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Let $(M, \omega)$ be a complex $n$-dimensional Hermitian symmetric space of compact type, where $\omega$ is the symplectic (Kaehler) form on $M$ normalized so that $[\omega]$ generates the integral cohomology class $H^2(M, Z)$. Let $A$ be the generator of $H_2(M, Z)$.

Problem: Find two submanifolds $X$ and $Y$ of $M$ such that: $$\dim_{\R} X+\dim_{\R} Y=4n-2c_1(TM)(A)$$ and $$\Phi_A([X], [Y], [p])\neq 0,$$ where $\Phi_A([X], [Y], [p])$ is the genus-zero Gromov--Witten invariant of the triple $[p], [X], [Y]$ and $[p]$, $[X]$ and $[Y]$ denote the homology classes of a point $p\in M$, $X$ and $Y$ respectively.)

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  • $\begingroup$ What kind of answer are you looking for? There is a classification of irreducible, compact Hermitian symmetric spaces. There is also a theorem of Kollár and Ruan that guarantees the existence of a nonzero, one-point Gromov-Witten invariant. In principle, you could go through the list and find nonzero Gromov-Witten invariants in each case (simplified by the many results of experts such as Buch-Kresch-Tamvakis on GW theory of homogeneous spaces). Is that what you want? $\endgroup$
    – Jason Starr
    Commented Dec 25, 2012 at 22:21
  • $\begingroup$ Thank you for your answer. Yes I think you answer to my question. My problem was the following (with the same notation as in my question). Let J be an almost complex structure of M tamed by \omega and p be a point of M. Does there exists a J-holomorphic curve in the class of A which pass through p? Therefore, if I can find a non-zero one-point genus-zero Gromov-Witten invariant I believe I can giva a positiva answer to my question. Thank you $\endgroup$
    – Andrea Loi
    Commented Dec 26, 2012 at 18:23

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I will just consider the simplest example, maybe someone will give the answer in complete generality. So let $M=\mathbb CP^n$. Then $4n-2c_1(M)(A)=2n-2$. This means that we are in good shape. Basically we can take for $X$ and $Y$ any complex submanifolds of $M$ satisfying your condition. Indeed in this case for a generic point $p$ in $\mathbb CP^n$ there will be $deg X\cdot deg Y$ lines that contain $p$ and intersect both $X$ and $Y$. I assumed $X$ (or $Y$) is not zero dimensional, in which case there is only one line and also that $X$ and $Y$ are in general position, but this does not matter for GW, of course.

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