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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  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
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  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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