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I know this question is over 8 years old, but if you are still interested in this question you may be interested in a series of papers I wrote with Giancarlo Lucchini:

Smooth quotients of abelian varieties by finite groups (A characterization of smooth quotients of abelian varieties by finite groups that fix the origin of dimension $\geq3$)

Smooth quotients of principally polarized abelian varieties (A characterization of smooth quotients of abelian varieties by finite groups that fix the origin as well as a principal polarization)

Smooth quotients of complex tori by finite groups (with an appendix by Stephen Griffeth) (Here we provide a "bridge" between the case of smooth quotients where the group fixes the origin and the case of a free action. Essentially general smooth quotients of complex tori by finite groups are fibrations in products of projective spaces over an étale quotient of a complex torus)

There is a lot to be done in the case of étale quotients, and as the other posts say, much work has been done in this case by Uchida and Yoshihara, Lange, Catanese and Demleitner.

You may be interested in a series of papers I wrote with Giancarlo Lucchini Arteche:

Smooth quotients of abelian varieties by finite groups (A characterization of smooth quotients of abelian varieties of dimension $\geq3$ by finite groups that fix the origin)

Smooth quotients of principally polarized abelian varieties (A characterization of smooth quotients of abelian varieties by finite groups that fix the origin as well as a principal polarization; this gives a moduli-theoretic spin on the problem)

Smooth quotients of complex tori by finite groups (with an appendix by Stephen Griffeth) (Here we provide a "bridge" between the case of smooth quotients where the group fixes the origin and the case of a free action. Essentially general smooth quotients of complex tori by finite groups are fibrations in products of projective spaces over an étale quotient of a complex torus)

There is a lot to be done in the case of étale quotients, and as the other posts say, much work has been done in this case by Uchida and Yoshihara, Lange, Catanese and Demleitner.

I know this question is over 8 years old, but if you are still interested in this question you may be interested in a paper I wrote with Giancarlo Lucchini where we classify smooth quotients of abelian varieties by finite groups that fix the origin in the case when the fixed locus is 0-dimensional, and we characterize all smooth quotients of Jacobians. It can be found on the arxiv at https://arxiv.org/abs/1801.00028.

I know this question is over 8 years old, but if you are still interested in this question you may be interested in a series of papers I wrote with Giancarlo Lucchini:

Smooth quotients of abelian varieties by finite groups (A characterization of smooth quotients of abelian varieties by finite groups that fix the origin of dimension $\geq3$)

Smooth quotients of principally polarized abelian varieties (A characterization of smooth quotients of abelian varieties by finite groups that fix the origin as well as a principal polarization)

Smooth quotients of complex tori by finite groups (with an appendix by Stephen Griffeth) (Here we provide a "bridge" between the case of smooth quotients where the group fixes the origin and the case of a free action. Essentially general smooth quotients of complex tori by finite groups are fibrations in products of projective spaces over an étale quotient of a complex torus)

There is a lot to be done in the case of étale quotients, and as the other posts say, much work has been done in this case by Uchida and Yoshihara, Lange, Catanese and Demleitner.

I know this question is over 8 years old, but if you are still interested in this question you may be interested in a paper I wrote with Giancarlo Lucchini where we classify smooth quotients of abelian varieties by finite groups in the case when the fixed locus is 0-dimensional, and we characterize all smooth quotients of Jacobians. It can be found on the arxiv at https://arxiv.org/abs/1801.00028.

I know this question is over 8 years old, but if you are still interested in this question you may be interested in a paper I wrote with Giancarlo Lucchini where we classify smooth quotients of abelian varieties by finite groups that fix the origin in the case when the fixed locus is 0-dimensional, and we characterize all smooth quotients of Jacobians. It can be found on the arxiv at https://arxiv.org/abs/1801.00028.