Is 'many' used in positive sentences or not?

There are some senses of "much" and "many" that are restricted to non-affirmative contexts, and when "much" and "many" are used in that way then they are negatively-oriented polarity-sensitive items (NPIs).

There are some senses of "much" and "many" that can be used in positive affirmative environments, but then those instances of "much" and "many" are not NPIs.

And then, to make things even more confusing, some words (such as "much" and "many") may be an NPI in some styles or registers and not in others.

This issue with "much" and "many" is discussed in the 2002 CGEL, pages 823, 826-7. There might be some related info about NPIs on the internet, such as in wikipedia or on linguistics sites, but you've gotta be careful of info found on the internet.

If you want more info here in this post, then leave me a comment and I'll try to come back later tonight. (The reason why this post is now so brief is that many topics are involved and I'll have to discuss them as a foundation for this topic that you are asking about.)

Note that CGEL is the 20002 reference grammar by Huddleston and Pullum et al., The Cambridge Grammar of the English Language.

I know of no such rule:

There are many stars in the sky.

This is grammatically correct and, in the absence of clouds, astronomically correct as well.

The rule one can find in some grammars is formulated in the wrong way. Fact is in spoken language "many" in positive sentences is nearly always replaced by "a lot of" and similar expressions. But it is wrong to see this as a matter of grammar. In written language you can find hundreds of examples where "many" is used in positive sentences.The use of "a lot of" is a current colloquialism, it's a matter of choice of words, but not a thing of grammar. It's up to you whether you want to speak in a colloquial or a more neutral way.