Macbook Pro Crashing when using Chrome

A shutdown cause of -128 is a catch-all code that means that the SMC does not know the reason for the power loss. Given SMC's extensive heuristics for determining shutdown causes, this is basically a 100% certain indication of either a software or firmware bug or a hardware failure. (More often the latter.)

Diagnosing root cause will be impossible due to the specialized debugging tools and access to proprietary schematics and source code which you will need. Only Apple can do this.

The best you can do is continue gathering circumstantial data. Start paying close attention to the apps you are running, the tasks you are performing, the peripherals you have attached, and other environmental variables. In addition to following your own intuition, here are a couple of questions I would ask:

  • How old is your battery? (I.e., what model & year is your machine, presuming that you haven't done a battery replacement?) Old batteries can experience voltage sag under load and lead to -128 shutdowns due to the sudden loss of power.
  • You specifically mention using Chrome. Chrome is known to have many problems on Mac due to its nonstandard implementation of resource management, among other reaons, and has often been a contributing factor to a lot of problems. One possible involvement in your case may be to stress the GPU during page rendering if its hardware acceleration feature is enabled, which puts load on the battery and can cause the power loss I mentioned above. Try not using Chrome and switch to Safari for a while to see if the problem goes away.
  • Do you see new logs appearing in /Library/Logs/DiagnosticReports? Specifically, anything ending in .crash, .hang, or .panic.

UPDATE:

From comments:

Battery:
 Cycle Count: 1045
 Condition: Service Recommended

Based on your cycle count, the battery needs to be replaced. This will likely fix your issue (and also give you longer battery life as a bonus).