The genus 0 Gromov–Witten invariants of projective complete intersections

Zinger, Aleksey¹

¹ Department of Mathematics, SUNY Stony Brook, Stony Brook, NY 11794-3651, USA

Geometry & topology, Tome 18 (2014) no. 2, pp. 1035-1114.

Résumé

We describe the structure of mirror formulas for genus $0$ Gromov–Witten invariants of projective complete intersections with any number of marked points and provide an explicit algorithm for obtaining the relevant structure coefficients. As an application, we give explicit closed formulas for the genus $0$ Gromov–Witten invariants of Calabi–Yau complete intersections with $3$ and $4$ constraints. The structural description alone suffices for some qualitative applications, such as vanishing results and the bounds on the growth of these invariants predicted by R Pandharipande. The resulting theorems suggest intriguing conjectures relating GW–invariants to the energy of pseudoholomorphic maps and the expected dimensions of their moduli spaces.

Détail

DOI : 10.2140/gt.2014.18.1035

Classification : 14N35, 53D45
Keywords: Gromov–Witten invariants, complete intersections

Bibliographie

[1] T M Apostol, Calculus, Vol. II: Multi-variable calculus and linear algebra, with applications to differential equations and probability, Blaisdell Publishing (1969)

[2] M Artin, Algebra, Prentice Hall (1991)

[3] M F Atiyah, R Bott, The moment map and equivariant cohomology, Topology 23 (1984) 1

[4] S Barannikov, Generalized periods and mirror symmetry in dimensions $n\gt 3$,

[5] A Bertram, Another way to enumerate rational curves with torus actions, Invent. Math. 142 (2000) 487

[6] A Bertram, H P Kley, New recursions for genus-zero Gromov–Witten invariants, Topology 44 (2005) 1

[7] P Candelas, X C De La Ossa, P S Green, L Parkes, A pair of Calabi–Yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B 359 (1991) 21

[8] L Cherveny, Genus-zero mirror principle with two marked points,

[9] P Di Francesco, C Itzykson, Quantum intersection rings, from: "RCP 25, Vol. 46" (editors R Dijkgraaf, C Faber, G van der Geer), Prépubl. Inst. Rech. Math. Av. 1994/29, Univ. Louis Pasteur (1994) 153

[10] G Ellingsrud, S A Strømme, Bott's formula and enumerative geometry, J. Amer. Math. Soc. 9 (1996) 175

[11] A Gathmann, Absolute and relative Gromov–Witten invariants of very ample hypersurfaces, Duke Math. J. 115 (2002) 171

[12] A B Givental, Equivariant Gromov–Witten invariants, Internat. Math. Res. Notices (1996) 613

[13] A B Givental, The mirror formula for quintic threefolds, from: "Northern California Symplectic Geometry Seminar", Amer. Math. Soc. Transl. Ser. 2 196, Amer. Math. Soc. (1999) 49

[14] A B Givental, Semisimple Frobenius structures at higher genus, Internat. Math. Res. Notices (2001) 1265

[15] B R Greene, D R Morrison, M R Plesser, Mirror manifolds in higher dimension, from: "Mirror symmetry, II" (editors B Greene, S T Yau), AMS/IP Stud. Adv. Math. 1, Amer. Math. Soc. (1997) 745

[16] P Griffiths, J Harris, Principles of algebraic geometry, Wiley Classics Library, Wiley (1994)

[17] K Hori, S Katz, A Klemm, R Pandharipande, R Thomas, C Vafa, R Vakil, E Zaslow, Mirror symmetry, Clay Mathematics Monographs 1, Amer. Math. Soc. (2003)

[18] S Katz, Rational curves on Calabi–Yau manifolds: Verifying predictions of mirror symmetry, from: "Projective geometry with applications" (editor E Ballico), Lecture Notes in Pure and Appl. Math. 166, Dekker (1994) 231

[19] A Klemm, R Pandharipande, Enumerative geometry of Calabi–Yau $4$–folds, Comm. Math. Phys. 281 (2008) 621

[20] Y P Lee, Quantum Lefschetz hyperplane theorem, Invent. Math. 145 (2001) 121

[21] Y P Lee, R Pandharipande, A reconstruction theorem in quantum cohomology and quantum $K$–theory, Amer. J. Math. 126 (2004) 1367

[22] B H Lian, K Liu, S T Yau, Mirror principle, I, Asian J. Math. 1 (1997) 729

[23] D Maulik, R Pandharipande, in progress

[24] D Maulik, R Pandharipande, A topological view of Gromov–Witten theory, Topology 45 (2006) 887

[25] D Mcduff, D Salamon, $J$–holomorphic curves and symplectic topology, AMS Colloquium Publications 52, Amer. Math. Soc. (2004)

[26] A Popa, The genus one Gromov–Witten invariants of Calabi–Yau complete intersections, Trans. Amer. Math. Soc. 365 (2013) 1149

[27] A Popa, A Zinger, Mirror symmetry for closed, open, and unoriented Gromov–Witten invariants,

[28] Y Ruan, G Tian, A mathematical theory of quantum cohomology, J. Differential Geom. 42 (1995) 259

[29] D Zagier, A Zinger, Some properties of hypergeometric series associated with mirror symmetry, from: "Modular forms and string duality" (editors N Yui, H Verrill, C F Doran), Fields Inst. Commun. 54, Amer. Math. Soc. (2008) 163

[30] A Zinger, Genus–zero two–point hyperplane integrals in the Gromov–Witten theory, Comm. Anal. Geom. 17 (2009) 955

[31] A Zinger, The reduced genus $1$ Gromov–Witten invariants of Calabi–Yau hypersurfaces, J. Amer. Math. Soc. 22 (2009) 691