What does naturally mean here? We often cross the sentence 

"Kahler geometry emerges naturally in sugra". 

I have always wondered what does this mean; actually what does naturally mean in that sentence?
 A: "Naturally" means in this case that the Kähler manifold appears out of the other ingredients of the theory without us putting in a Kähler structure by hand. "Natural" in such mathematical contexts often means that there is no choice for a certain structure, it just appears from the other parts of the theory.
In this case, the Kähler geometry appears out of the potential on some complex scalar fields. These fields were introduced without the slightest idea of a Kähler geometry, but it turns out that their potential has the right form to be interpreted as a Kähler metric on a manifold for which those fields provide the coordinates. That is, the target space of the collection of scalar fields carries a natural Kähler structure in the sense that it is prescribed by the theory already - there is no choice of this structure to be made once the fields and their potential are fixed.
A: Natural -
I take it to mean you can reasonable accept it as reasonable method to explain things within a reasonable scope. The orders of magnitude seem acceptable and the outcomes seem probable.
Unnatural -
Would be mean one can't wrap ones mind around the method comfortably it is seems unreasonable. For example in the terms of scales. The orders of magnitude seem off. For example when examining gravity as compared to the weak force. In particle physics, the most important hierarchy problem is the question that asks why the weak force is $10^{32}$ times stronger than gravity. Often added dimension of space and or time are needed to find a reasonableness of explanation.
Let's use a story from link below to catch the concept of Naturalness. 
http://profmattstrassler.com/articles-and-posts/particle-physics-basics/the-hierarchy-problem/naturalness/
"The Notion of “Natural” and “Unnatural”
I think the concept of naturalness is best illuminated by a bit of story-telling.
A couple of friends of mine from college (I’ll call them Ann and Steve) got married, and now have two teenage children. Back when their kids were younger — say, 4 and 7 years old — they were pretty wild. They often played rough, got mad at each other, threw things, and generally needed at lot of supervision.
One day, Ann bought some beautiful flowers and put them in her favorite glass vase. But before she put the vase on the kitchen table, the doorbell rang. She ran to the front, carrying the vase, and as she made her way to the door, she absent-mindedly put the vase down on the small, rickety table that sits by the wall of the kids’ play room.
Half an hour later, Steve returned home with the kids, and sent them into the play room to occupy themselves while he and Ann settled in from the day and prepared dinner. They heard the usual sounds: bumps and crashes, the sounds of bouncing balls and falling blocks, yells of “no fair” and “ow! stop that!”, a moment of screaming that blissfully stopped almost as soon as it started…
It was forty-five minutes later when Ann noticed the vase with the flowers wasn’t on the kitchen table. After a moment searching the kitchen and dining room, she suddenly realized that she’d put it down and forgotten it in the most dangerous place in the house.
So she went running into the play room, hoping she wasn’t too late. And what do you think she found when she opened the door?
Guess. You get three options (Figure 1). Choose the most plausible.
The vase was exactly where she’d left it, comfortably placed at the center of the table.
The vase was smashed, and the flowers crushed, down on the floor.
The vase was hanging off the table, right at the edge, within a millimeter of disaster."

"Well, the answer is #3. There it was, just hanging there.
Somehow I suspect you don’t believe me. Or at least, if you do believe me, you probably are assuming there must be some complicated explanation that I’m about to give you as to how this happened. It can’t possibly be that two young kids were playing wildly in the room and somehow managed to get the vase into this extremely precarious position just by accident, can it? For the vase to end up just so — not firmly on the table, not falling off the table, but just in between — that’s … that’s not natural!
There must (mustn’t there?) be an explanation.  
Maybe there was glue on the side of the table and the vase stuck to it before falling off? Maybe one of the kids was hiding behind the table and holding the vase there as a practical joke on his mom? Maybe her husband had somehow tied a string around the vase and attached it to the table, or to the ceiling, so that the vase couldn’t fall off?  Maybe the table and vase are both magnetized somehow…?
Something so unnatural as that can’t just end up that way on its own… especially not in a room with two young children playing rough and throwing things around."
