Overriding == operator. How to compare to null?

you could alway override and put

(Object)(person1)==null

I'd imagine this would work, not sure though.

Easier than any of those approaches would be to just use

public static bool operator ==(Person person1, Person person2)   
{   
    EqualityComparer<Person>.Default.Equals(person1, person2)
} 

This has the same null equality semantics as the approaches that everyone else is proposing, but it's the framework's problem to figure out the details :)

I've always done it this way (for the == and != operators) and I reuse this code for every object I create:

public static bool operator ==(Person lhs, Person rhs)
{
    // If left hand side is null...
    if (System.Object.ReferenceEquals(lhs, null))
    {
        // ...and right hand side is null...
        if (System.Object.ReferenceEquals(rhs, null))
        {
            //...both are null and are Equal.
            return true;
        }

        // ...right hand side is not null, therefore not Equal.
        return false;
    }

    // Return true if the fields match:
    return lhs.Equals(rhs);
}

"!=" then goes like this:

public static bool operator !=(Person lhs, Person rhs)
{
    return !(lhs == rhs);
}

Edit
I modified the == operator function to match Microsoft's suggested implementation here.

Use object.ReferenceEquals(person1, null) or the new is operator instead of the == operator:

public static bool operator ==(Person person1, Person person2)
{
    if (person1 is null)
    {
         return person2 is null;
    }

    return person1.Equals(person2);
}