Stabilization of the cohomology of thickenings

Bhargav Bhatt, Manuel Blickle, Gennady Lyubeznik, Anurag K. Singh, Wenliang Zhang

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

For a local complete intersection subvariety X = V (I) in ℙn over a field of characteristic zero, we show that, in cohomological degrees smaller than the codimension of the singular locus of X, the cohomology of vector bundles on the formal completion of ℙn along X can be effectively computed as the cohomology on any sufficiently high thickening Xt = V (It); the main ingredient here is a positivity result for the normal bundle of X. Furthermore, we show that the Kodaira vanishing theorem holds for all thickenings Xt in the same range of cohomological degrees; this extends the known version of Kodaira vanishing on X, and the main new ingredient is a version of the Kodaira- Akizuki-Nakano vanishing theorem for X, formulated in terms of the cotangent complex.

Original languageEnglish (US)
Pages (from-to)531-561
Number of pages31
JournalAmerican Journal of Mathematics
Volume141
Issue number2
DOIs
StatePublished - Apr 2019

Bibliographical note

Funding Information:
Research of the first author supported by NSF grants DMS-1501461 and DMS-1522828, and by a Packard Fellowship; research of the second author supported by DFG grant SFB/TRR45; research of the third author supported by NSF grants DMS-1500264 and DMS-1800355; research of the fourth author supported by NSF grants DMS-1500613 and DMS-1801285; research of the fifth author supported by NSF grants DMS-1606414 and DMS-1752081. The authors are grateful to the American Institute of Mathematics (AIM) for supporting their collaboration. American Journal of Mathematics 141 (2019), 531–561. ©c 2019 by Johns Hopkins University Press.

Funding Information:
Research of the first author supported by NSF grants DMS-1501461 and DMS-1522828, and by a Packard Fellowship; research of the second author supported by DFG grant SFB/TRR45; research of the third author supported by NSF grants DMS-1500264 and DMS-1800355; research of the fourth author supported by NSF grants DMS-1500613 and DMS-1801285; research of the fifth author supported by NSF grants DMS-1606414 and DMS-1752081. The authors are grateful to the American Institute of Mathematics (AIM) for supporting their collaboration.

Publisher Copyright:
© 2019 by Johns Hopkins University Press.

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