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The Lattice Conjecture for smooth proper dg-categories

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Let C\mathcal{C} be a C\mathbb{C}-linear dg-category. Write K∗top⁡(C)K_*^{\operatorname{top}}(\mathcal{C}) for its topological KK-theory groups and HP∗(C)HP_*(\mathcal{C}) for its periodic cyclic homology groups. The topological Chern character is a map

ch⁡top⁡ ⁣:K∗top⁡(C)→HP∗(C).\operatorname{ch}^{\operatorname{top}}\colon K_*^{\operatorname{top}}(\mathcal{C})\to HP_*(\mathcal{C}).

A dg-category is smooth if it is perfect as a C\mathcal{C}-C\mathcal{C}-bimodule, and proper if the total cohomology of each morphism complex is finite-dimensional over C\mathbb{C}. The Lattice Conjecture. If C\mathcal{C} is smooth and proper, then the topological Chern character induces an isomorphism

K∗top⁡(C)⊗ZC→≅HP∗(C).K_*^{\operatorname{top}}(\mathcal{C})\otimes_{\mathbb{Z}}\mathbb{C}\xrightarrow{\cong}HP_*(\mathcal{C}).

This conjecture predicts that topological KK-theory and periodic cyclic homology agree after complexification for smooth proper dg-categories; the supplied source gives no evidence that it has been resolved.

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Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The lattice conjecture for smooth proper dg-categories

    Let TT be a smooth proper dg-category over C\mathbb{C}. The map

    Chtop∧SHC:Ktop(T)∧SHC⟶HP(T)\mathrm{Ch^{top}}\wedge_\mathbb{S} H\mathbb{C}: \mathbf{K}^{\mathrm{top}}(T)\wedge_{\mathbb{S}} H\mathbb{C} \longrightarrow \mathrm{HP}(T)

    Lattice conjecture. This map is an equivalence. It is an analog of known facts for smooth proper algebraic varieties and concerns the rational part of the hypothetical noncommutative Hodge structure of TT; its status is not resolved in the supplied source.

    source: Anthony Blanc, “Topological K-theory of complex noncommutative spaces”, arXiv:1211.7360 (2015).

References

Primary source

Michael K. Brown and Prashanth Sridhar, “Orlov's Theorem for dg-algebras”, arXiv:2312.17422 (2025).

cyclic-homologydg-categoriesnoncommutative-geometrytopological-k-theory
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