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This article is concerned with the infinite depth water wave equation in two space dimensions. We consider this problem expressed in position-velocity potential holomorphic coordinates. Viewing this problem as a quasilinear dispersive equation, we establish two results: (i) local well-posedness in Sobolev spaces, and (ii) almost global solutions for small localized data. Neither of these results are new; they have been recently obtained by Alazard-Burq-Zuily [1], respectively by Wu [23] using different coordinates and methods. Instead our goal is improve the understanding of this problem by providing a single setting for both problems, by proving sharper versions of the above results, as well as presenting new, simpler proofs. This article is self contained.For the derivation of the above equations, we refer the reader to Appendix A. In the real case these equations originate in [17]. The changes needed for the periodic case are also described in the same Appendix A. There are also other ways of expressing the equations, for instance in Cartesian coordinates using the Dirichlet to Neumann map associated to the water domain, see e.g. [1] . Here we prefer the holomorphic coordinates due to the simpler form of the equations; in particular, in these coordinates the Dirichlet to Neumann map is given in terms of the standard Hilbert transform.It is convenient to work with a new variable, namelyThe equations becomeThese equations are considered either in R × R or in R × S 1 .As the system (1.1) is fully nonlinear, a standard procedure is to convert it into a quasilinear system by differentiating it. Observing that almost no undifferentiated functions appear in (1.1), one sees that by differentiation we get a self-contained first order quasilinear system for (W α , Q α ). To write this system we introduce the auxiliary real function b, which we call the advection velocity, and is given byThe reason for this will be immediately apparent. Using b, the system (1.1) is written in the form

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