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Consider the rational curve (conic) given by image of the map $$ u([z,w])=[z^2,-z^2,w^2,-w^2,zw] \in \mathbb{P}^4 $$ which lies in quintic 3-fold $X: x_1^5+\cdots+x_5^5- x_1\cdots x_5=0$. By Groth …

It is possible to construct (in many ways) a family of Calabi-Yau quintics $\mathcal{X}\rightarrow \Delta$, over disk, such that the fiber over $0$ has a singularities locally given by the equation $x …

Let $\psi_t: X\to X$, $t \in [0,1]$, be a path Hamiltonian diffeomorphism on a symplectic manifold $X$, given by functions $H_t$. If $H_t \equiv H$ is independent of $t$ then $$ \psi_1 = \psi_{\frac …

I am looking for examples of (symplectic or not) 4-dimensional manifolds $X$ that have positive dimensional Seiberg-Witten moduli spaces (and $b^{2+}>1$). Of course, the result/conjecture is that the …

Let $f\!=\!(u\colon (\Sigma,p_1,\ldots,p_k) \to X)$ be an element of the moduli space of genus $g$ $k$-marked degree $A$ $J$-holomorphic maps $\mathcal{M}_{g,k}(X,A,J)$. For simplicity assume $C=(\Sig …

Blow-ups of points can also be performed in the symplectic category; for a given point $p\in (X,\omega)$ we choose a Darboux chart around $p$ and then use the symplectic cut corresponding to the stand …

Let $E\to C$ be a rank $2$, degree $2g-2$, holomorphic vector bundle over a curve of genus $g$. By Riemann-Roch theorem, $$H^0(E)-H^1(E)= \deg(E)+2.(1-g)=0. $$ Question: For which $g$, there is such …

The symplectic sum of Gompf and the symplectic cut of Lerman are known to be inverse of each other, in the sense that if you apply one of these first and the other one afterward, you obtain the origin …

Notations : Suppose V is a closed contact compact manifold with contact form $\alpha$, of dimension 2n+1. Consider the symplectic sub-bundle $ \xi \subset TV $ given by $ \xi=$ ker($\alpha$). So $ \xi …

Consider a Symplectic manifold D (with $H^1(D)=0$) with symplectic form $w$. Let V be the total space of a circle bundle over D with non-trivial Euler class $e\in H^2(D)$. You may think of V as the se …

A balanced smooth rational curve in a calabi-Yau X is a smooth rational curve whose normal bundle is $O(-1)\oplus O(-1)$. We usually like these curves because of their rigidity. But, Is there any t …

In mirror symmetry one usually considers a complexified kahler form $B+iw$ instead of kahler form $w$ itself.(Or their moduli) Here is the question: What does $B$ correspond to? what kind of informa …

Assume $(X,\omega)$ is a compact real $2n$-dimensional symplectic manifold with a Hamiltonian torus action corresponding to the moment map $\mu:X\to \mathfrak{t}^*\cong \mathbb{R}^k$. In this situati …

I am trying to gather a list of all known symplectic manifolds which don't have Kahler structure. If you know any please add to the list and give references for it. Please avoid giving repetitive exa …

If $(X,\omega)$ is a symplectic manifold and $J\colon TX \to TX$ is an almost complex structure, we know that $J$ is actually a complex structure if and only if the Nijenhuis tensor $N_J(\cdot,\cdot)$ …