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<p>We construct bordifications of the moduli spaces of tropical curves and of tropical abelian varieties, and show that the tropical Torelli map extends to their bordifications. We prove that the classical bi-invariant differential forms studied by Cartan and others extend to these bordifications by studying their behaviour at infinity, and consequently deduce infinitely many new non-zero unstable classes in the cohomology of the general and special linear groups <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="222_2025_1335_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">GL</mi> <mi>g</mi> </msub> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$\mathrm{GL}_{g}(\mathbb{Z})$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="222_2025_1335_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">SL</mi> <mi>g</mi> </msub> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$\mathrm{SL}_{g}(\mathbb{Z})$</EquationSource> </InlineEquation>. In particular, we obtain a new geometric proof of Borel’s theorem on the stable cohomology of these groups. We completely determine the cohomology of the link of the moduli space of tropical abelian varieties within a certain range, and show that it contains the stable cohomology of the general linear group. In addition, we define new transcendental invariants associated to the minimal vectors of quadratic forms, and show that a certain part of the cohomology of the general linear group <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="222_2025_1335_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">GL</mi> <mi>g</mi> </msub> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$\mathrm{GL}_{g}(\mathbb{Z})$</EquationSource> </InlineEquation> admits the structure of a motive. In an <InternalRef RefID="Sec95">Appendix</InternalRef>, we give an algebraic construction of the Borel-Serre compactification by embedding it in the real points of an iterated blow-up of a projective space along linear subspaces, which may have independent applications.</p>

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Bordifications of the moduli spaces of tropical curves and abelian varieties, and unstable cohomology of \(\mathrm{GL}_{g}(\mathbb{Z})\) and \(\mathrm{SL}_{g}(\mathbb{Z})\)

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  • 发表日期：

    2025年5月6日

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  • Francis Brown

摘要

We construct bordifications of the moduli spaces of tropical curves and of tropical abelian varieties, and show that the tropical Torelli map extends to their bordifications. We prove that the classical bi-invariant differential forms studied by Cartan and others extend to these bordifications by studying their behaviour at infinity, and consequently deduce infinitely many new non-zero unstable classes in the cohomology of the general and special linear groups GL g ( Z ) $\mathrm{GL}_{g}(\mathbb{Z})$ and SL g ( Z ) $\mathrm{SL}_{g}(\mathbb{Z})$ . In particular, we obtain a new geometric proof of Borel’s theorem on the stable cohomology of these groups. We completely determine the cohomology of the link of the moduli space of tropical abelian varieties within a certain range, and show that it contains the stable cohomology of the general linear group. In addition, we define new transcendental invariants associated to the minimal vectors of quadratic forms, and show that a certain part of the cohomology of the general linear group GL g ( Z ) $\mathrm{GL}_{g}(\mathbb{Z})$ admits the structure of a motive. In an Appendix, we give an algebraic construction of the Borel-Serre compactification by embedding it in the real points of an iterated blow-up of a projective space along linear subspaces, which may have independent applications.