jQuery make global variable

Your code looks fine except the possibility that if the variable declaration is inside a dom read handler then it will not be a global variable… it will be a closure variable

jQuery(function(){
    //here it is a closure variable
    var a_href;
    $('sth a').on('click', function(e){
        a_href = $(this).attr('href');

          console.log(a_href);  
         //output is "home"

        e.preventDefault();
    }
})

To make the variable global, one solution is to declare the variable in global scope

var a_href;
jQuery(function(){
    $('sth a').on('click', function(e){
        a_href = $(this).attr('href');

          console.log(a_href);  
         //output is "home"

        e.preventDefault();
    }
})

another is to set the variable as a property of the window object

window.a_href = $(this).attr('href')

Why console printing undefined

You are getting the output as undefined because even though the variable is declared, you have not initialized it with a value, the value of the variable is set only after the a element is clicked till that time the variable will have the value undefined. If you are not declaring the variable it will throw a ReferenceError