The analysis of multi-modality positron emission tomography and computed tomography (PET-CT) images for computer aided diagnosis applications requires combining the sensitivity of PET to detect abnormal regions with anatomical localization from CT. Current methods for PET-CT image analysis either process the modalities separately or fuse information from each modality based on knowledge about the image analysis task. These methods generally do not consider the spatially varying visual characteristics that encode different information across the different modalities, which have different priorities at different locations. For example, a high abnormal PET uptake in the lungs is more meaningful for tumor detection than physiological PET uptake in the heart. Our aim is to improve fusion of the complementary information in multi-modality PET-CT with a new supervised convolutional neural network (CNN) that learns to fuse complementary information for multi-modality medical image analysis. Our CNN first encodes modality-specific features and then uses them to derive a spatially varying fusion map that quantifies the relative importance of each modalitys features across different spatial locations. These fusion maps are then multiplied with the modality-specific feature maps to obtain a representation of the complementary multi-modality information at different locations, which can then be used for image analysis. We evaluated the ability of our CNN to detect and segment multiple regions with different fusion requirements using a dataset of PET-CT images of lung cancer. We compared our method to baseline techniques for multi-modality image fusion and segmentation. Our findings show that our CNN had a significantly higher foreground detection accuracy (99.29%, p < 0.05) than the fusion baselines and a significantly higher Dice score (63.85%) than recent PET-CT tumor segmentation methods.