How do I make SVG files more transferable from Inkscape to Illustrator and viewable in Chrome?

I've created a SVG file in Inkscape [1] and I've noticed that viewing it in Chrome or Illustrator causes a number of parts of the diagram to be invisible or otherwise poorly rendered (e.g., in Chrome some of the arrows are oriented the wrong way, whereas in Illustrator CS6 the line segments in arrows are invisible, but the arrow heads are still visible). Is there any way of making the file more compatible with these pieces of software whilst not loosing any of the existing parts of the file (e.g., I want to keep all the arrows, all the shapes, all the gradient colourings, etc.)