No Kronos do not have any kind of LAN support.

That is however to be expected. If you read on the description in the sales page. You can read:

"Multiplayer notice: In order to access the multiplayer portion of the game, you must first enter your unique CD-key and set up an account with King Art Games via the in-game interface. Your CD-key is located on your game shelf."

That mens the Multi-player part of the game comes with DRM (Digital Right Management) so you can only play the game with others after you have registered the game at "King Art Games"

From that it follows that you can not play a game of KRONOS with a family member at home without buying several copies of the game (except for hot seat).

Exactly as you wold wold have to buy several chess boards if you wanted to play chess with him :P
Well that is how it wold have been if software companies was to sell chess boards, I think.

Now some people cold say "But you can use hot seat"
That is true but not not a good argument.

Who wold play chess the hot set way:

1 you sit on a char at a table and move your chess piece
2 you turn the chess board around indicating it is your friends turn
3 you raise from the chair and goes away so that your friend can sit down and play his move.
4 you friend takes the chair and start to think over his next move.
5 your friend move his chess piece
6 your friend turn the chess board around indicating it is your turn now
7 ...
8 ....
Goto 1.

This is how hot seat wold be with chess. If one did it in a manner simulair withh hot seat on most computer-games
Now I might perhaps have exaggerated slightly here. But you can probably understand the reasoning for my general standpoint on this issue from it.

Furthermore, there is no way you can start a multi player locally, the games DRM is in the way.
So once the servers at "King Art Games" are dead/down/Gone. Multi player gaming with Kronos will likely no longer be possible.

So I do think that multi player DRM sucks approximately as much as single player DRM does.

Now that all this negative stuff have been said.
I did buy the game. And it seems to be real nice to play.