What question did this study set out to answer?

To construct boundary compactifications for moduli spaces of tropical curves and abelian varieties, and to determine new unstable cohomology classes in integer linear groups.

May 6, 2025Inventiones mathematicaeOpen Access

Bordifications of the moduli spaces of tropical curves and abelian varieties, and unstable cohomology of GL₆ (Z) and SL₆ (Z)

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FBFrancis BrownUniversity of Oxford

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Implication

Geometric analysis uncovers unstable cohomology classes in general linear groups using tropical bordifications, highlighting new geometric proofs for Borel's theorem and motivic structures.

Key Points

To construct boundary compactifications for moduli spaces of tropical curves and abelian varieties, and to determine new unstable cohomology classes in integer linear groups.
Constructed bordifications of moduli spaces for tropical curves and tropical abelian varieties, extending the tropical Torelli map to these boundaries.
Analyzed the asymptotic behavior at infinity of classical bi-invariant differential forms and computed the link cohomology of the tropical abelian moduli space.
Formulated an algebraic construction of the Borel-Serre compactification using iterated projective blow-ups along linear subspaces.
Generated infinitely many new non-zero unstable cohomology classes in the arithmetic linear groups GL_g(Z) and SL_g(Z).
Produced an alternative geometric proof of Borel's theorem on stable cohomology and demonstrated that the tropical abelian link contains the stable cohomology of GL_g(Z).
Introduced transcendental invariants for minimal vectors of quadratic forms and established a motivic structure on a portion of GL_g(Z) cohomology.

Cite This Study

Francis Brown (2025) studied this question.

synapsesocial.com/papers/6aaa3c88fb629eaa3d19580d https://doi.org/10.1007/s00222-025-01335-y

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Implication

Geometric analysis uncovers unstable cohomology classes in general linear groups using tropical bordifications, highlighting new geometric proofs for Borel's theorem and motivic structures.

Bordifications of the moduli spaces of tropical curves and abelian varieties, and unstable cohomology of GL₆ (Z) and SL₆ (Z)

Discussion

Member takes

Implication

Key Points

Cite This Study

Bordifications of the moduli spaces of tropical curves and abelian varieties, and unstable cohomology of GL₆ (Z) and SL₆ (Z)

Discussion

Member takes

Implication

Key Points

Cite This Study

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