$A^1$ curves on quasi-projective varieties

In this talk, I will present the recent joint work with Yi Zhu on $A^1$-connectedness for quasi-projective varieties. The theory of $A^1$-connectedness for quasi-projective varieties is an analogue of rationally connectedness for projective varieties. To study curves on a quasi-projective variety $U$, we compactify $U$ by a log smooth pair $(X,D)$. Using the theory of stable log maps to $(X,D)$, we were able to produce $A^1$ curves on $U$ from degeneration. This provides many interesting examples of $A^1$-connected varieties. Some applications to arithmetic geometry and usual rationally connectedness will be discussed as well.