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 $2\pi i$.

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?

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I fixed some of the grammar/spelling. I hope that I haven't changed the intended meaning of anything. – Kevin H. Lin Mar 24 '10 at 9:50
Just checking: your expression $tr(exp(-\nabla_i^2))$ is, up to some factors of $2\pi i$-stuff, the Chern character of the complexification of $E_i$? So, what you're interested in is the difference of [the top-dimensional-parts of] the Chern characters of $E_1$ and $E_2$, given that there's a bundle map from $E_1$ to $E_2$ which is singular only on the finite set $X$? This isn't stuff I know about, but I'm intrigued. Some simple observations: the information about $X$ revealed by this quantity is invariant under (1) adding more points to $X$; (2) perturbing $X$ slightly. – macbeth Mar 24 '10 at 16:44
Here I allows assume $E_i$ is complex vector bundle. In fact, $X$ may be expressed as a finite disjoint union of closed and connected subsets. I want to kown the value of the integration on only one subset. – Chen Mar 25 '10 at 10:50
My question comes from this paper:arxiv.org/abs/0908.3335 – Chen Mar 29 '10 at 9:50