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A Small and Non-simple Geometric Transition

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Abstract

Following notation introduced in the recent paper (Rossi Int. J. Geom. Methods Mod. Phys. 12(5), 2015), this paper is aimed to present in detail an example of a small geometric transition which is not a simple one i.e. a deformation of a conifold transition. This is realized by means of a detailed analysis of the Kuranishi space of a Namikawa cuspidal fiber product, which in particular improves the conclusion of Y. Namikawa in Remark 2.8 and Example 1.11 of Namikawa (Topology 41(6), 1219–1237, 2002). The physical interest of this example is presenting a geometric transition which can’t be immediately explained as a massive black hole condensation to a massless one, as described by Strominger (Nucl. Phys. B451, 97–109, 1995).

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References

  1. Arbarello, E., Cornalba, M., Griffiths, P.A.: Geometry of algebraic curves II Grundlehren der Mathematischen Wissenschaften, vol. 268. Springer, Heidelberg (2011)

  2. Barth, W., Peters, C., Van de Ven, A.: Compact complex surfaces, vol. 4. Springer–Verlag, E.M.G (1984)

  3. Bogomolov, F.: Hamiltonian manifolds. Dokl.Akad.Nauk.SSSR 243/5, 1101–1104 (1978)

    MathSciNet  MATH  Google Scholar 

  4. Candelas, P., Green, P.S., Hübsch, T.: Rolling among vacua. Nucl. Phys. B330, 49–102 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Clemens, C.H.: Double Solids. Adv. in Math. 47, 107–230 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  6. Douady, A.: Le problème des modules locaux pour les espaces C–analytiques compacts. Ann. Scient. Éc. Norm. Sup. 4e Sér. 7, 569–602 (1974)

    MATH  Google Scholar 

  7. Friedman, R.: Simultaneous resolution of threefold double points. Math. Ann. 247, 671–689 (1986)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Godement, R.: Topologie algébrique et théorie des faisceaux. Hermann, Paris (1958)

  9. Grauert, H.: Über die Deformation isolierter Singularitäten analytischer Mengen. Invent. Math. 15, 171–198 (1972)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Grauert, H.: Der Satz von Kuranishi für Kompakte Komplexe Räume. Invent. Math. 25, 107–142 (1974)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Greuel, G.-M., Lossen, C., Shustin, E.: Introduction to singularities and deformations Springer Monographs in Mathematics. Springer-Verlag, Berlin (2007)

  12. Gross, M.: Primitive threefolds. J. Diff. Geom. 45, 288–318 (1997). math.AG/9512002

  13. Grothendieck, A.: Sur quelques points d’algèbre homologique. Tôhoku Math. J. 9, 199–221 (1957)

    MATH  Google Scholar 

  14. Heckman, G., Looijenga, E.: The moduli space of rational elliptic surfaces, Algebraic geometry 2000, Azumino (Hotaka). Adv. Stud. Pure Math. 36, 185–248 (2002)

    MATH  Google Scholar 

  15. Joyce, D.: Compact manifolds with Special Holonomy. Oxford Science Publications, Oxford–New York (2000)

    MATH  Google Scholar 

  16. Kapustka G.M.: Fiber products of elliptic surfaces with sections and associated Kummer fibrations. Manuscripta Math. 130, 121–135 (2009). arXiv:0802.3760 [math.AG]

    Article  MathSciNet  Google Scholar 

  17. Kas, A.: Weierstrass normal forms and invariants of elliptic surfaces. Trans. Amer. Math. Soc. 225, 259–266 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kodaira, K.: On compact analytic surface. II. Ann. Math. 77(2), 563–626 (1963)

    Article  MATH  Google Scholar 

  19. Kollár, J.: Minimal models of algebraic threefolds: Mori’s program. Séminaire Bourbaki, Astérisque 177-178, Exp. No. 712, 303–326 (1989)

    MATH  Google Scholar 

  20. Kollár, J., Mori, S.: Classification of three–dimensional flips. J. Amer. Math. Soc. 5, 533–703 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  21. Laufer, H.: On \(\mathbb {C} P^1\) as an exceptional set” in Recente developments in several complex variables. Ann. Math. Stud. 100, 261–276 (1981)

    Google Scholar 

  22. Lichtenbaum, S., Schlessinger, M.: On the cotangent complex of a morphism. Trans. A.M.S. 128, 41–70 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  23. Looijenga, E.J.N.: Isolated singular points on complete intersections London Mathematical Society Lecture Note Series, vol. 77. Cambridge University Press, Cambridge (1984)

