There could be a couple of things at issue. Is 'strColumn_Name' the name of the field or a variable that hold the name of the field? If it's a variable, then you should use
pFDef.DefinitionExpression = strColumn_Name + "='" + myObject.ToString() + "'";
Is the field numeric instead of string? Then you would remove the single quotes from the string.
I would suggest comparing the DefinitionExpression value (use the debugging mode to see what it is exactly) with the expression that works properly in ArcMap