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Rational connectedness implies finiteness of quantum \(K\)-theory. (English) Zbl 1364.14044

This short but multifaceted paper is concerned with several topics that play central roles in the contemporary mathematics, namely the geometry and the cohomology of homogeneous spaces, the theory of rational connected varieties, and the very well studied and developed Gromov-Witten theory. To be more precise, in the paper under review the authors cope with the Gromov-Witten variety of rational curves of degree \(d\) in a generalized flag variety \(X\), that meet three points of \(X\) in general position. Recall that the Grothendieck ring of coherent sheaves on a smooth complex compact variety admits a generalization to a quantum (and also equivariant) framework. If \(\alpha,\beta\) are any two elements of \(K(X)\), their quantum product in \(KQ(X)\) is given by \[ \alpha\star\beta=\sum_{d\geq 0}(\alpha\star \beta)_d\, q^d\in K(X)[[q]]. \tag{*} \] In other words, it is a deformation of the classical \(K\)-theoretic product \((\alpha\star\beta)_0\) with corrections for each degree \(d \geq 0\), which are encoded as coefficients of a formal power series, where \((\alpha\star\beta)_d\) is defined using the \(K\)-theory ring of the Kontsevich moduli space \(\overline{M}_{0,3}(X,d)\) of stable maps \(({\mathbb P}^1, P_1,P_2,P_3)\rightarrow X\) of degree \(d\).
An important issue when dealing with quantum products is to establish the finiteness of sums like (*), encoding the quantum deformation.
The main beautiful result of this short, although conceptual and dense paper, is that if \(\overline{M}_{0,3}(X,d)\) is rationally connected for sufficiently large degree \(d\), i.e. if there exists an integer \(d_{rc}\) such that \(\overline{M}_{0,3}(X,d)\) is rational connected for all \(d\geq d_{rc}\), and said \(d_{lc}\) the minimum length of a chain of rational curve connecting two general points, then the product \((\alpha\star\beta)_d\) vanishes for all \(d\geq d_{rc}+d_{lc}\).
This paper is so especially well written that the reviewer’s job may have reduced to merely pointing out its occurrence in the literature and recommending the interested audience to directly read the enlightening introduction. The introduction explains the content of the paper certainly better than any reviewer could do.

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14M15 Grassmannians, Schubert varieties, flag manifolds
14M20 Rational and unirational varieties
14M22 Rationally connected varieties
14N15 Classical problems, Schubert calculus
19E08 \(K\)-theory of schemes
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