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# Asian Journal of Mathematics

## Volume 20 (2016)

### Number 1

### Rational connectedness implies finiteness of quantum $K$-theory

Pages: 117 – 122

DOI: http://dx.doi.org/10.4310/AJM.2016.v20.n1.a5

#### Authors

#### Abstract

Let $X$ be any generalized flag variety with Picard group of rank one. Given a degree $d$, consider the Gromov–Witten variety of rational curves of degree $d$ in $X$ that meet three general points. We prove that, if this Gromov–Witten variety is rationally connected for all large degrees $d$, then the structure constants of the small quantum $K$-theory ring of $X$ vanish for large degrees.

#### Keywords

quantum $K$ theory, rational connected varieties, Gromov–Witten variety

#### 2010 Mathematics Subject Classification

Primary 14N35. Secondary 14M15, 14M20, 14M22, 14N15, 19E08.