Intrinsic mirror symmetry

We associate a ring <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to a log Calabi-Yau pair <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper X comma upper D right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(X,D)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or a degeneration of Calabi-Yau manifolds <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X right-arrow upper B"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>B</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">X\to B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The vector space underlying <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is determined by the tropicalization of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper X comma upper D right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(X,D)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X right-arrow upper B"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>B</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">X\to B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , while the product rule is defined using punctured Gromov-Witten invariants, defined in joint work with Abramovich and Chen. In the log Calabi-Yau case, if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is maximally degenerate, then we propose that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S p e c upper R"> <mml:semantics> <mml:mrow> <mml:mi>Spec</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mi>R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Spec} R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the mirror to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X minus upper D"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo class="MJX-variant"> ∖ </mml:mo> <mml:mi>D</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">X\setminus D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , while in the Calabi-Yau degeneration case, if the degeneration is maximally unipotent, the mirror is expected to be <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P r o j upper R"> <mml:semantics> <mml:mrow> <mml:mi>Proj</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mi>R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Proj} R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The main result in this paper is that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as defined is an associative, commutative ring with unit, with associativity the most difficult part.