Counterexamples to Kollár's conjecture

Question

In "Singularities of theta divisors, and birational geometry of irregular varieties", Jour. AMS 10, 1 (1997), 243–258, L. Ein and R. Lazarsfeld provided a counterexample to Kollár's conjecture, i.e. an irregular 3-fold of general type of maximal Albanese dimension with Euler characteristic = 0. Their example can be trivially generalized to higher dimension.

My question is: are other counterexamples to Kollár's conjecture known? Where can I find them?

Thank you very much for the time dedicated to me Best regards

Answer 1

I just saw this question (better late than never). The following references come to mind (the first one has an example which somewhat improves on the example of Ein-Lazarsfeld, the others have a variety of related results):

The Example at the end of Chen and Hacon "On the irregularity of the image of the Iitaka fibration. Comm. in Algebra Vol. 32, No. 1, pp. 203-215 (2004), doi: 10.1081/AGB-120027861 .

Example 5.6 in arXiv:1111.6279 On the Iitaka fibration of varieties of maximal Albanese dimension. Zhi Jiang, Martí Lahoz, Sofia Tirabassi.

arXiv:1105.3418 Varieties with vanishing holomorphic Euler characteristic. J. A. Chen, O. Debarre, Z. Jiang.

Cai, Jin-Xing; Chen, Jungkai Alfred: A note on characterizations of Abelian varieties by topological invariants. Manuscripta Math. 112 (2003), no. 1, 15–19, doi: 10.1007/s00229-003-0384-2.