Write an idea first. I may start with some general methods of studying surfaces in mathematics, such as classical embedding into high-dimensional space and giving an algebraic equation. Then the definition of K3 is given, that is, H (1, 0) = 0 and its canonical bundle is the beginning of a complex two-dimensional surface. Explain why these can determine most properties of this surface, such as Euler number and fundamental group. Finally, in order to apply it to string theory, especially to dual functions, I will talk about the module space of K3 surface, especially the module space of complex structures. Because this is far from my present job, and I need to turn over books to be sure, it is impossible to finish it at once. But I will update it from time to time as soon as I have time, and try to write more easily. (Actually, I personally think Riemannian surfaces are more interesting. K3 surface plays an important role in communicating physics and geometry. Physicists (Dr. Kurosaki, Uji Li Chuan and their teacher Kawaguchi) found that the Laurent expansion coefficient was the same as the irreducible representation coefficient of Mathieu Group when calculating the elliptic genus of K3, and then Mathieu speculated that Duncan and Miranda Cheng extended this work to umbra moonlight. The modular space of K3 is the modular space of the compactness of hybrid strings:, because M theory is dual on K3 and hybrid string theory. The ADE singularity of K3 can correspond to the ADE classification of Lie algebras. This can be understood by considering the K 3 compatibility of m theory/type IIA. Ade-type enhanced gauge symmetry can be achieved by winding 2 films around the disappearing 2 turns of k3. Although I grew up listening to this statement, my personal opinion/feeling is that the relationship between physics and mathematics is not a "problem and toolbox" relationship, or it has been 2 1 century, and this statement/view should not be emphasized any more. Mathematics and physics are complete academic systems, each with its own set of values, methodology and channels for generating and solving problems.