Algebraic fiber spaces over abelian varieties: Around a recent theorem by Cao and Păun

Christopher Hacon; Mihnea Popa; Christian Schnell

doi:10.1090/conm/712/14346

Algebraic fiber spaces over abelian varieties: Around a recent theorem by Cao and Păun

Christopher Hacon, Mihnea Popa, Christian Schnell

Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Chapter

59 Scopus citations

Abstract

We present a simplified proof for a recent theorem by Junyan Cao and Mihai Păun, confirming a special case of Iitaka’s C_n,m conjecture: if f X → Y is an algebraic fiber space, and if the Albanese mapping of Y is generically finite over its image, then we have the inequality of Kodaira dimensions κ(X) ≥ κ(Y) + κ(F), where F denotes a general fiber of f. We include a detailed survey of the main algebraic and analytic techniques, especially the construction of singular hermitian metrics on pushforwards of adjoint bundles (due to Berndtsson, Păun, and Takayama).

Original language	English (US)
Title of host publication	Contemporary Mathematics
Publisher	American Mathematical Society
Pages	143-195
Number of pages	53
DOIs	https://doi.org/10.1090/conm/712/14346
State	Published - 2018

Publication series

Name	Contemporary Mathematics
Volume	712
ISSN (Print)	0271-4132
ISSN (Electronic)	1098-3627

Funding

this paper, and for many useful discussions and advice about its contents. We also thank Dano Kim and Luigi Lombardi for reading and commenting on a draft version. During the preparation of the paper, CH was partially supported by NSF grants DMS-1300750 and DMS-1265285 and by a grant from the Simons Foundation (Award #256202). MP was partially supported by NSF grant DMS-1405516 and by a Simons Fellowship. CS was partially supported by NSF grants DMS-1404947 and DMS-1551677, and by a Centennial Fellowship from the American Mathematical Society.

ASJC Scopus subject areas

General Mathematics

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10.1090/conm/712/14346

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title = "Algebraic fiber spaces over abelian varieties: Around a recent theorem by Cao and P{\u a}un",

abstract = "We present a simplified proof for a recent theorem by Junyan Cao and Mihai P{\u a}un, confirming a special case of Iitaka{\textquoteright}s Cn,m conjecture: if f X → Y is an algebraic fiber space, and if the Albanese mapping of Y is generically finite over its image, then we have the inequality of Kodaira dimensions κ(X) ≥ κ(Y) + κ(F), where F denotes a general fiber of f. We include a detailed survey of the main algebraic and analytic techniques, especially the construction of singular hermitian metrics on pushforwards of adjoint bundles (due to Berndtsson, P{\u a}un, and Takayama).",

author = "Christopher Hacon and Mihnea Popa and Christian Schnell",

note = "Funding Information: this paper, and for many useful discussions and advice about its contents. We also thank Dano Kim and Luigi Lombardi for reading and commenting on a draft version. During the preparation of the paper, CH was partially supported by NSF grants DMS-1300750 and DMS-1265285 and by a grant from the Simons Foundation (Award #256202). MP was partially supported by NSF grant DMS-1405516 and by a Simons Fellowship. CS was partially supported by NSF grants DMS-1404947 and DMS-1551677, and by a Centennial Fellowship from the American Mathematical Society. Publisher Copyright: {\textcopyright} 2018 American Mathematical Society.",

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doi = "10.1090/conm/712/14346",

language = "English (US)",

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N1 - Funding Information: this paper, and for many useful discussions and advice about its contents. We also thank Dano Kim and Luigi Lombardi for reading and commenting on a draft version. During the preparation of the paper, CH was partially supported by NSF grants DMS-1300750 and DMS-1265285 and by a grant from the Simons Foundation (Award #256202). MP was partially supported by NSF grant DMS-1405516 and by a Simons Fellowship. CS was partially supported by NSF grants DMS-1404947 and DMS-1551677, and by a Centennial Fellowship from the American Mathematical Society. Publisher Copyright: © 2018 American Mathematical Society.

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AB - We present a simplified proof for a recent theorem by Junyan Cao and Mihai Păun, confirming a special case of Iitaka’s Cn,m conjecture: if f X → Y is an algebraic fiber space, and if the Albanese mapping of Y is generically finite over its image, then we have the inequality of Kodaira dimensions κ(X) ≥ κ(Y) + κ(F), where F denotes a general fiber of f. We include a detailed survey of the main algebraic and analytic techniques, especially the construction of singular hermitian metrics on pushforwards of adjoint bundles (due to Berndtsson, Păun, and Takayama).

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Algebraic fiber spaces over abelian varieties: Around a recent theorem by Cao and Păun

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