An upper bound of the heat kernel along the harmonic-Ricci flow

Let \({c : C \rightarrow X \times X}\) be a correspondence with C and X quasi-projective schemes over an algebraically closed field k. We show that if \({u_\ell : c_1^*\mathbb{Q}_\ell \rightarrow c_2^!\mathbb{Q}_\ell}\) is an action defined by the localized Chern classes of a c ₂-perfect complex of vector bundles on C, where ? is a prime invertible in k, then the local terms of u _? are given by the class of an algebraic cycle independent of ?. We also prove some related results for quasi-finite correspondences. The proofs are based on the work of Cisinski and Deglise on triangulated categories of motives.