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Let M be a 2n-dimensional closed symplectic manifold. Then is there a constant c such that , for any real 2-dimensional embedded J-holomorphic disk u, the symplectic volume of u is bounded by c ? I …

Hi. I have a question. When $X$ is a symplectic manifold which is diffeomorphic to $T^2$-bundle over $T^{2n}$, then does the first Chern class $c_1(X)$ vanishes in $H^2(X;\mathbb{R})$? (i.e. a sym …

Hi. I want to find an example of symplectic (smooth compact) manifold $(M,\omega)$ whose Betti numbers are not unimodal, i.e. $b_i(M) \leq b_{i+2}(M)$ fails for some $i < n$. ($ \dim{M} = 2n$.) ( …

Let $M$ be a simply connected smooth compact four manifold. Now I am trying to find an example such that (1) there is two symplectic forms $\omega$ and $\sigma$ on $M$, and (2) the intersection pro …

Let $(M^{2n},\omega)$ be a closed connected symplectic manifold and let $HR : H^2(M;R) \times H^2(M;R) \rightarrow R$ be the Hodge-Riemann form defined by $HR(\alpha,\beta) = \int_M \alpha \beta \ome …

Let $\omega$ a symplectic(may be Kahler) forms on $M^{2n}$. Then we have a symmetric bilnear two form on $H^2(M,\mathbb{R})$ given by $ HR_\omega (\alpha,\beta) := < \alpha\beta[\omega]^{n-2}, [M] …

Let $M$ be an even dimensional smooth manifold. I want to find an example $M$ satisfying the following conditions, $M$ admits a Kahler structure. $\omega$ is a symplectic form on $M$. There is no …

Hi. I have a question. Let $(M,\omega)$ be a closed symplectic 4-manifold equipped with a free circle action which preserves $\omega$ (symplectic circle action). My question is , is there an examp …

Let $(M^{2n},\omega)$ be a symplectic manifold with an integral symplectic form $\omega$. Due to the work of M.Gromov and D.Tischler (M.Gromov "A topological technique for the construction of solution …

Hi. I have a stupid question. Let $M$ be a blow-up of the complex projective plane at $k$ generic points. Then we can choose an orthoginal basis (with respect to the cup product) $H, E_1, \cdots, E_ …

Hi. I have a question about the notion "symplectic Fano". Let $(M,\omega)$ be a symplectic manifold with a $\omega$-tamed almost complex structure $J$. According to "J-holomorphic curves and symplect …

I have a question about Chern class of symplectic reduction. Let $(M,\omega)$ be a smooth compact symplectic manifold with a Hamiltonian circle action. Let $H : M \rightarrow \mathbb{R}$ be the corre …

Hi. I am studying Hodge theory on Kahler manifolds. I have several questions. Is Hodge number a topological invariant? (I mean, is it independent of the choice of Kahler structure?) If the questio …

Hi. I have a question about automorphisms of smooth ratonal surfaces. (I am not an algebraic geometer, so my question could be stupid. I am sorry about that.) Let $X_k$ be a blow-up of $\mathbb{P}^2$ …

Let $M$ be a smooth manifold (may be almost complex, almost Kahler, Kahler..). and Let $\phi : T^*M \rightarrow T^*M$ be a cotangent bundle automorphism. (the restriction of $\phi$ on the base $M$ is …