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Mirror symmetry and rational curves on quintic threefolds: A guide for mathematicians. Appendix A: Proofs of the monodromy lemmas. Appendix B: Resolution of certain quotient singularities. Appendix C: The monodromy of the quintic-mirrors. (English) Zbl 0843.14005
J. Am. Math. Soc. 6, No. 1, 223-247 (1993); Appendices: A: 238–241; B: 241–243; C: 243–245 (1993).
The article is a survey and an introduction to the conjectures around mirror symmetries, which tries to explain the mathematical context. The conjectures were stated and some examples of evidence were discussed in the late eighties by physicists and since then there is a continuous interest among mathematicians in this questions.
The first three sections (1. Variation of Hodge structure arising from families of Calabi-Yau manifolds; 2. The asymptotic behavior of the periods; 3. The \(q\)-expansion of the Yukawa coupling) are mathematical rigorous. They are based on well established mathematical topics like variation of Hodge-structures and deformation of complex structures. – The Yukawa coupling is explained mathematically as the map \(\text{Sym}^n H^1 (\Theta_X) \to \operatorname{Hom} (H^{n,0} (X)\), \(H^{0,n} (X)) \simeq H^{n,0} (X)^{* \otimes 2}\), obtained on an \(n\)-dimensional compact Kähler variety (in particular Calabi-Yau manifold) by iteration of cup products and contractions with vector fields. For families of such varieties, parametrized by a quasiprojective variety \(C\), with suitable compactification of the family over \(\overline C \supset C = \overline C \backslash B\) \((B\) a divisor with normal crossings) it is a map \(\text{Sym}^n (\Theta_{\overline C} (- \log B)) \to \overline {\mathcal F}^n\), where \((\overline {\mathcal F}^r)\) is the natural extension on \(\overline C\) of the Hodge filtration on the bundle \(R^n \pi_* \mathbb{C} \otimes {\mathcal O}_C\) for the family \(X @>\pi>> C\). Over \(C\) it is obtained by composing the previous map with the Kodaira-Spencer map of the family. The choice of a section of \((\overline {\mathcal F}^n)^{\otimes 2}\) defines then a map (normalized Yukawa map) \(\text{Sym}^n (\Theta_C (- \log B) \to {\mathcal O}_{\overline C}\) and in the second and third section the choice of such a normalization is discussed and the asymptotic behaviour near boundary points \((P \in B = \overline C \backslash C)\) where the monodromy is maximal unipotent (i.e. unipotent with Jordan blocks of length \(n + 1)\).
Section 4 discusses the mathematical version of the mirror symmetry conjecture: Conjectural, for Calabi-Yau 3-folds \(X\) equipped with some “extra structure” \(S\) [not yet precisely defined in geometric terms, something like a domain \(U \subset H^{1,1} (X)\) with a group \(\Gamma\) of automorphisms of \(U\) for example; the first order deformations should correspond to elements of \(H^{1,1} (X)]\) there should be associated a “mirror pair” \((X',S')\), equipped with natural isomorphisms, \(H^1 (X, \Theta_X)\to H^{1,1}(X)\) and \(H^{1,1} (X) @>\sim>> H^1 (X', \Theta_{X'})\), compatible with some cubic forms [the suitable normalized Yukawa map on \(H^1(X,\Theta_X) ]\) on the spaces.
In the last two sections the example of quintics is discussed.
There are three appendices explaining some more technical details on mathematical results.
Reviewer: H.Kurke (Berlin)

MSC:
14D07 Variation of Hodge structures (algebro-geometric aspects)
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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