtom-ORD><mml:mi class=MJX-tex-caligraphic mathvariant=script>E</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy=false>(</mml:mo><mml:mi>L</mml:mi><mml:mo>,</mml:mo><mml:mrow class=MJX-TeXAtom-ORD><mml:mrow class=MJX-TeXAtom-ORD><mml:mi class=MJX-tex-caligraphic mathvariant=script>L</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy=false>)</mml:mo></mml:mrow><mml:annotation encoding=application/x-tex>{mathcal E}(L,{mathcal L})</mml:annotation></mml:semantics></mml:math></inline-formula>. We construct a canonical isomorphism in the case when a certain transversality condition is satisfied. Our isomorphism then is functorial." @default.
• W2100024891 created "2016-06-24" @default.
• W2100024891 creator A5046412940 @default.
• W2100024891 date "2002-02-27" @default.
• W2100024891 modified "2023-10-09" @default.
• W2100024891 title "Mirror symmetry of abelian varieties and multi-theta functions" @default.
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• W2100024891 doi "https://doi.org/10.1090/s1056-3911-02-00329-6" @default.
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