## 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
Full Text:

### References:

 [1] P. S. Aspinwall and C. A. Lütken, Geometry of mirror manifolds, Nuclear Phys. B 353 (1991), no. 2, 427 – 461. · doi:10.1016/0550-3213(91)90343-V [2] P. S. Aspinwall and C. A. Lütken, Quantum algebraic geometry of superstring compactifications, Nuclear Phys. B 355 (1991), no. 2, 482 – 510. · doi:10.1016/0550-3213(91)90123-F [3] P. S. Aspinwall, C. A. Lütken, and G. G. Ross, Construction and couplings of mirror manifolds, Phys. Lett. B 241 (1990), no. 3, 373 – 380. · doi:10.1016/0370-2693(90)91659-Y [4] F. A. Bogomolov, Hamiltonian Kählerian manifolds, Dokl. Akad. Nauk SSSR 243 (1978), no. 5, 1101 – 1104 (Russian). [5] P. Candelas, Yukawa couplings between (2,1)-forms, Nuclear Phys. B 298 (1988), no. 3, 458 – 492. · Zbl 0659.53059 · doi:10.1016/0550-3213(88)90351-3 [6] Philip Candelas, Xenia C. de la Ossa, Paul S. Green, and Linda Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B 359 (1991), no. 1, 21 – 74. · Zbl 1098.32506 · doi:10.1016/0550-3213(91)90292-6 [7] P. Candelas, M. Lynker, and R. Schimmrigk, Calabi-Yau manifolds in weighted \?$$_{4}$$, Nuclear Phys. B 341 (1990), no. 2, 383 – 402. · Zbl 0962.14029 · doi:10.1016/0550-3213(90)90185-G [8] James Carlson, Mark Green, Phillip Griffiths, and Joe Harris, Infinitesimal variations of Hodge structure. I, Compositio Math. 50 (1983), no. 2-3, 109 – 205. Phillip Griffiths and Joe Harris, Infinitesimal variations of Hodge structure. II. An infinitesimal invariant of Hodge classes, Compositio Math. 50 (1983), no. 2-3, 207 – 265. Phillip A. Griffiths, Infinitesimal variations of Hodge structure. III. Determinantal varieties and the infinitesimal invariant of normal functions, Compositio Math. 50 (1983), no. 2-3, 267 – 324. · Zbl 0531.14006 [9] S. Cecotti, \?=2 supergravity, type \?\?\? superstrings, and algebraic geometry, Comm. Math. Phys. 131 (1990), no. 3, 517 – 536. · Zbl 0712.53045 [10] S. Cecotti, \?=2 Landau-Ginzburg vs. Calabi-Yau \?-models: nonperturbative aspects, Internat. J. Modern Phys. A 6 (1991), no. 10, 1749 – 1813. · Zbl 0743.57022 · doi:10.1142/S0217751X91000939 [11] J. H. Conway and S. P. Norton, Monstrous moonshine, Bull. London Math. Soc. 11 (1979), no. 3, 308 – 339. · Zbl 0424.20010 · doi:10.1112/blms/11.3.308 [12] Robbert Dijkgraaf, Erik Verlinde, and Herman Verlinde, On moduli spaces of conformal field theories with \?\ge 1, Perspectives in string theory (Copenhagen, 1987) World Sci. Publ., Teaneck, NJ, 1988, pp. 117 – 137. · Zbl 0649.32019 [13] M. Dine, N. Seiberg, X.-G. Wen, and E. Witten, Nonperturbative effects on the string world sheet. II, Nuclear Phys. B 289 (1987), no. 2, 319 – 363. · doi:10.1016/0550-3213(87)90383-X [14] Jacques Distler and Brian Greene, Some exact results on the superpotential from Calabi-Yau compactifications, Nuclear Phys. B 309 (1988), no. 2, 295 – 316. · doi:10.1016/0550-3213(88)90084-3 [15] Lance J. Dixon, Some world-sheet properties of superstring compactifications, on orbifolds and otherwise, Superstrings, unified theories and cosmology 1987 (Trieste, 1987) ICTP Ser. Theoret. Phys., vol. 4, World Sci. Publ., Teaneck, NJ, 1988, pp. 67 – 126. [16] G. Ellingsrud and S. A. Strømme, The number of twisted cubic curves on the general quintic threefold, University of Bergen Report no. 63-7-2-1992. · Zbl 0824.14047 [17] Robert Friedman and Francesco Scattone, Type \?\?\? degenerations of \?3 surfaces, Invent. Math. 83 (1986), no. 1, 1 – 39. · Zbl 0613.14030 · doi:10.1007/BF01388751 [18] Doron Gepner, Exactly solvable string compactifications on manifolds of \?\?