The delete ex command will work nicely.
:+,$d
This will delete all the lines from current +1 till the end ($)
To delete the next 2 lines the follow range would work, +1,+2 or shorthand +,+2
:+,+2d
As @ib mentioned the :delete or :d command will move the cursor to the start of the line next to the deleted text. (Even with nostartofline set). To overcome this we can issue the `` normal mode command. `` will jump back to the exact position before the last jump, in this case the :d command. Our command is now
:+,+2denter``
Or as one ex command
:+,+2d|norm! ``
To make this easier we wrap this all up in a command:
command! -count=1 -register D :+,+<count>d <reg><bar>norm! ``
Now to delete the next following 3 lines:
:3D
This command can also take a {reg} like :delete and :yank do. So deleting the next 4 lines into register a would be:
:4D a
For more information
:h :d
:h :command
:h :command-register
:h :command-count
:h ``