## Quantum cohomology of complete intersections (1995)

Citations: | 32 - 0 self |

### BibTeX

@TECHREPORT{Beauville95quantumcohomology,

author = {Arnaud Beauville},

title = {Quantum cohomology of complete intersections},

institution = {},

year = {1995}

}

### Years of Citing Articles

### OpenURL

### Abstract

The quantum cohomology algebra of a projective manifold X is the cohomology of X endowed with a different algebra structure, which takes into account the geometry of rational curves in X. This structure has been first defined heuristically

### Citations

220 |
A mathematical theory of quantum cohomology
- Ruan, Tian
- 1995
(Show Context)
Citation Context ...een first defined heuristically by the mathematical physicists [V,W]; a rigorous construction (and proof of the associativity, which is highly non trivial) has been achieved recently by Ruan and Tian =-=[R-T]-=-. When computed e.g. for surfaces, the quantum cohomology looks rather complicated [C-M]. The aim of this note is to show that the situation improves considerably when the dimension becomes high with ... |

181 | Topological sigma models - Witten - 1988 |

131 | The Intermediate Jacobian of the Cubic Threefold Ann
- Clemens, Griffiths
- 1972
(Show Context)
Citation Context ...amma is a smooth curve; the map ' : H 3 (X; Z) ! H 1 (\Gamma; Z) gives rise to a morphism \Phi : JX ! J\Gamma , where J\Gamma is the Jacobian of \Gamma and JX the intermediate Jacobian of X (see e.g. =-=[C-G]-=-); the formula ('(ff) j '(fi)) = \Gamma2 (ff j fi) for ff; fi 2 H 3 (X; Z) given by Proposition 5 means that the principal polarization of J\Gamma induces twice the principal polarization of JX . One ... |

35 |
Characterizations of complex projective spaces and hyperquadrics
- Kobayashi, Ochiai
- 1973
(Show Context)
Citation Context ...d. Let us look at conics. Let p; q; r be positive integers such that p + q + r = n + 2k ; as above we assume psqsrsn . Moreover we will assume k ! n , which excludes only the trivial case of quadrics =-=[K-O]-=-. This implies p ! k and therefore 2ksp + q ! 3k . We have as before H p \Delta H q = ( n\Gammaq X i=0 ` i ) H p+q\Gammak ; since H p+q\Gammak = H p+q\Gammak + ( n\Gammar X j=0 ` j ) Hn\Gammar , we ob... |

30 | Topological mirrors and quantum rings. In Essays in mirror symmetry. Ed. S.-T.Yau - Vafa - 1992 |

23 |
Sur quelques proprietes fondamentales en theorie des intersections
- Grothendieck
- 1958
(Show Context)
Citation Context ...ness. Put c p = ff p+1 \Gamma fi p+1 ff \Gamma fi for all p . The (usual!) cohomology algebra of G is the algebra of symmetric polynomials in ff; fi , modulo the ideal generated by c N\Gamma1 and c N =-=[G]-=-. Consider the linear form which associates to a symmetric polynomial P(ff; fi) the coefficient of ff N\Gamma1 fi N\Gamma1 in \Gamma 1 2 (ff \Gamma fi) 2 P(ff; fi) . It vanishes on the ideal (c N\Gamm... |

9 | Cohomologie des intersections complètes - DELIGNE - 1973 |

8 |
R.: Quantum cohomology of rational surfaces
- Crauder, Miranda
- 1995
(Show Context)
Citation Context ...ction (and proof of the associativity, which is highly non trivial) has been achieved recently by Ruan and Tian [R-T]. When computed e.g. for surfaces, the quantum cohomology looks rather complicated =-=[C-M]-=-. The aim of this note is to show that the situation improves considerably when the dimension becomes high with respect to the degree. Our main result is: Theorem :\Gamma Let X ae P n+r be a smooth co... |

5 |
Cohomologie des intersections completes (SGA 7
- Deligne
- 1973
(Show Context)
Citation Context ... theorem. One has KX = \GammakH , with k = n + 1 \Gamma P (d i \Gamma 1) ; therefore the inequality on n ensures that (ii) holds. The space 4 H n (X; Q) is nonzero except for odd-dimensional quadrics =-=[D]-=-, so condition (iii) holds as well. Finally if H n (X; Q) is of dimension 2 for n even, it is of type ( n 2 ; n 2 ) ; by [D] this is possible only for even-dimensional quadrics, which gives (iv). Ther... |

4 |
Bott’s formula and enumerative geometry, preprint
- Ellingsrud, Stromme
- 1994
(Show Context)
Citation Context ... 1 . To prove the theorem, we can assume in view of (1:1) that X is general; then the variety of lines (resp. conics, resp. twisted cubics) contained in X has the expected dimension: see for instance =-=[E-S]-=-, where the proof (given for the case of twisted cubics) adapts immediately to the easier cases of lines and conics. Let us check that the hypotheses of Proposition 1 are satisfied. Condition (i) hold... |

4 |
characteristics of systems of straight lines on complete intersections
- Libgober
(Show Context)
Citation Context ... P ` p . Recall that d` p is the number of lines in X meeting two general linear spaces of codimensionsn \Gamma p and k + p \Gamma 1 respectively (Remark 1). This number has been computed by Libgober =-=[L]-=-; I will give here a different proof. Let V be a complex vector space, of dimension N ; let us denote by G = G(2; V) the Grassmannian of lines in the projective space P(V) 1 . On G we have a tautologi... |