# Weak Fano varieties and small transformations

A projective normal and $$\mathbb{Q}$$-factorial variety $$X$$ is said to be log Fano if there exists and effective divisor $$D$$ on $$X$$ such $$-K_X-D$$ is ample and the pair $$(X,D)$$ is klt.

Now, let $$f:X\dashrightarrow Y$$ birational map which is an isomorphism in codimension one between two projective normal and $$\mathbb{Q}$$-factorial varieties. Assume that the strict transform of $$-K_X$$ on $$Y$$ is nef and big. Can we conclude that then $$X$$ is log Fano? I know that this holds if the strict transform of $$-K_X$$ on $$Y$$ is ample.

Thank you very much.

You need to know that $$Y$$ as at worst klt singularities. In this case $$Y$$ is a weak Fano variety with klt singularities and then it is log Fano.
Now, you are done since a small $$\mathbb{Q}$$-factorial transfomation of a log Fano variety is log Fano.