They are wonderful looking babies, Uma, congratulations on growing them.
For the next year I would leave them in the seed compost that you have them in. Keep it just moist for the moment andf watch the leaves and corms grow. The leaves are likely to die back at the begining of the summer; reduce watering but don't let the compost dry out completely. I would give a weak feed of liquid tomato fertiliser once they start growing again and perhaps once a month afterwards. By next summer you should have some corms that are of a handlable size. Round about August/September 2012, just before they start into growth again, you could look at pricking them out into individual pots of a more gritty plant compost and keep growing them on. Throughout all of this keep the seedlings at a cool temperature, above freezing but well below sitting room temperature. They will need light but not direct sunlight, which will scorch them.
Good luck with the cyclamen.

Hello, Bulbaholic
My belated (very busy at the moment) MANY THANKS for your quick answer! I'm glad you like my little seedlings. :)
Do you think the corms will be big enough to let them die? Summer is very hot in my region and I'm afraid they will dry in my garden. What if I will place them in the refrigerator with +4 temperature when the leaves die or do they need a warm summer resting?
Anyway I'm so grateful for your help, Bulbaholic! :)

Cyclamen persicum come from hot, Mediteranean regions like southern Turkey, Isreal and Jordan, Uma. I have seen mature corms growing in the crevaces of baking hot rocks in Cyprus so they are accustumed to heat.
I am not familiar with growing conditions in Ukraine, unfortunately, but I would not put them in the fridge, they need plent of ventilation. For the hot period I would keep them in full shade. In the wild the corms probably have a hot, dry summer but I would not let these little babies dry out completely, keep them just moist.
In my first post I refered to a cool temperature for growing. This related to winter conditions where they would not like typical British sitting room temperatures.
good luck

Thank you very much, Bulbaholic for your detailed answer!
Will try to grow them in right conditions. Thanks again! :)

Your welcome, Uma, I hope that my ideas work in the Ukrain.

I'm sure they will. :)

Wonderful, I hope they all grow strong long. Such cheering little flowers. xx