dg.differential geometry - virtual bundle with compact support

A virtual bundle with compact support is a triple $E$={$E_1$,$E_2$,$a$},
where $E_1$ and $E_2$ are bundles over $M$ of the same dimension, $M$ is a closed manifold, $a$ is a bundle map from $E_1$ to $E_2$, which is an isomorphism of bundles over $M smallsetminus X$, where $X$ is a compact set in M.

Let $nabla_1$ be a connection on $E_1$, $nabla_2$ a connection on $E_2$. If $X$ is finite number of points in $M$, and Ch($E$) is defined by $text{tr}(exp(-nabla_1)^{2})-text{tr}(exp(-nabla_2)^{2})$, we will find Ch($E$) integrated on $M$ is an integer multiple of a power of $2pi i$.

I want to know, if $X$ is a compact submanifold, then how to get the value of the integration? Maybe we can't get the value but how can we analyze the information about $X$ revealed by the integration?