Let $X$ be a smooth projective variety. By Definition 24.4.2 in the 2003 book [Mirror Symmetry](https://www.claymath.org/library/monographs/cmim01c.pdf), $X$ is called convex if $h^1\left( \Sigma, f^\*T\_X \right) = 0$ for every genus zero stable map $f:\Sigma \to X$.


**Question:** Why is a convex variety called convex?


On page 505 of the book, it was said that $H^1\left( \Sigma, f^\*T\_X \right) $ measures obstructions to the deformations of the map $f$ (when the structure of the source curve $\Sigma$ is fixed). But I don't know what the obstructions mean, or what zero obstruction means. So I'm also asking the following


**Related question:** What do zero and nonzero obstructions to the deformations of the map $f$ mean?


**My background and what answer I'm looking for:** I have close-to-zero background in deformation theory. So I don't expect to fully understand the precise mathematical meaning over one night. I would appreciate it very much if someone could explain what zero and nonzero obstructions mean in an intuitive way. If this can be further supported by concrete examples for zero and nonzero obstructions, then it will be perfect.



Why is a convex variety called convex?
If there are no obstructions to deformation, i.e. $h^1=0$, then Kodaira's deformation theorem says that the map admits a family of deformations, forming a manifold, near that given map, with tangent space given by $H^0$. You think of genus zero curves as like lines, or line segments, and deforming a line while staying in the variety should be like trying to move around on a set of points in Euclidean space while still being able to move your line segment with you, something you can do when points of the set are connected by line segments, i.e. in a convex set.


For example, if $f$ is the identity map of the projective line, then the space of deformations of $f$ is the space of maps of degree 1, i.e. the space of projective linear transformations of the projective line. The tangent space at the identity element of the group of projective linear transformations is the Lie algebra of vector fields on the projective line, i.e. $H^0$ is the space of linear vector fields, i.e. the space of $2 \times 2$ traceless matrices, so 3 dimensional. 


Another example: if $f$ is constant, then the deformations are constant maps, and if the target $X$ is smooth at the image point $x=f(\mathbb{P}^1)$, then $H^0=T\_x X$.


Kodaira's book on deformation theory is where I learned about this stuff. Maybe there are better references, especially in the algebraic category.


