Sounds to me like your video card is outputting a signal/resolution not supported by your monitor. Assuming it's only losing the signal when you launch game, try tweaking the resolution manually:
in Documents/Starcraft II/Variables.txt
or whatever settings that you want to try.
Also, here are some other settings you can try tweaking in that file and see if you can gain some more performance:
foliagedensity (Default 0.750000)
It shows how dense the different
foliage elements like grass, leaves
will appear. You can decrease the
value to something that fit your needs
and also increase the performance of
parallaxshadow (Default 1)
Parallax Shadows give depth to game
textures and geometry at the expense
of CPU/GPU performance. Switching it
off by giving it ’0′ value may
increase performance of the game.
particlelod (Default 3)
particleobjects (Default 1.000000)
particleterrain (Default 1.000000)
These three commands govern the
quality of the particles in game.
Lower the values to get performance
shadowmapsize (Default 2048)
How shodows will be rendered in game,
lowering this value to 1024/512 will
increase the performance of the game
and also take off some load on GPU and
TerrainTextureHiResCacheSize (Default 24)
TerrainTextureLowResCacheSize (Default 50)
These two variables control the amount
of memory the game uses to cache and
stream textures high and low
resolution textures separately.
TerrainTextureSize (Default 1024)
This is the memory of your graphic
card, memory available at the disposal
for the game to utilize.
transparentshadows (Default 1)
This is an eye candy tweak, if enabled
it will give shadows a transparent
touch enhancing colors in the process.
Its OK to change the value to ’0′, its
not something you need really. Infact,
its an effect you will hardly notice
while playing the game.
watercaustics (Default 1)
It makes the water look real, waves
effects and splashes on the surface of
water. Not required if you have low
end GPU. Set this value to ’0′ to turn
waterreflection (Default 1)
This variable enabled the reflection
on the surface of water. Again, not
required for people on low end graphic
waterrendertargetsize (Default 1024)
How the game will render everything
water, lower the value to boast