Asian Journal of Mathematics

Volume 20 (2016)

Number 1

Rational connectedness implies finiteness of quantum $K$-theory

Pages: 117 – 122



Anders S. Buch (Department of Mathematics, Rutgers University, Piscataway, New Jersey, U.S.A.)

Pierre-Emmanuel Chaput (Domaine Scientifique Victor Grignard, Université Henri Poincaré Nancy 1, Vandoeuvre-lès-Nancy, France)

Leonardo C. Mihalcea (Department of Mathematics, Virginia Tech University, Blacksburg, Va., U.S.A.)

Nicolas Perrin (Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay, Versailles, France)


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.


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

2010 Mathematics Subject Classification

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

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Published 28 January 2016