北京雁栖湖应用数学研究院

Derived Algebraic Geometry is a machinery regarded as an extension of algebraic geometry, whose goal is to study exotic geometric settings and situations that might occur in algebraic geometry where algebraic geometry might not be able to rigorously study. Take for instance the example of intersection of two subvarieties, X, Y, in a fixed ambient smooth algebraic variety Z. We have the notion of "nice intersection" (or generic intersection) of X and Y in Z, which is equivalent to transverse intersection of the X and Y. In this case the span of the tangent space generated by tangent spaces of X and Y is equal to the tangent space of Z and their locus of intersection X∩Y will be of expected dimension. Now take for instance a “bad intersection” of X and Y, that is non-generic or non-transverse intersection of X and Y in Z. The latter situation might occur if X and Y intersect over points or loci with higher multiplicities or when their locus of intersection is not of expected dimension. Certainly such situation has been addressed in algebraic geometry, using cohomology theory, that is, one realizes the locus of intersection, X∩Y, as a cohomology class in the ambient cohomology theory of Z (for instance an element in de Rham cohomology of Z, or in complex cobordism ring of Z, or an element of the K-theory in the intersection ring of Z) and studies the intersection of X and Y cohomologically. The drawback of this approach is that X ∩Y is realized only as a cohomology class and not as a geometric object any more. The Derived Algebraic Geometry allows one to construct a geometric object associated to the non-generic locus of intersection of X and Y, which is called the “Derived Scheme”. It is roughly speaking the homotipical perturbation of the naive locus of intersection of X and Y, and contains the data of higher multiplicity components of X∩Y or components with defects of expected dimension. The machinery of homotopy theory in derived algebraic geometry enables one to identify the points in derived intersection of X and Y as points which lie in X and Y respectively, together with certain continuous homotopy maps between them (as opposed to generic intersection of X and Y where points in X∩Y are given by points which lie both in X and Y simultaneously). Similarly taking “bad” quotients of algebraic schemes/varieties by non-proper or non-free actions is yet another example which can be modeled geometrically and rigorously by derived algebraic geometry. In usual setting the points in quotient of a variety X by the action of a free proper group G are the ones that lie in the orbit space of elements of G. However in instances where the group G is acting non-freely on X, the derived algebraic geometry enables one to realize the points in the "bad quotient" as points which lie in the orbit of elements of G up to homotopy, that is two points, a and b, lie in the orbit of an element of G if they are related to each other by a homotopy path or a path with homotopical structure in that orbit. The course sets foundations to such theory of derived schemes, and follows by discussing the derived moduli spaces, and specially the derived structure of the moduli spaces of coherent sheaves, and some applications of derived geometry in enumerative geometry (specially the Donaldson-Thomas theory) of Calabi-Yau 3 folds and 4 folds will be discussed at the end.