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I first got concious of the notion of normal varieties around 3 years ago and despite the fact that by now I can manipulate with it a bit, this notion still puzzles me a lot. One thing that strikes me …

Let $M$ be a connected finite dimensional topological manifold and $g$ be any metric on it that induces the topology of $M$ ($g$ is not a Riemannian metric). How to prove that the group of isometries …

Take a convex polyhedron $P$ in $\mathbb R^3$ and remove all the faces, i.e. leave only the edges. Call this graph $E$. Let us now try to continuously deform $E$ in $\mathbb R^3$ so that all the edges …

Let $X$ be a smooth complex projective variety with an ample line bundle $L$, and let $D\subset X$ be a smooth divisor. Suppose in an analytic neighborhood $U$ of $D$ there is a Kahler form $\omega$ s …

Let $M$ be smooth compact riemannian manifold with boundary and $\varphi_0: S^1\to M$ be a rectifiable curve (or a smooth one). I would like to find a reference to the following statement: Statement. …

I would like to know if following is correct. Statement. Suppose we have a smooth (i.e., $C^\infty$) almost complex structure on $\mathbb R^4$ and $C_1, C_2$ are two $J$-holomorphic curves passing t …

This question is quite subtle, I don't believe the answer is known in the general situation. But if you consider the case of $1$-forms on surfaces, one can completely characterise those that are harmo …

Is there an example of a smooth vector field $v$ on $S^3$ such that $v$ preserves a volume form and $v$ is not a Reeb vector field? Recall that $v$ is a Reeb vector field if there exists a contact $ …

Let $S$ be a smooth complex projective surface and $D\subset S$ be a smooth complex curve. Fix an integer $m>1$ and consider $(S,D,m)$ as an orbifold with orbi-locus $D$ with stabilizer $\mathbb Z_m$ …

Suppose $S$ is a smooth compact oriented surface without boundary. Let $g_0$ and $g_1$ be two smooth Riemannian metrics on $S$. Consider the interpolating path of metrics $g_t=g_1t+g_0(1-t)$. Recall t …

Let $(X,\omega)$ be a smooth Kähler manifold (not necessarily compact) with an isometric $S^1$-action with a Hamiltonian $H$. It is a well known fact that 1) The reduced spaces $X(c)=H^{-1}(c)/S^1$ …

I would like to find an appropriate reference for the following statement: Statement. Let $M$ be a compact Riemannian manifold with non-negative Ricci curvature. Then $\pi_1(M)$ is virtually abelian. …

Let $X$ be a Kähler manifold with an isometric $S^1$-action (which of course complexifies to a $\mathbb C^*$ action). Consider the corresponding Hamiltonian $H$ and let $X_0=H^{-1}(0)/S^1$ be the redu …

I am certain that the following result holds, but was not able to find a reference. Do you know one? Or maybe you can give a short proof? Statement. Let $(M^4,\omega)$ be a compact symlectic manifold …

Take $\mathbb C^2$ with coordinates $(z,w)$. Suppose that $J$ is a $C^{\infty}$ almost complex structure on $\mathbb C^2$ such that the line $w=0$ is $J$-holomorphic and $J(0,0)$ is given by $(z,w)\to …