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Question: Suppose $(M,J)$ is an almost complex manifold, and $X$ and $Y$ are two almost complex submanifolds (i.e. $J(T(X)) \subset T(X)$ and $J(T(Y)) \subset T(Y)$). Then must $X \cap Y$ also be an almost complex submanifold?

Clarification: If $X$ and $Y$ intersect transversally, then the answer is yes.

If the intersection is not transverse, however, it might be the case that $X \cap Y$ is not even a submanifold. In this case, if we revise our notion of tangent space, e.g. if $T$ is the tangent cone, then do we still have $J(T(X \cap Y)) \subset T(X \cap Y)$?

If the intersection is a submanifold, it might fail to be an almost complex manifold for other reasons, e.g. it might be of odd dimension. Is there an example of this (with odd-dimensional intersection, or otherwise)?

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  • $\begingroup$ Even when $J$ is complex, the intersection of complex submanifolds need not be a submanifold, as the intersection of smooth complex submanifolds can be a singular variety, so maybe you want to modify your question. In the case that $(M,J)$ is real-analytic (whether $J$ is integrable or not), the intersection of two almost complex submanifolds is a disjoint union of almost complex submanifolds (of varying dimensions). Do you actually know of a case in which the intersection has odd dimension or are you just guessing that there might be such an example (which would necessarily be non-analytic). $\endgroup$ – Robert Bryant Oct 24 '14 at 8:48
  • $\begingroup$ Oops, I had removed the part that addressed the singular case in an edit, which I've now re-added. I do not have an example of the intersection being odd-dimensional. The real analytic case, however, is of great interest to me. Can you sketch why that fact is true, or provide a source for it? Thanks! $\endgroup$ – user2028 Oct 28 '14 at 20:57
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Let me answer the easy part now and then come back later for the (slightly) more delicate singular case: If $(M,J)$ is a real-analytic almost-complex manifold and $X,Y\subset M$ are almost-complex submanifolds of $M$ whose intersection $X\cap Y$ is a submanifold, then $X\cap Y$ is an almost-complex manifold.

To see this, recall the following easy consequence of elliptic theory: Suppose that $(M,J)$ is a real-analytic almost-complex manifold and that $u:\mathbb{R}\to M$ is a real-analytic curve in $M$. Then there exists an open neighborhood $W\subset\mathbb{C}$ of $\mathbb{R}$ in $\mathbb{C}$ and a map $h:W\to M$ that is pseudoholomorphic (i.e., $h'(iv) = Jh'(v)$ for all tangent vectors $v\in TW$) such that $h(x) = u(x)$ for all $x\in\mathbb{R}$. Moreover, $h$ is locally unique in the sense that any two such extensions $h_i:W_i\to M$ agree on some open neighborhood of $\mathbb{R}\subset W_1\cap W_2$.

This immediately implies the claim: Since $X$ and $Y$ are almost-complex manifolds and $(M,J)$ is real-analytic, they are real-analytic submanifolds. If $X\cap Y$ is a submanifold (necessarily real-analytic) and $v$ is a tangent vector to $X\cap Y$, then there will be a real-analytic $u:\mathbb{R}\to X\cap Y$ such that $u'(0) = v$. Let $h:W\to M$ be a pseudo-holomorphic extension as above. Since $u:\mathbb{R}\to X$ and $u:\mathbb{R}\to Y$ are real-analytic, and $J$ restricts to each to be an almost-complex structure, it follows by the local uniqueness that, by shrinking $W$ if necessary, it can be assumed that $h(W)\subset X\cap Y$. But now $h'(0)(\partial_x) = u'(0) = v$, so $h'(0)(\partial_y) = h'(0)(i\partial_x) = Jv$. Thus, $Jv$ is also tangent to $X\cap Y$. Thus, $J\bigl(T(X\cap Y)\bigr) = T(X\cap Y)$, so $X\cap Y$ is almost-complex.

Added remark about the singular case: I still don't have time to write out a careful argument for this (and my real-analytic singularity theory is a little rusty), but I thought I'd go ahead and put in the sketch of the argument for what happens when $X\cap Y$ is not assumed to be a submanifold (but I am still assuming that $(M,J)$ is real-analytic). The point is that $X\cap Y$ is still a real-analytic variety and, as such, has a Whitney stratification into a disjoint union of real analytic submanifolds (not necessarily closed).

Recall that the analytic tangent variety $\mathsf{A\!T}(X\cap Y)\subset TM$ is defined to be the closure of the set of vectors $v\in TM$ that are the velocities of real-analytic curves $u:\mathbb{R}\to M$ whose images lie in $X\cap Y$. The argument given above in the case that $X\cap Y$ is a submanifold clearly extends to the more general case, showing that $\mathsf{A\!T}(X\cap Y)$ is invariant under $J$. This fact, coupled with the properties of the Whitney stratification (i.e., that the WS satisfies Whitney's Conditions A and B), should show that each of the Whitney strata of $X\cap Y$ is an almost-complex submanifold of $M$, thus implying that $X\cap Y$ is a disjoint union of almost-complex submanifolds of $M$. I'm pretty sure that this is will go through, but it does appear that the details are somewhat tedious.

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