  24. Miranda, R.: The moduli of Weierstrass fibrations over ℙ1. Math. Ann. 255(3), 379–394 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  25. Miranda, R., Persson, U.: On extremal rational elliptic surfaces. Math. Z. 193(4), 537–558 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  26. Morrison, D.R.: The birational geometry of surfaces with rational double points. Math. Ann. 271, 415–438 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  27. Namikawa, Y.: On deformations of Calabi-Yau 3-folds with terminal singularities. Topology 33(3), 429–446 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  28. Namikawa, Y.: Stratified local moduli of Calabi-Yau threefolds. Topology 41 (6), 1219–1237 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  29. Palamodov, V.P.: The existence of versal deformations of complex spaces. Dokl. Akad. Nauk. SSSR 206, 538–541 (1972)

    MathSciNet  MATH  Google Scholar 

  30. Palamodov, V.P.: Deformations of complex spaces. Russ. Math. Surv. 31(3), 129–197 (1976). from russian Uspekhi Mat. Nauk 31(3), 129194 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  31. Palamodov, V.P.: Deformations of complex spaces Several Complex Variables IV, Encyclopedia of Mathematics Sciences, vol. 10. Springer (1990)

  32. Pinkham, H.: Factorization of birational maps in dimension three. Singularities Proc. Symp. Pure Math. 40, 343–372 (1981)

    Google Scholar 

  33. Ran, Z.: Deformations of maps. In: Ballico, E., Ciliberto, C. (eds.) Algebraic Curves and Projective Geometry. LNM 1389, Springer–Verlag (1989)

  34. Ran, Z.: Deformations of manifolds with torsion or negative canonical bundle. J. Alg. Geom. 1, 279–291 (1992)

    MathSciNet  MATH  Google Scholar 

  35. Reid, M.: Canonical 3–folds Journées de Géométrie Algébrique D’Angers, pp. 671–689. Sijthoff and Norddhoff (1980)

  36. Reid, M.: Minimal model of canonical 3–folds Algebraic Varieties and Analytic Varieties, vol. 1, pp. 131–180. Advanced Studies of Pure Mathematics, North–Holland (1983)

  37. Reid, M.: The moduli space of 3–folds with K = 0 may neverthless be irreducible. Math. Ann. 287, 329–334 (1987)

    Article  MATH  Google Scholar 

  38. Rossi, M.: Geometric transitions. J. Geom. Phys. 56(9), 1940–1983 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. Rossi, M.: Homological type of geometric transitions. Geom. Dedicata 151, 323–359 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  40. Rossi, M.: Deforming geometric transition. Int. J. Geom. Methods Mod. Phys. 12,5 (2015)

  41. Schoen, C.: On fiber products of rational elliptic surfaces with section. Math. Z. 197(2), 177–199 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  42. Stevens, J.: Deformations of singularities Lecture Notes in Mathematics, vol. 1811. Springer-Verlag, Berlin (2003)

  43. Strominger, A.: Massless black holes and conifolds in string theory. Nucl. Phys. B451, 97–109 (1995). arXiv:hep-th/9504145

  44. Tian, G.: Smoothness of the universal deformation space of compact manifolds and its Weil–Petersson metric. In: Yau, S.-T. (ed.) Mathematical Aspects of String Theory, pp. 629–646. World Scientific, Singapore (1987)

    Chapter  Google Scholar 

  45. Todorov, A.: The Weil-Petersson geometry of the moduli space of (n3) (Calabi–Yau) manifolds. Comm. Math. Phys. 126, 325–346 (1989)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  46. Wahl, J.M.: Equisingular deformations of normal surface singularities, I. Ann. Math. 104, 325–356 (1976)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

A first draft of the present paper was written on a visit to the Dipartimento di Matematica e Applicazioni of the Università di Milano Bicocca and the Department of Mathematics of the University of Pennsylvania. I would like to thank the Faculties of both Departments for warm hospitality and in particular F. Magri, R. Paoletti and S. Terracini from the first Department and R. Donagi, A. Grassi, T. Pantev and J. Shaneson from the second Department. Special thanks are due to A. Grassi for beautiful and stimulating conversations. I am also greatly indebted to B. van Geemen for useful suggestions.

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  1. Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123, Torino, Italy

    Michele Rossi

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  1. Michele Rossi
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Correspondence to Michele Rossi.

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The author was partially supported by MIUR-PRIN 2010-11 Research Funds “Geometria delle Varietà Algebriche” and by the I.N.D.A.M. as a member of the G.N.S.A.G.A.

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Rossi, M. A Small and Non-simple Geometric Transition. Math Phys Anal Geom 20, 15 (2017). https://doi.org/10.1007/s11040-017-9247-z

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