The topology of the space of J–holomorphic maps to ℂP2

Miller, Jeremy¹

¹ Department of Mathematics, Stanford University, Building 380, Room 383A, Stanford, CA 94305, USA

Geometry & topology, Tome 19 (2015) no. 4, pp. 1829-1894.

Résumé

The purpose of this paper is to generalize a theorem of Segal proving that the space of holomorphic maps from a Riemann surface to a complex projective space is homology equivalent to the corresponding space of continuous maps through a range of dimensions increasing with degree. We will address if a similar result holds when other almost-complex structures are put on a projective space. For any compatible almost-complex structure $J$ on $ℂP^{\mathfrak{2}}$, we prove that the inclusion map from the space of $J$–holomorphic maps to the space of continuous maps induces a homology surjection through a range of dimensions tending to infinity with degree. The proof involves comparing the scanning map of topological chiral homology with analytic gluing maps for $J$–holomorphic curves . This is an extension of the author’s work regarding genus-zero case.

Détail

DOI : 10.2140/gt.2015.19.1829

Classification : 53D05, 55P48
Keywords: almost-complex structure, little disks operad, gluing

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