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Enumeration of rational curves via torus actions. (English) Zbl 0885.14028
Dijkgraaf, R. H. (ed.) et al., The moduli space of curves. Proceedings of the conference held on Texel Island, Netherlands, during the last week of April 1994. Basel: Birkhäuser. Prog. Math. 129, 335-368 (1995).
This paper provides a very detailed survey on the author’s recent pioneering work concerning the interrelations between topological quantum field theory, intersection theory on moduli spaces of curves, and enumerative geometry. Kontsevich’s spectacular work consisted in formulating rigorously, and checking affirmatively predictions in the enumerative geometry of curves on certain algebraic manifolds, which conjecturally followed from physical reasoning (in quantum field theory) and from the mirror conjecture in complex geometry [cf. M. Kontsevich, Commun. Math. Phys. 147, No. 1, 1-253 (1992; Zbl 0756.35081)].
The text of the paper consists of five parts. The first part introduces stable maps of \(n\)-pointed curves, establishes several basic properties of their moduli spaces, and gives an outline of a rigorous construction of Gromov-Witten invariants in quantum cohomology. Section 2 discusses a few example of counting problems in the enumerative geometry of curves, with a special emphasis on rational curves on quintics. The author’s comparatively simple algebro-geometric definition of the number of those rational curves works without assuming the validity of the Clemens conjecture or using symplectic methods.
Section 3 contains the main body of the crucial computations of intersection and counting numbers. The basic assumption on the target manifold, in which the rational curves are to be counted, is that it is equipped with a torus action. This allows to reduce the counting problems to questions concerning Chern classes of the certain space of rational curves lying in a projective space, and then to apply Bott’s residue formula for torus actions. Section 4 describes a general formula, well-known in physics and combinatorics, which gives – via Feynman rules – explicit values of certain infinite sums over special Feynman diagrams, the so-called “trees”. These values appear as the critical values of some action functionals, and in a very tricky way the author manages to reduce all the exemplary counting problems (from section 2) to the problem of inverting certain explicit square matrices with hypergeometric coefficients. However, this last, crucial step is not yet completely accomplished. The author points out that this inversion problem might be related to some hidden structure of an integrable system and Sato’s infinite Grassmannians. The concluding section 5 gives a brief outlook to possible extensions of the computation scheme, developed so far, to other enumerative problems involving rational curves. This includes curves in Calabi-Yau manifolds and Fano varieties of arbitrary dimension, toric varieties, and generalized flag varieties.
The paper gives an excellent introduction and overview concerning this kind of problems in contemporary geometry and physics. It is written in a very lucid and detailed style, and of highly enlightening and inspiring character. As for a more and further-going work in this direction, the reader is referred to the paper “Gromov-Witten classes, quantum cohomology, and enumerative geometry” by M. Kontsevich and Yu. Manin [Commun. Math. Phys. 164, No. 3, 525-562 (1994; Zbl 0853.14020)].
For the entire collection see [Zbl 0827.00037].

MSC:
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14H45 Special algebraic curves and curves of low genus
14D20 Algebraic moduli problems, moduli of vector bundles
14M20 Rational and unirational varieties
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