NOVELTY - Raw material (A) containing element (A), element (M), element (X), and element (Z), is heated to obtain molten material. The molten material is cooled, and resultant solidified material is pulverized to obtain primary ground material. Primary ground material is heated, and resultant compound and carbon source(s) chosen from organic compound and carbon-type electroconductivity substance, are crushed to obtain secondary ground material. The resultant material is heated under inert gas or reducing gas to obtain positive electrode active material.    USE - Manufacture of positive electrode active material used for manufacturing positive electrode for manufacturing secondary battery (all claimed) used as power supply for plug-in hybrid motor vehicles, electric vehicles and electric power storage.    ADVANTAGE - The method efficiently provides positive electrode active material having excellent electrical conductivity, and productivity with reduced generation of peeling of carbon coating layer due to volume changes associated with crystallization during production. The secondary battery using the positive electrode active material has excellent reliability, and can be manufactured economically by simple method.    DETAILED DESCRIPTION - Raw material (A) containing element (A), element (M), element (X), and element (Z), is heated to obtain molten material. The raw material (A) is of formula: AaMbXcOdZe (I). The obtained molten material is cooled, and resultant solidified material is pulverized to obtain primary ground material. The primary ground material is heated under inert gas or reducing gas atmosphere, and resultant compound and carbon source(s) chosen from organic compound and carbon-type electroconductivity substance, are crushed to obtain secondary ground material. The resultant material is heated under inert gas or reducing gas to obtain positive electrode active material.    A=Li, Na or K;    M=Fe, Mn, Co or Ni;    X=Si, P, S, B, Al, V, Mo, W, As, Ge or Sb;    Z=halo;    a=0.8-2.7;    b=0.6-1.4;    c=0.9-1.1;    d=integer depending on b, c and e;and    e=0-2.2.    INDEPENDENT CLAIMS are included for the following:    (1) manufacture of positive electrode, which involves using positive electrode active material;    (2) positive electrode; and    (3) manufacture of secondary battery.