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Computing the quantum cohomology of some Fano threefolds and its semisimplicity

Gianni Ciolli

Bollettino dell'Unione Matematica Italiana (2004)

  • Volume: 7-B, Issue: 2, page 511-517
  • ISSN: 0392-4041

Abstract

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We compute explicit presentations for the small Quantum Cohomology ring of some Fano threefolds which are obtained as one- or two-curve blow-ups from P 3 or the smooth quadric. Systematic usage of the associativity property of quantum product implies that only a very small and enumerative subset of Gromov- Witten invariants is needed. Then, for these threefolds the Dubrovin conjecture on the semisimplicity of Quantum Cohomology is proven by checking the computed Quantum Cohomology rings and by showing that a smooth Fano threefold X with b 3 ⁢ X = 0 admits a complete exceptional set of the appropriate length. Details are contained in the preprint [4] and will be published elsewhere.

How to cite

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Ciolli, Gianni. "Computing the quantum cohomology of some Fano threefolds and its semisimplicity." Bollettino dell'Unione Matematica Italiana 7-B.2 (2004): 511-517. <http://eudml.org/doc/195165>.

@article{Ciolli2004,
abstract = {We compute explicit presentations for the small Quantum Cohomology ring of some Fano threefolds which are obtained as one- or two-curve blow-ups from $\mathbb\{P\}^\{3\}$ or the smooth quadric. Systematic usage of the associativity property of quantum product implies that only a very small and enumerative subset of Gromov- Witten invariants is needed. Then, for these threefolds the Dubrovin conjecture on the semisimplicity of Quantum Cohomology is proven by checking the computed Quantum Cohomology rings and by showing that a smooth Fano threefold $X$ with $b_\{3\}(X)=0$ admits a complete exceptional set of the appropriate length. Details are contained in the preprint [4] and will be published elsewhere.},
author = {Ciolli, Gianni},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {511-517},
publisher = {Unione Matematica Italiana},
title = {Computing the quantum cohomology of some Fano threefolds and its semisimplicity},
url = {http://eudml.org/doc/195165},
volume = {7-B},
year = {2004},
}

TY - JOUR
AU - Ciolli, Gianni
TI - Computing the quantum cohomology of some Fano threefolds and its semisimplicity
JO - Bollettino dell'Unione Matematica Italiana
DA - 2004/6//
PB - Unione Matematica Italiana
VL - 7-B
IS - 2
SP - 511
EP - 517
AB - We compute explicit presentations for the small Quantum Cohomology ring of some Fano threefolds which are obtained as one- or two-curve blow-ups from $\mathbb{P}^{3}$ or the smooth quadric. Systematic usage of the associativity property of quantum product implies that only a very small and enumerative subset of Gromov- Witten invariants is needed. Then, for these threefolds the Dubrovin conjecture on the semisimplicity of Quantum Cohomology is proven by checking the computed Quantum Cohomology rings and by showing that a smooth Fano threefold $X$ with $b_{3}(X)=0$ admits a complete exceptional set of the appropriate length. Details are contained in the preprint [4] and will be published elsewhere.
LA - eng
UR - http://eudml.org/doc/195165
ER -

References

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  1. ANCONA, VINCENZO- MAGGESI, MARCO, Quantum Cohomology of some Fano threefolds, to appear in Advances in Geometry, 2002. Zbl1076.14077MR2110460
  2. BAYER, AREND, Semisimple Quantum Cohomology and Blow-ups, Preprint arXiv:math.AG/0403260, 2004. Zbl1080.14063MR2064316
  3. BAYER, AREND- MANIN, YURI I., (Semi)simple exercises in Quantum Cohomology, Preprint arXiv:math.AG/0103164, 2001. MR2112573
  4. CIOLLI, GIANNI, On the Quantum Cohomology of some Fano threefolds and a conjecture of Dubrovin, 2004, Preprint Dip. Mat. «U. Dini» n. 3/2004. Zbl1081.14075MR2168069
  5. COSTA, L.- MIRÒ-ROIG, R. M., Quantum cohomology of projective bundles over P n 1 × … × P n s , International J. of Math., 11, no. 6 (2000), 761-797. Zbl0969.14038MR1785517
  6. DUBROVIN, BORIS, Geometry and analytic theory of Frobenius manifolds, Proceedings of the International Congress of Mathematicians, Vol. II, Doc. Math. 1998, no. Extra Vol. II, Berlin, 1998, pp. 315-326. Zbl0916.32018MR1648082
  7. FULTON, W.- PANDHARIPANDE, R., Notes on stable maps and quantum cohomology, Algebraic geometry - Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 45-96. Zbl0898.14018MR1492534
  8. ISKOVSKIH, V. A., Fano threefolds. I, Izv. Akad. Nauk SSSR Ser. Mat., 41, no. 3 (1977), 516-562, 717. Zbl0363.14010MR463151
  9. ISKOVSKIH, V. A., Fano threefolds. II, Izv. Akad. Nauk SSSR Ser. Mat., 42, no. 3 (1978), 506-549. Zbl0407.14016MR503430
  10. MORI, SHIGEFUMI- MUKAI, SHIGERU, Classification of Fano 3 -folds with B 2 ≥ 2 , Manuscripta Math., 36, no. 2 (1981/82), 147-162. Zbl0478.14033MR641971
  11. MORI, SHIGEFUMI- MUKAI, SHIGERU, Erratum to «classification of Fano 3-folds with B 2 ≥ 2 », Manuscripta Math., 110 (2003), 407. Zbl0478.14033MR1969009
  12. ORLOV, D. O., Projective bundles, monoidal transformations, and derived categories of coherent sheaves, Russian Acad. Sci. Izv. Math., 41, no. 1 (1993), 133-141. Zbl0798.14007MR1208153
  13. QIN, Z.- RUAN, Y., Quantum cohomology of projective bundles over P n , Transactions of the Am. Math. Soc., 350, no. 9 (1998), 3615-3638. Zbl0932.14030MR1422617
  14. SPIELBERG, HOLGER, The Gromov-Witten invariants of symplectic toric manifolds, and their quantum cohomology ring, C. R. Acad. Sci. Paris Sér. I Math., 329, no. 8 (1999), 699-704. Zbl1004.14014MR1724149

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