I guess it depends on what you mean by varied. My boys can have quite a short menu of foods that they will eat, but they do encompass Hot Dogs, Pizza, as well as Sushi, Pasta, and Chinese Dumplings. Overall it's more important you get nutrition, your Pediatrician should be able to guide you on what the needs are as they change over time. If you feel like they are really lacking you can always try and supplement foods with PediaSure or those other types of nutrition drinks, my 4 year old LOVES those and many times wakes up at night and won't go to sleep without one. Although we buy the cheaper, wholesale club version of those drinks and he doesn't really notice, but my wallet does. In general we try to offer them a wide variety of foods, my wife is Chinese so we do have a wide variety of foods that they do eat even if it's only a few from American and Chinese cooking.
We push fruits in our house, we always try to have in season fruit and eat it all the time. I made a big deal of it when the kids were younger "it's natures candy!" I always said. Eventually they picked up on it. We don't keep many sweets in the house, so my kids have very little desire for chocolates and such - at most they get their candy fix from ice cream. Chips and sodas are for outside the house, so we keep that sort of foods out of the cupboard and they don't miss it. Same for cookies, when we do have them we make them at home, and it gives them something to do. I don't limit snacks exclusively, as I find that makes the kids want them more, but I preach moderation. A little is fine, but we aren't buying huge amounts of snacks like that for our house, and often by limiting snacks to more healthy options, and limiting it altogether on some days, we find they eat dinners better and more food. No drinks during the beginning of dinner, so they eat more and don't fill up on juices and water, they usually get a drink when they are halfway done. I find most of this gets them to eat and by not catering to their specific meal whims (home is not a restaurant) they learn to eat foods you like and sometimes widen their options more.
I don't agree that picky eaters end up as picky adults, I was an extremely picky eater when I was younger (as were some of my friends) but we grew up fine and not partake in stuff we'd never have though of having when we were kids. While this may be true for some people it's not everyone, it's more of a sense of adventure and willingness to try new things.
You can always check out the Food Pyramid through Google, as well as looking for Healthy options for kids. Overall nutrition studies get updated and amended every few years, so I am not sure what you would get from a study other than knowing a balanced diet is important.