NOVELTY - A lithium source, a manganese source, a compound containing element (M1) and a compound containing element (M2) are pulverize-mixed in a solvent. The obtained slurry is dried, and precursor adjustment process is carried out to obtain a precursor. The precursor is baked at 600-1200 degrees C to obtain a primary baked product. The baked product is washed, and baked at 600-1200 degrees C to obtain lithium-manganese complex oxide (I). The element (M1) is chosen from nickel, cobalt, magnesium, iron, aluminum and chromium. The element (M2) is chosen from boron, phosphorus, lead, antimony and vanadium.    USE - Manufacture of lithium-manganese complex oxide used for active material for positive electrode (all claimed) of lithium-ion secondary battery. Uses include but are not limited to mobile telephone, personal handy-phone system, small computer, and power supply for electric power storage and electric vehicles.    ADVANTAGE - The lithium-ion secondary battery containing the electrode comprising active material has high discharge capacitance and excellent cycle characteristics.    DETAILED DESCRIPTION - A lithium source, a manganese source, a compound containing element (M1) and a compound containing element (M2) are pulverize-mixed in a solvent. The obtained slurry is dried, and precursor adjustment process is carried out to obtain a precursor. The precursor is baked at 600-1200 degrees C to obtain a primary baked product. The primary baked product is washed, and baked at 600-1200 degrees C to obtain lithium-manganese complex oxide of formula: Li(x+y)Mn(2-y-p)M1pO(4-a) (I). The element (M1) is chosen from nickel, cobalt, magnesium, iron, aluminum and chromium. The element (M2) is chosen from boron, phosphorus, lead, antimony and vanadium. The ratio of lithium source, a manganese source, a compound containing element (M1) and a compound containing element (M2) is ( alpha + beta ):(2- beta - gamma )-( delta ):( gamma ):( delta ), where alpha is 1 or more and less than 1.2, beta is more than 0 and 0.2 or less, alpha + beta is more than 1 and 1.2 or less, gamma is 0-1 and delta is more than 0 and 1 or less.    x=1 or more and less than 1.2;    y=more than 0 and 0.2 or less;    x+y=more than 0 and 1.2 or less;    p=0-1;and    a=0-1.    INDEPENDENT CLAIMS are included for the following:    (1) positive electrode active material, which contains the lithium-manganese complex oxide; and    (2) positive electrode, which contains the positive electrode active material.