[Paper Review] Künneth projectors for open varieties

Spencer Bloch, Hélène Esnault|ArXiv.org|Feb 21, 2005

Advanced Algebra and Geometry5 references3 citations

TL;DR

This paper constructs an algebraic Künneth projector for the first cohomology group $ H^1(U) $ of a smooth quasi-projective variety $ U $, assuming a smooth projective completion with normal crossings at infinity. Using relative motivic cohomology, it proves integrality and independence of $ \ u $ for traces of open correspondences, generalizing the Lefschetz trace formula to open varieties under good position conditions.

ABSTRACT

We consider correspondences on smooth quasiprojective varieties $U$. An algebraic cycle inducing the Künneth projector onto $H^1(U)$ is constructed. Assuming normal crossings at infinity, the existence of relative motivic cohomology is shown to imply the independence of $\ell$ for traces of open correspondences.

Motivation & Objective

• To construct an algebraic cycle inducing the Künneth projector onto $ H^1(U) $ for smooth quasi-projective varieties $ U $ with a smooth projective completion.
• To establish the existence of relative motivic cohomology for such varieties under normal crossings at infinity.
• To prove integrality and $ \ell $-independence of the trace of an algebraic correspondence on $ U \times U $, generalizing trace formula techniques to open varieties.
• To extend the Lefschetz trace formula to open varieties via Gysin maps and Mayer-Vietoris spectral sequences on strata of the divisor at infinity.

Proposed method

• Constructs a cycle on $ X \times U $, where $ X $ is a smooth projective completion of $ U $, that induces the Künneth projector on $ H^1(U) $, though only partially trivialized on $ (X\setminus U) \times U $.
• Uses the Poincaré bundle on $ C \times J(C,\Delta) $ to realize the first Chern class in $ H^1(C,\Delta) \otimes H^1(J(C,\Delta))(1) $, linking it to the cohomology of $ U $.
• Applies the Mayer-Vietoris spectral sequence $ E_1^{a,b} = \bigoplus_{|I|=a} H^b(D_I) \Rightarrow H^{a+b}_c(U) $ to relate cohomology of $ U $ to that of strata $ D_I $ of the divisor at infinity.
• Defines $ Z_I = \bar{\Gamma} \cdot (D_I \times X) $ as the restriction of the correspondence $ \Gamma $ to strata, and uses Gysin maps and restriction maps to compute traces.
• Imposes a scheme-theoretic good position condition: $ \bar{\Gamma}_j \cap (D_I \times X) \subset D_I \times D_I $, ensuring intersection multiplicities are one and cancellation of boundary contributions.
• Applies the trace formula $ \operatorname{Tr}(\Gamma_*) = \sum_{r=0}^d (-1)^r \sum_{|I|=r} \deg(Z_I \cdot \Delta_I) $, with $ \Delta_I $ the diagonal in $ D_I \times D_I $.

Experimental results

Research questions

• RQ1 Can an algebraic cycle be constructed that induces the Künneth projector on $ H^1(U) $ for a smooth quasi-projective variety $ U $?
• RQ2 Under what conditions does the trace of an open correspondence on $ U \times U $ satisfy integrality and independence of $ \ell $?
• RQ3 How can the Lefschetz trace formula be generalized to open varieties with normal crossings at infinity?
• RQ4 When does the contribution from the boundary at infinity cancel in the trace formula, so that the trace depends only on fixed points in $ U $?
• RQ5 What role does relative motivic cohomology play in proving $ \ell $-independence of traces?

Key findings

• An algebraic cycle on $ X \times U $ induces the Künneth projector on $ H^1(U) $, though it is only partially trivialized on $ (X\setminus U) \times U $, not fully trivialized.
• The trace of a correspondence $ \Gamma $ on $ H^*_{c}(U) $ is given by $ \sum_{r=0}^d (-1)^r \sum_{|I|=r} \deg(Z_I \cdot \Delta_I) $, where $ Z_I = \bar{\Gamma} \cdot (D_I \times X) $.
• Under the scheme-theoretic good position condition and transversality, the trace $ \operatorname{Tr}(\Gamma_*) $ equals $ \deg(\Delta_U \cdot \Gamma) $, i.e., the number of fixed points in $ U $.
• The formula holds even when $ \Gamma $ is the graph of Frobenius in characteristic $ p $, recovering known results via new motivic methods.
• The independence of $ \ell $ for traces of open correspondences is established under the assumption of relative motivic cohomology, via the trace formula and cancellation of boundary terms.
• The example $ \Gamma_{pq} \subset \mathbb{A}^1 \times \mathbb{A}^1 $ with $ p \neq q $ shows that the trace is $ p $, while $ \deg(\Delta_U \cdot \Gamma) = \max(p,q) $, illustrating the cancellation mechanism in the formula.