Let $Y\subset X$ be a Lagrangian submanifold in a holomorphic symplectic manifold $X$. We know that there exists a local moduli space $M$, which parametrizes lagrangian submanifolds in $X$(there are no obstructions), and such that there is a natural isomorphism $T_{[Y]} \cong H^0(Y,N_{Y/X}) $. Let us fix a section $s\in H^0(Y,N_{Y/X})$. Suppose that $s$ is nowhere vanishing. Is it true that for a sufficiently small disk $U\subset \mathbb{C}$ there exists a holomorphic map $\gamma: U\to M$, such that $\gamma(0)=[Y]$ and $\frac{d\gamma}{dz}|_{z=0}=s$ and such that submanifolds $Y_{\gamma(t)}$ don't intersect each other? Here by $Y_{\gamma(t)}$ we denote the Lagragian submanifold in $X$ which corresponds to a point $\gamma(t)$ in the moduli space $M$ . Or more generally: if $s$ has zeros, is it true that $Y_{\gamma(t)}$ intersect each other only by zeros of $s$?
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1$\begingroup$ Do you have some hidden compactness assumption here? What if you take $X$ to be the cotangent bundle of the complex line (with coordinate $x$), $Y$ to be the zero section and and $Y_{\gamma(t)}$ to be the graph of the closed one form $t(x-t)dx$. In this case the section $s$ is $xdx$ so it vanishes at $x=0$. But $Y_{\gamma(t)}$ intersects $Y$ at the point $x =t$. $\endgroup$– Tony PantevCommented Mar 22, 2016 at 0:13
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$\begingroup$ @TonyPantev. Since the OP asserts that there is a local moduli space, I suspect that $Y$ is intended to be compact, and $X$ is assumed to be smooth and separated at every point of $Y$. $\endgroup$– Jason StarrCommented Mar 22, 2016 at 0:25
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1$\begingroup$ Still, I am not sure why it is expected that the first order deformation determines the intersections. What if we take a linear pencil of high genus curves in a K3, and then take a curve $\Lambda$ in the parameter space of the linear system which is tangent to the line parameterizing the pencil at some point $p$. Then the divisors corresponding to points $t \in \Lambda$ near $p$ will intersect the divisor corresponding to $p$ at a locus that varies with $t$ and has nothing to do with the pencil. $\endgroup$– Tony PantevCommented Mar 22, 2016 at 0:56
1 Answer
No for the first question. A counter-example is given by the Fano variety $X$ of lines in a cubic fourfold $V\subset \mathbb{P}^5$: for each hyperplane $H$ of $\mathbb{P}^5$, the lines contained in $H\cap V$ form a Lagrangian surface $Y_H$ (the Fano surface of the cubic threefold $H\cap V$); varying $H$ gives a complete family of deformations of $Y_H$. For $H$ and $H'$ general, the surfaces $Y_H$ and $Y_{H'}$ intersect in 27 points, corresponding to the 27 lines of the cubic surface $V\cap H\cap H'$.
I think that the answer to the last question (more generally...) is yes. This should follow from general deformation theory, given that the deformations of $Y$ are unobstructed. Unfortunately I don't have the adequate references on hand at this moment.
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1$\begingroup$ But existence of a non-vanishing section is quite strong condition: in particular $ c_{2}(N)=0$ and surfaces can't have only zero-dimensional intersection. In your example $c_{2}(N)=27$. Sorry, maybe I should have written nowhere vanishing section instead of non-vanishing. $\endgroup$– LyaCommented Mar 22, 2016 at 10:25
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$\begingroup$ Oh, I see. Then, as I said, it should follow from deformation theory. $\endgroup$– abxCommented Mar 22, 2016 at 11:15
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$\begingroup$ Nice example with the Fano variety of lines! I still don't understand why you say that the answer of the second question is 'yes'? Your example shows that the answer to the second question is 'no'. and I think my elementary example with a non-linear family of curves on a K3 (in the comment above) also shows that the answer is 'no'. The first one has a chance of being true though. $\endgroup$ Commented Mar 22, 2016 at 14:39
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$\begingroup$ Note that the OP has edited her(?) question, she wants to start with a nowhere vanishing section of the normal bundle. So my example does not apply. $\endgroup$– abxCommented Mar 23, 2016 at 8:05