Notes on stable maps and quantum cohomology. (English) Zbl 0898.14018

Kollár, János (ed.) et al., Algebraic geometry. Proceedings of the Summer Research Institute, Santa Cruz, CA, USA, July 9–29, 1995. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 62(pt.2), 45-96 (1997).
This is an introduction to a new development in classical enumerative geometry based on the theory of quantum cohomology. The authors discuss the construction of the moduli space \(M_{g,n}(X,\beta)\) of isomorphism classes of pointed maps \((C,p_1,\ldots,p_n,\mu)\), where \(C\) is a smooth projective curve of genus \(g\), with fixed distinct \(n\) points on it, and \(\mu\) is a map \(C\to X\) to a fixed smooth algebraic variety such that \(\mu_*([C])\) is equal to the fixed algebraic 1-cycle \(\beta\) on \(X\). This moduli space admits a natural compactification \(\overline M_{g,n}(X,\beta)\) which corresponds to adding stable degenerate curves and stable maps. Based on these spaces, one can define the Gromov-Witten invariants \(I_\beta(\gamma_1,\ldots,\gamma_n)\). They assign to arbitrary algebraic cycles \(\gamma_1,\ldots,\gamma_n\) the top dimensional part of the cohomology class \(\rho_1^*(\gamma_1)\cup\ldots\cup\rho_n^*(\gamma_n)\) on \(\overline M_{0,n}(X,\beta)\), where \(\rho_i:\overline M_{0,n}(X,\beta)\to X\) is defined by sending \((C,p_1,\ldots,p_n,\mu)\) to \(\mu(p_i)\). When \(g=0\) and \(n =3\) the Gromov-Witten invariants lead to the definition of quantum cohomology of \(X\). When the target space is a homogeneous space with automorphism group \(G\) (e.g. projective space) and \(\gamma_i = [\Gamma_i]\) for some pure-dimensional subvarieties of \(X\) with \(\sum_{i=1}^n \text{ codim}(\Gamma_i) = \dim(X)+\int_\beta c_1(T_X)+n-3\), one can interpret \(I_\beta(\gamma_1,\ldots,\gamma_n)\) as the number of rational curves \(R\) on \(X\) which intersect general translates of the \(\Gamma_i\) at a fixed point \(q_i\in R\). The property of the associativity of the quantum cohomology gives some non-trivial relations for enumerative problems on \(X\). For example, the authors discuss the solutions of the following classical problems: Find the degree of the locus of plane rational curves of given degree; find the number of degree \(d\) rational curves in \(\mathbb{P}^3\) meeting \(a\) general lines and \(b\) general points, where \(a=2b=3d\).
For the entire collection see [Zbl 0882.00033].


14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14H10 Families, moduli of curves (algebraic)
14E99 Birational geometry
81T20 Quantum field theory on curved space or space-time backgrounds
81T70 Quantization in field theory; cohomological methods
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