Logarithmic asymptotics of the genus zero Gromov–Witten invariants of the blown up plane

Itenberg, Ilia; Kharlamov, Viatcheslav; Shustin, Eugenii

doi:10.2140/gt.2005.9.483

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Logarithmic asymptotics of the genus zero Gromov–Witten invariants of the blown up plane

Ilia Itenberg, Viatcheslav Kharlamov and Eugenii Shustin

Geometry & Topology 9 (2005) 483–491

DOI: 10.2140/gt.2005.9.483

arXiv: math.AG/0412533

Abstract

We study the growth of the genus zero Gromov–Witten invariants $G W_{n D}$ of the projective plane $P_{k}^{2}$ blown up at $k$ points (where $D$ is a class in the second homology group of $P_{k}^{2}$ ). We prove that, under some natural restrictions on $D$ , the sequence $log G W_{n D}$ is equivalent to $λ n log n$ , where $λ = D \cdot c_{1} (P_{k}^{2})$ .

Keywords

Gromov–Witten invariants, rational, ruled algebraic surfaces, rational, ruled symplectic 4–manifolds, tropical enumerative geometry

Mathematical Subject Classification 2000

Primary: 14N35

Secondary: 14J26, 53D45

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Publication

Received: 30 December 2004

Accepted: 25 March 2005

Published: 7 April 2005

Proposed: Yasha Eliashberg

Seconded: Leonid Polterovich, Simon Donaldson

Authors

Ilia Itenberg
Université Louis Pasteur et IRMA 7, rue René Descartes 67084 Strasbourg Cedex France

Viatcheslav Kharlamov
Université Louis Pasteur et IRMA 7, rue René Descartes 67084 Strasbourg Cedex France

Eugenii Shustin
School of Mathematical Sciences Tel Aviv University Ramat Aviv 69978 Tel Aviv Israel

Volume 9, issue 1 (2005)