DECORATED SHEAVES AND MORPHISMS IN TILTED HEARTS

We identify stable pairs and stable framed sheaves as epi-morphisms and monomorphisms in the hearts of tilted t-structures under appropriate conditions. We then identify the moduli spaces with the corresponding Quot spaces. As a result, we obtain the projectivity of the Quot spaces in absolute cases. In addition, we prove a formula in a motivic Hall algebra, which relates together the Quot spaces under a tilt.