TY - BOOK

T1 - Cohomology of the moduli space of cubic threefolds and its smooth models

AU - Hulek, Klaus

AU - Casalaina-Martin, Sebastian

AU - Laza, Radu

AU - Grushevsky, Samuel

N1 - Funding Information: Research of the first author is supported in part by grants from the Simons Foundation (317572) and the NSA (H98230-16-1-0053). Research of the second author is supported in part by NSF grants DMS-15-01265 and DMS-18-02116. Research of the third author is supported in part by DFG grant Hu-337/7-1. Research of the fourth author is supported in part by NSF grants DMS-12-54812 and DMS-18-02128. The first author would like to thank the Institut für Algebraische Geometrie at Leibniz Universität for support during the Fall Semester 2017. The first and third authors are also grateful to MSRI Berkeley, which is supported by NSF Grant DMS-14-40140, for providing excellent working conditions in the Spring Semester 2019.

PY - 2023/2

Y1 - 2023/2

N2 - We compute and compare the (intersection) cohomology of various natural geometric compactifications of the moduli space of cubic threefolds: the GIT compactification and its Kirwan blowup, as well as the Baily–Borel and toroidal compactifications of the ball quotient model, due to Allcock–Carlson–Toledo. Our starting point is Kirwan’s method. We then follow by investigating the behavior of the cohomology under the birational maps relating the various models, using the decomposition theorem in different ways, and via a detailed study of the boundary of the ball quotient model. As an easy illustration of our methods, the simpler case of the moduli space of cubic surfaces is discussed in an appendix.

AB - We compute and compare the (intersection) cohomology of various natural geometric compactifications of the moduli space of cubic threefolds: the GIT compactification and its Kirwan blowup, as well as the Baily–Borel and toroidal compactifications of the ball quotient model, due to Allcock–Carlson–Toledo. Our starting point is Kirwan’s method. We then follow by investigating the behavior of the cohomology under the birational maps relating the various models, using the decomposition theorem in different ways, and via a detailed study of the boundary of the ball quotient model. As an easy illustration of our methods, the simpler case of the moduli space of cubic surfaces is discussed in an appendix.

KW - math.AG

KW - 14J30, 14J10, 14L24, 14F25, 55N33, 55N25

UR - http://www.scopus.com/inward/record.url?scp=85147336822&partnerID=8YFLogxK

U2 - 10.1090/memo/1395

DO - 10.1090/memo/1395

M3 - Monograph

SN - 978-1-4704-6020-4

VL - 282

T3 - Memoirs of the American Mathematical Society

BT - Cohomology of the moduli space of cubic threefolds and its smooth models

ER -