Torus actions on small blowups of ℂℙ2

Liat Kessler

doi:10.2140/pjm.2010.244.133

Torus actions on small blowups of ℂℙ2

Research output: Contribution to journal › Article › peer-review

Abstract

A manifold obtained by k simultaneous symplectic blowups of C{double-struck}P{double-struck}₂ of equal sizes ∈ (where the size of C{double-struck}P{double-struck}¹ ≤ C{double-struck}P{double-struck}² is one) admits an effective two dimensional torus action if k ≤ 3. We show that it does not admit such an action if k ≥ 4 and ∈ ≤ 1/(3k2^2k). For the proof, we show a correspondence between the geometry of a symplectic toric four-manifold and the combinatorics of its moment map image. We also use techniques from the theory of J-holomorphic curves.

Original language	English
Pages (from-to)	133-154
Number of pages	22
Journal	Pacific Journal of Mathematics
Volume	244
Issue number	1
DOIs	https://doi.org/10.2140/pjm.2010.244.133
State	Published - 2010
Externally published	Yes

Keywords

Delzant polygon
Dimension four
J-holomorphic curve
Moment map
Symplectic blowup
Symplectic manifold
Torus action

ASJC Scopus subject areas

General Mathematics

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10.2140/pjm.2010.244.133

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abstract = "A manifold obtained by k simultaneous symplectic blowups of C{double-struck}P{double-struck}2 of equal sizes ∈ (where the size of C{double-struck}P{double-struck}1 ≤ C{double-struck}P{double-struck}2 is one) admits an effective two dimensional torus action if k ≤ 3. We show that it does not admit such an action if k ≥ 4 and ∈ ≤ 1/(3k22k). For the proof, we show a correspondence between the geometry of a symplectic toric four-manifold and the combinatorics of its moment map image. We also use techniques from the theory of J-holomorphic curves.",

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AB - A manifold obtained by k simultaneous symplectic blowups of C{double-struck}P{double-struck}2 of equal sizes ∈ (where the size of C{double-struck}P{double-struck}1 ≤ C{double-struck}P{double-struck}2 is one) admits an effective two dimensional torus action if k ≤ 3. We show that it does not admit such an action if k ≥ 4 and ∈ ≤ 1/(3k22k). For the proof, we show a correspondence between the geometry of a symplectic toric four-manifold and the combinatorics of its moment map image. We also use techniques from the theory of J-holomorphic curves.

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Torus actions on small blowups of ℂℙ2

Abstract

Keywords

ASJC Scopus subject areas

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