File descriptor and fork

When a child is forked then it inherits parent's file descriptors, if child closes the file descriptor what will happen ?

It inherits a copy of the file descriptor. So closing the descriptor in the child will close it for the child, but not the parent, and vice versa.

If child starts writing what shall happen to the file at the parent's end ? Who manages these inconsistencies , kernel or user ?

It's exactly (as in, exactly literally) the same as two processes writing to the same file. The kernel schedules the processes independently, so you will likely get interleaved data in the file.

However, POSIX (to which *nix systems largely or completely conform), stipulates that read() and write() functions from the C API (which map to system calls) are "atomic with respect to each other [...] when they operate on regular files or symbolic links". The GNU C manually also provisionally promises this with regard to pipes (note the default PIPE_BUF, which is part of the proviso, is 64 kiB). This means that calls in other languages/tools, such as use of echo or cat, should be included in that contract, so if two indepenedent process try to write "hello" and "world" simultaneously to the same pipe, what will come out the other end is either "helloworld" or "worldhello", and never something like "hweolrllod".

when a process call close function to close a particular open file through file descriptor.The file table of process decrement the reference count by one.But since parent and child both are holding the same file(there refrence count is 2 and after close it reduces to 1)since it is not zero so process still continue to use file without any problem.

There are TWO processes, the parent and the child. There is no "reference count" common to both of them. They are independent. WRT what happens when one of them closes a file descriptor, see the answer to the first question.

Whenever fork() makes a new child, the file descriptors are not retained at all - they are changed.

Although the file will be a duplicate, it will have a different file descriptor.