Cohomology of the Moduli Space of Cubic Threefolds and Its Smooth Models

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–To...

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Hauptverfasser: Casalaina-Martin, Sebastian, Grushevsky, Samuel, Hulek, Klaus, Laza, Radu
Format: E-Book Buch
Sprache:Englisch
Veröffentlicht: Providence, Rhode Island American Mathematical Society 2023
Ausgabe:1
Schriftenreihe:Memoirs of the American Mathematical Society
Schlagworte:
Cohomology operations
Moduli theory
Threefolds (Algebraic geometry)
ISBN:9781470460204, 1470460203
ISSN:0065-9266, 1947-6221
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Abstract 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.
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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.
Author Grushevsky, Samuel
Hulek, Klaus
Casalaina-Martin, Sebastian
Laza, Radu
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BackLink https://cir.nii.ac.jp/crid/1130577121818944658$$DView record in CiNii
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Copyright Copyright 2023 American Mathematical Society
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Notes Other authors: Samuel Grushevsky, Klaus Hulek, Radu Laza
February 2023, volume 282, number 1395 (fourth of 6 numbers)
Includes bibliographical references (p. 97-100)
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Snippet We compute and compare the (intersection) cohomology of various natural geometric compactifications of the moduli space of cubic threefolds: the GIT...
View the abstract.
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SubjectTerms Cohomology operations
Moduli theory
Threefolds (Algebraic geometry)
TableOfContents Introduction -- Preliminaries -- The cohomology of the Kirwan blowup, part I: equivariant cohomology of the semi-stable locus -- The cohomology of the Kirwan blowup, part II -- The intersection cohomology of the GIT moduli space <inline-formula content-type="math/mathml"> M GIT \mathcal {M}^{\operatorname {GIT}} </inline-formula> -- The intersection cohomology of the ball quotient -- The cohomology of the toroidal compactification -- Equivariant cohomology -- Stabilizers, normalizers, and fixed loci for cubic threefolds -- The moduli space of cubic surfaces
Cover -- Title page -- Chapter 1. Introduction -- Acknowledgments -- Chapter 2. Preliminaries -- 2.1. Notation and conventions -- 2.2. Moduli space of cubic threefolds and its standard compactifications \GIT and \BG -- 2.3. The Kirwan blowup \MK of the moduli space of cubic threefolds -- 2.4. The toroidal compactification -- Chapter 3. The cohomology of the Kirwan blowup, part I: equivariant cohomology of the semi-stable locus -- 3.1. The equivariantly perfect stratification and the equivariant cohomology of the semi-stable locus in general -- 3.2. The equivariant cohomology of the locus of semi-stable cubic threefolds -- Chapter 4. The cohomology of the Kirwan blowup, part II -- 4.1. The correction terms in general -- 4.2. The main correction terms for cubic threefolds -- 4.3. The extra correction terms for cubic threefolds -- 4.4. Putting the terms together to compute the cohomology of \calM^{ } -- Chapter 5. The intersection cohomology of the GIT moduli space \GIT -- 5.1. Obtaining the intersection cohomology of the GIT quotient from the cohomology of the Kirwan blowup, in general -- 5.2. The intersection cohomology of the GIT quotient for cubic threefolds -- 5.3. Putting the terms together to compute the cohomology of \GIT -- 5.4. The intersection cohomology of ̂\calM -- Chapter 6. The intersection cohomology of the ball quotient -- 6.1. A special case of the decomposition theorem -- 6.2. The intersection cohomology of the ball quotient -- Chapter 7. The cohomology of the toroidal compactification -- 7.1. The arithmetic of the two cusps of \calB/Γ -- 7.2. The cohomology of the toroidal boundary divisors -- 7.3. The cohomology of the toroidal compactification -- Appendix A. Equivariant cohomology -- A.1. Review of Atiyah-Bott -- A.2. Compact and complex Lie groups -- A.3. Kirwan's result for compact groups acting on symplectic manifolds
A.4. Fibrations -- Appendix B. Stabilizers, normalizers, and fixed loci for cubic threefolds -- B.1. Connected component \CC* -- B.2. Connected component \PGL(2,\CC) -- B.3. Connected component (\CC*)² -- Appendix C. The moduli space of cubic surfaces -- C.1. The moduli space of cubic curves -- C.2. The moduli space of cubic surfaces -- C.3. The proof of Theorem C.1 -- C.4. The cohomology of the Naruki compactification -- Bibliography -- Back Cover
Title Cohomology of the Moduli Space of Cubic Threefolds and Its Smooth Models
URI https://www.ams.org/memo/1395/
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Volume 282
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