Computing the quantum cohomology of some Fano threefolds and its semisimplicity

Ciolli, Gianni

Bollettino della Unione matematica italiana, Série 8, 7B (2004) no. 2, pp. 511-517.

Abstract (VO)

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.

Riassunto

Nella presente nota si calcola una presentazione esplicita dell'anello di coomologia quantica «small» per alcune threefold di Fano, ottenute scoppiando una o due curve lisce in $\mathbb{P}^{3}$ o nella quadrica liscia. Usando sistematicamente l'associatività del prodotto quantico, si rende necessario calcolare esplicitamente soltanto un sottoinsieme molto piccolo ed enumerativo della famiglia degli invarianti di Gromov-Witten. Successivamente, si mostra che tali varietà soddisfano la congettura di Dubrovin sulla semisemplicità della coomologia quantica, sia mediante una semplice verifica sulle presentazioni in precedenza calcolate, sia mostrando che una threefold di Fano liscia $X$ con $b_{3}(X)=0$ ammette un sistema eccezionale completo di generatori per la categoria derivata dei fasci coerenti. I dettagli si trovano nel preprint [4] e saranno pubblicati altrove.

Bibliographie

[1] Vincenzo Ancona-Marco Maggesi, Quantum Cohomology of some Fano threefolds, to appear in Advances in Geometry, 2002. | MR | Zbl

[2] Arend Bayer, Semisimple Quantum Cohomology and Blow-ups, Preprint arXiv:math.AG/0403260, 2004. | fulltext mini-dml | MR | Zbl

[3] Arend Bayer-Yuri I. Manin, (Semi)simple exercises in Quantum Cohomology, Preprint arXiv:math.AG/0103164, 2001. | fulltext mini-dml | MR

[4] Gianni Ciolli, On the Quantum Cohomology of some Fano threefolds and a conjecture of Dubrovin, 2004, Preprint Dip. Mat. «U. Dini» n. 3/2004. | fulltext mini-dml | MR | Zbl

[5] L. Costa-R. M. Mirò-Roig, Quantum cohomology of projective bundles over $\mathbb{P}^{n_1}\times\ldots\times\mathbb{P}^{n_s}$, International J. of Math., 11, no. 6 (2000), 761-797. | MR | Zbl

[6] Boris Dubrovin, 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. | MR | Zbl

[7] W. Fulton-R. Pandharipande, 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. | MR | Zbl

[8] V. A. Iskovskih, Fano threefolds. I, Izv. Akad. Nauk SSSR Ser. Mat., 41, no. 3 (1977), 516-562, 717. | MR | Zbl

[9] V. A. Iskovskih, Fano threefolds. II, Izv. Akad. Nauk SSSR Ser. Mat., 42, no. 3 (1978), 506-549. | MR | Zbl

[10] Shigefumi Mori-Shigeru Mukai, Classification of Fano $3$-folds with $B_2 \ge 2$, Manuscripta Math., 36, no. 2 (1981/82), 147-162. | MR | Zbl

[11] Shigefumi Mori-Shigeru Mukai, Erratum to «classification of Fano 3-folds with $B_2 \ge 2$», Manuscripta Math., 110 (2003), 407. | MR | Zbl

[12] D. O. Orlov, Projective bundles, monoidal transformations, and derived categories of coherent sheaves, Russian Acad. Sci. Izv. Math., 41, no. 1 (1993), 133-141. | MR | Zbl

[13] Z. Qin-Y. Ruan, Quantum cohomology of projective bundles over $\mathbb{P}^n$, Transactions of the Am. Math. Soc., 350, no. 9 (1998), 3615-3638. | MR | Zbl

[14] Holger Spielberg, 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. | MR | Zbl