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Kodaira dimensions of almost complex manifolds I
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Chen, Haojie and Zhang, Weiyi (2023) Kodaira dimensions of almost complex manifolds I. American Journal of Mathematics, 145 (2). pp. 477-514. doi:10.1353/ajm.2023.0011 ISSN 0002-9327.
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Official URL: https://doi.org/10.1353/ajm.2023.0011
Abstract
This is the first of a series of papers in which we study the plurigenera, the Kodaira dimension, and more generally the Iitaka dimension on compact almost complex manifolds.
Based on the Hodge theory on almost complex manifolds, we introduce the plurigenera, Kodaira dimension and Iitaka dimension on compact almost complex manifolds. We show that plurigenera and the Kodaira dimension are birational invariants in almost complex category, at least in dimension $4$, where a birational morphism is defined to be a degree one pseudoholomorphic map. However, they are no longer deformation invariants, even in dimension $4$ or under tameness assumption. On the way to establish the birational invariance, we prove the Hartogs extension theorem in the almost complex setting by the foliation-by-disks technique.
Some interesting phenomena of these invariants are shown through examples. In particular, we construct non-integrable compact almost complex manifolds with large Kodaira dimensions. Hodge numbers and plurigenera are computed for the standard almost complex structure on the six sphere $S^6$, which are different from the data of a hypothetical complex structure.
Item Type: | Journal Article | ||||||
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Subjects: | Q Science > QA Mathematics | ||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
Library of Congress Subject Headings (LCSH): | Complex manifolds, Surfaces, Algebraic | ||||||
Journal or Publication Title: | American Journal of Mathematics | ||||||
Publisher: | The Johns Hopkins University Press | ||||||
ISSN: | 0002-9327 | ||||||
Official Date: | 3 April 2023 | ||||||
Dates: |
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Volume: | 145 | ||||||
Number: | 2 | ||||||
Page Range: | pp. 477-514 | ||||||
DOI: | 10.1353/ajm.2023.0011 | ||||||
Status: | Peer Reviewed | ||||||
Publication Status: | Published | ||||||
Re-use Statement: | This article appeared in the American Journal of Mathematics, Volume 145 Issue 02, Year, pages 477-514, Copyright © 2022, Johns Hopkins University Press. | ||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||
Date of first compliant deposit: | 25 April 2022 | ||||||
Date of first compliant Open Access: | 25 April 2022 | ||||||
Related URLs: | |||||||
Open Access Version: |
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