Asymptotic behaviour of bigraded components of local cohomology modules

Let C be a commutative Noetherian ring containing a field K of characteristic zero. Let R=C[X_1, \ldots, X_n, Y_1, \ldots, Y_m] be a polynomial ring over C with \mathrm{bideg}~ c=(0,0) for all c \in C, \mathrm{bideg}~ X_i=(1,0) and \mathrm{bideg}~ Y_j=(0,1) for i=1, \ldots, n and j=1, \ldots, m. Let I be a bihomogeneous ideal in R. In this article, we study asymptotic behaviour of bigraded pieces of the local cohomology module H^i_I(R). Moreover, under the extra assumption that C is regular, we investigate the asymptotic stability of invariants associated to its bigraded components. Consequently, we obtain certain properties of components of the bigraded local cohomology module H^i_I(R), where C=K is a field and I is a binomial edge ideal.