# Two smooth tangent almost complex curves in a $4$-manifold

I would like to know if following is correct.

Statement. Suppose we have a smooth (i.e., $C^\infty$) almost complex structure on $\mathbb R^4$ and $C_1, C_2$ are two $J$-holomorphic curves passing through $(0,0)$, tangent at $(0,0)$ and regular at $(0,0)$. Then there exist $C^{\infty}$ smooth complex coordinates $(z,w)$ such that $C_1$ is locally given by $w=0$ and $C_2$ by $w=z^n$.

PS. I think it would be enough for me to know that $C_2$ can be given by $w=z^n+O|z^{n+1}|$. However everything must be $C^{\infty}$ - the coordinates, and the $O|z^{n+1}|$ term. If, on the other hand one can not expect to have a $C^{\infty}$ diffeo, what is the best one can expect?

• It should follow from Lemma 1.2.2 in McDuff's "Singularities and positivity of intersections of J-holomorphic curves" that we can find such coordinates except that $C_2$ is locally $w=z^n+O(|z|^{n+1})$. I'm not sure we have enough freedom to get rid of the higher order terms, unless you perturb $C_2$ (but try McDuff's other related papers on local properties of $J$-holomorphic curves). – Chris Gerig Mar 28 '18 at 23:43
• Dear Chris, thank you for your comment. I had a look into Lemma 1.2.2. Unfortunately, I don't see how one deduces from Lemma 1.2.2 what you are claiming...(Is this something written in the beginning of the proof of Lemma 1.3.1?) In fact, if you could give me a more less explicit reference to the statement that you give it will be very useful. – aglearner Mar 29 '18 at 15:39
• Chris, I don't understand what you mean by multiply covered. Both $C_1$ and $C_2$ are smooth, and I just consider them as surfaces embedded in $\mathbb R^4$ (so why speak about multiply covered?) – aglearner Mar 30 '18 at 17:29
• Typically in J-holomorphic theory, “curve” means map, whereas if we just care about the image we say “surface” or “subvariety”. So based on what you’re looking for, I’m just writing $C_2$ as a graph of $C_1$ as in McDuff’s paper. – Chris Gerig Mar 30 '18 at 19:14
• Thanks for explaining Chris. I corrected the formulation of the question, hope that it is clear now what I mean. If I understand correctly what you write you can answer the weaker version of this question (with the term $O|z|^{n+1}$). Would you be so kind to write this down as an answer (and not just a comment)? – aglearner Mar 30 '18 at 20:08

This follows from theorem 6.2 (and the first sentence in the proof) of Mario J. Micallef and Brian White, The structure of branch points in minimal surfaces and in pseudoholomorphic curves, Ann. of Math. (2) 141 (1995), no. 1, 35–85.

• Thanks for the reference Ben! I had a look. The theorem that you cite is speaking about $C^2$ and $C^1$ diffeos. At the same time I want a smooth -- $C^{\infty}$ diffeo. It is not clear for me if the article answers my question, unfortunately. Is this the case? – aglearner Mar 29 '18 at 11:48
• @aglearner: I am sorry, I don't know. – Ben McKay Mar 29 '18 at 13:08
• In case you're interested, Appendix E (written by Lazzarini) in McDuff-Salamon's book clarifies the differentiability issues. – Chris Gerig Mar 31 '18 at 22:16
• I guess, I'll need to buy the book, it is not in our library. In the end maybe $C^2$ or $C^1$ diffeo is enough... But it would be nice to have an example where $C^{\infty}$ diffeo does not exist. – aglearner Apr 1 '18 at 9:03
• I'll accept the answer, since it helped me to answer the question that I needed to answer (with the error term) and prompted a useful discussion – aglearner Apr 7 '18 at 10:12

It turns out that it is nice to read books. The answer to the weaker version of my question with $O|z|^{n+1}$ term is contained on page 17 of McDuff-Salamon book [MS] (no need of Micallef-White!): https://people.math.ethz.ch/~salamon/PREPRINTS/jholsm.pdf

Proof. In the proof of Lemma 2.2.3 of [MS] one uses coordinates in $\mathbb C^2$ such that $C_1$ is given by $w=0$ and the almost complex structure $J$ along the line $(z,0)$ is the standard one. Then it is explained that the almost complex map $z\to \mathbb C^2$ corresponding to $C_2$ is given by

$$z\to (p(z)+O(|z^{n+1}|), az^n+O(|z^{n+1}|))$$

where $p(z)$ is a polynomial of order at most $n$, $a\ne 0$. In our case of course $p'(0)\ne 0$. It is now clear that in these coordinates $C_2$ is as need. QED.

Comment. The above proof is elementary and does not use Micallef-White. Similarly to Micallef-White's, statement it can be used to answer the original question with a $C^1$-smooth change of coordinates (instead of $C^{\infty}$). Indeed, after a smooth reparameterization in $z$ and scaling in $w$ the above map for $C_2$ looks as $$z\to (z, z^n+O(|z^{n+1}|)).$$ Denote the second term by $f(z)$. Then the map $(z,w)\to (z,w-(f(z)-z^n)/z^n)$ is $C^1$ and it sends the couple $C_1,C_2$ to the couple $(w=0, w=z^n)$.

I wonder still if one can make this last change of coordinates $C^{\infty}$...

• This is what I wrote in my comment (following the beginning of the proof of McDuff's Lemma 1.3.1 in the quoted paper of my comment), and is just an adaptation of Micallef-White's theorem which Ben quotes in his answer; but I thought the real question was about other $C^\infty$-smooth coordinates for the images. – Chris Gerig Apr 2 '18 at 16:11
• Chris, thanks for your comment! I am neophyte with $J$-holomorphic maps, and (partially) for this reason had a problem to understand what you've written (also, I guess the book is better written than the article). But it is great that you say that this is your reasoning, since now I know that you consider it correct. I am asking this question for a particular purpose and in the end it turned out that having this $O|z^{n+1}|$ error is fine for me so solving it in full form is not urgent anymore for me. But I am still curious if it is possible to kill the $O|z^{n+1}|$ term. – aglearner Apr 2 '18 at 16:15
• No worries at all -- I've been confused by the difference with choosing local coordinates for the maps versus just the subspaces, so I'm glad we're both focusing in on it. I believe we can get rid of the $O$-term by perturbing $C_2$ while preserving tangency at the origin, but if you need $C_2$ fixed then I don't know but my guess is no (there are some nice examples which may be your counterexamples at the very end of McDuff's 1992 paper "Singularities of J-holomorphic curves in almost complex 4-manifolds"). – Chris Gerig Apr 2 '18 at 16:20
• I just realized that the lemma you quote is not similar but precisely the same Lemma 1.3.1 of McDuff's paper I quote. This paper precedes production of those lecture notes you refer to, which by the way is an old version, the new version is the "big book" (2nd edition) that I also referenced containing Lazzarini's Appendix E. – Chris Gerig Apr 2 '18 at 22:54