Logarithmic motives with compact support

We develop a theory of motives with compact support for logarithmic schemes over a field. Starting from the notion of finite logarithmic correspondences with compact support, we define the logarithmic motive with compact support analogous to the classical case for schemes. We then establish an analog of a Gysin sequence and, assuming resolution of singularities, a K{ü}nneth formula. This implies that our theory is $\bar{\mathrm{\square }}$-invariant, which presents a critical feature that is absent in the classical case. Further assuming resolution of singularities, we prove a duality theorem for log schemes which we use to establish a cancellation theorem for log schemes whose underlying scheme is proper. Moreover, we discuss new homology and cohomology theories for log smooth fs logarithmic schemes based on our results.