The Advanced Telescope for High ENergy Astrophysics (Athena) is the X-ray observatory mission selected by ESA within its Cosmic Vision 2015-2025 programme to address the Hot and Energetic Universe scientific theme. The ESO-Athena Synergy Team (EAST) has been tasked to single out the potential scientific synergies between Athena and optical/near-infrared (NIR) and sub/mm ground based facilities, in particular those of ESO (i.e., the VLT and ELT, ALMA and APEX), by producing a White Paper to identify and develop the: 1. needs to access ESO ground-based facilities to achieve the formulated Athena science objectives; 2. needs to access Athena to achieve the formulated science objectives of ESO facilities contemporary to Athena; 3. science areas where the synergetic use of Athena and ESO facilities in the late 2020s will result in scientific added value. Community input to the process happened primarily via a dedicated ESO - Athena Synergy Workshop that took place on Sept. 14 - 16, 2016 at ESO, Garching. This White Paper presents the results of the EASTs work, sorted by synergy area, and deals with the following topics: 1. the Hot Universe: Early groups and clusters and their evolution, Physics of the Intracluster medium, Missing baryons in cosmic filaments; 2. the Energetic Universe: Supermassive black hole (SMBH) history, SMBH accretion disks, Active Galactic Nuclei feedback - Molecular outflows, Ultra-fast outflows, Accretion Physics, Transient Science; 3. Observatory Science: Star Formation, Stars. It then discusses the optical-NIR-sub-mm perspective by providing details on VLT/MOONS, the E-ELT instruments, in particular the MOS, VISTA/4MOST, the ESO and ALMA archives, future ALMA and ESO developments, and finally the (likely) ESO - Athena astronomical scene in the 2020s. (abridged)
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