Let H be the complex line 1+z 1 +z 2 = 0 in (C * ) 2 and H trop the associated phase tropical line. We show that H and H trop are isotopic as topological submanifolds, by explicitly constructing the isotopy map. arXiv:1705.05664v1 [math.AG] 16 May 2017image is H 1trop . We call this isotopy map Φ t 1 . In order to do that, we subdivide H 1 and H 1trop in two parts, see Figure 8 and Definition 4.13. In Section 5 we write down at first the isotopy map Φ t 1 and then we prove Theorem 1.1. Acknowledgements. The content of this paper is part of the Master's thesis I wrote at the University of Hamburg under the supervision of Professor Bernd Siebert, who I would like to thank for the many inspiring discussions on the subject.

We show that every compact complex analytic space endowed with a fine logarithmic structure and every morphism between such spaces admit a semi-universal deformation. These results generalize the analogous results in complex analytic geometry first independently proved by A. Douady and H. Grauert in the ’70. We follow Douady’s two steps process approach consisting of an infinite-dimensional construction of the deformation space followed by a finite-dimensional reduction.