(\?) holonomy, Phys. Lett. B 199 (1987), no. 3, 380 – 388. · doi:10.1016/0370-2693(87)90938-5 [19] B. R. Greene and M. R. Plesser, Duality in Calabi-Yau moduli space, Nuclear Phys. B 338 (1990), no. 1, 15 – 37. · doi:10.1016/0550-3213(90)90622-K [20] B. R. Greene, C. Vafa, and N. P. Warner, Calabi-Yau manifolds and renormalization group flows, Nuclear Phys. B 324 (1989), no. 2, 371 – 390. · Zbl 0744.53044 · doi:10.1016/0550-3213(89)90471-9 [21] Phillip Griffiths , Topics in transcendental algebraic geometry, Annals of Mathematics Studies, vol. 106, Princeton University Press, Princeton, NJ, 1984. · Zbl 0528.00004 [22] Sheldon Katz, On the finiteness of rational curves on quintic threefolds, Compositio Math. 60 (1986), no. 2, 151 – 162. · Zbl 0606.14039 [23] G. Kempf, Finn Faye Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings. I, Lecture Notes in Mathematics, Vol. 339, Springer-Verlag, Berlin-New York, 1973. · Zbl 0271.14017 [24] Alan Landman, On the Picard-Lefschetz transformation for algebraic manifolds acquiring general singularities, Trans. Amer. Math. Soc. 181 (1973), 89 – 126. · Zbl 0284.14005 [25] Wolfgang Lerche, Cumrun Vafa, and Nicholas P. Warner, Chiral rings in \?=2 superconformal theories, Nuclear Phys. B 324 (1989), no. 2, 427 – 474. · doi:10.1016/0550-3213(89)90474-4 [26] D. G. Markushevich, M. A. Olshanetsky, and A. M. Perelomov, Description of a class of superstring compactifications related to semisimple Lie algebras, Comm. Math. Phys. 111 (1987), no. 2, 247 – 274. · Zbl 0628.53065 [27] Emil J. Martinec, Algebraic geometry and effective Lagrangians, Phys. Lett. B 217 (1989), no. 4, 431 – 437. · doi:10.1016/0370-2693(89)90074-9 [28] Emil J. Martinec, Criticality, catastrophes, and compactifications, Physics and mathematics of strings, World Sci. Publ., Teaneck, NJ, 1990, pp. 389 – 433. · Zbl 0737.58060 [29] Tadao Oda, Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15, Springer-Verlag, Berlin, 1988. An introduction to the theory of toric varieties; Translated from the Japanese. · Zbl 0628.52002 [30] Miles Reid, Minimal models of canonical 3-folds, Algebraic varieties and analytic varieties (Tokyo, 1981) Adv. Stud. Pure Math., vol. 1, North-Holland, Amsterdam, 1983, pp. 131 – 180. · Zbl 0558.14028 [31] Miles Reid, The moduli space of 3-folds with \?=0 may nevertheless be irreducible, Math. Ann. 278 (1987), no. 1-4, 329 – 334. · Zbl 0649.14021 · doi:10.1007/BF01458074 [32] Shi-Shyr Roan, On the generalization of Kummer surfaces, J. Differential Geom. 30 (1989), no. 2, 523 – 537. · Zbl 0661.14031 [33] Shi-Shyr Roan, On Calabi-Yau orbifolds in weighted projective spaces, Internat. J. Math. 1 (1990), no. 2, 211 – 232. · Zbl 0793.14031 · doi:10.1142/S0129167X90000137 [34] Shi-Shyr Roan, The mirror of Calabi-Yau orbifold, Internat. J. Math. 2 (1991), no. 4, 439 – 455. · Zbl 0817.14018 · doi:10.1142/S0129167X91000259 [35] Wilfried Schmid, Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973), 211 – 319. · Zbl 0278.14003 · doi:10.1007/BF01389674 [36] Chad Schoen, On the geometry of a special determinantal hypersurface associated to the Mumford-Horrocks vector bundle, J. Reine Angew. Math. 364 (1986), 85 – 111. · Zbl 0568.14022 · doi:10.1515/crll.1986.364.85 [37] Andrew Strominger and Edward Witten, New manifolds for superstring compactification, Comm. Math. Phys. 101 (1985), no. 3, 341 – 361. [38] J. G. Thompson, Some numerology between the Fischer-Griess Monster and the elliptic modular function, Bull. London Math. Soc. 11 (1979), no. 3, 352 – 353. · Zbl 0425.20016 · doi:10.1112/blms/11.3.352 [39] Gang Tian, Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric, Mathematical aspects of string theory (San Diego, Calif., 1986) Adv. Ser. Math. Phys., vol. 1, World Sci. Publishing, Singapore, 1987, pp. 629 – 646. [40] Andrey N. Todorov, The Weil-Petersson geometry of the moduli space of \?\?(\?\ge 3) (Calabi-Yau) manifolds. I, Comm. Math. Phys. 126 (1989), no. 2, 325 – 346. · Zbl 0688.53030 [41] Shing Tung Yau, Calabi’s conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 5, 1798 – 1799. · Zbl 0355.32028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.