Momentum Graphs, Chinese Remainder Theorem and Surjectivity of Restriction Map

This paper concerns the momentum graph of a GKM-action of an algebraic torus \(T\cong ({\mathbb C}^*)^k\) on a projective variety X over \({\mathbb C}\) . That is, an algebraic action (T, X) such that the fixed point set \(X^T\) and the set of T-stable curves in X are both finite. We begin by explaining some basic results on the momentum graph which are particular to the projective setting which may not be well-known. As examples, we discuss these graphs for T-orbit closures and the variety of complete flags. After that we consider a Chinese remainder theorem for (labeled) graphs and use it to give a condition for surjectivity of the restriction map \(H^*(X, {\mathbb C}) \rightarrow H^*(Y, {\mathbb C})\) , where Y is a T-stable subvariety of X. In particular, this applies to certain invariant subvarieties in a smooth toric variety and Richardson varieties.