Abstract. I will report about my recent joint work with Angel Rios Ortiz about the SYZ conjecture for a special class of singular symplectic varieties. The SYZ conjecture predicts that nef and isotropic line bundles are associated to lagrangian fibrations - thanks to results of Matsushita and Kawamata, this can be rephrased by saying that nef line bundles are semiample, thus settling the abundance conjecture too. After having recalled some generalities about symplectic varieties and the SYZ conjecture, I will state the main result and explain the main ideas behind its proof.