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It was proven by Batyrev in 1981 http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=1581&option_lang=eng that there exist exactly 18 smooth toric Fano three-folds. I would like to …

I've heard a few times that "the time was imaginary before the Big Bang". I am guessing it means that at this stage, the space-time was a Riemannian $4$-manifold, but I am not sure this guess is corre …

Suppose that $(M,\omega)$ is a (connected compact) symplectic manifold with a Hamiltonian $S^1$-action given by Hamiltonian $H$. I would like to find a reference for the fact that every level set of …

Let $n$ be a positive integer. Consider the family of extensions $$0\to O(-n)\to E\to O(n)\to 0,$$ parameterized by $H^1(O(-2n))$. For each element $e\in H^1(O(-2n))$ we get a rank two bundle $E$ tha …

In Mori program in dimension $3$ there is a class of Mori contractions $\phi: X\to C$ called quadric bundles, where $X$ is a three-dimensional manifold and $C$ is a curve. As far as I understand, such …

I believe that the following is true: Statement. A holomorphic $\mathbb C^*$-action on a complex projective manifold is algebraic if and only if it has a fixed point. Where can I find a proof of th …

A Log del Pezzo surface is a normal complex surface with ample anti-canonical bundle and at worst quotient singularities. It is known that such surfaces are rational. This is proven, for example, he …

Suppose $M$ is a connected smooth manifold with a smooth $S^1$-action that fixes a point in $M$. Let $X$ be a $K(\pi,1)$-space and let $\varphi: M\to X$ be a continuous map. Question. How to prove t …

Let $A_r$ be a complex annulus of modulus $r>0$ obtained from a $1\times r$ rectangle in $\mathbb C$ with vertices $A=0$, $B=r$, $C=r+i$, $D=i$, by identifying isomterically $AB$ with $DC$. Let us no …

Let $M^{2n}$ be a symplectic manifold and let $M^{2n-2}$ be a symplectic submanifold. How to construct a non-trivial Hamiltonian $S^1$-action on $M^{2n-2}$ on a small neighborhood of $M^{2n-2}$, that …

Let $X$ be a smooth compact complex manifold of dimension $n$. Suppose $L$ is a line bundle on $X$ such that $dim(H^0(X,L^k))>c\cdot k^n$ for $c>0$ and $k>>0$. Question. Is it true that $X$ is Moishe …

Recall that Novikov additivity of signature of compact oriented smooth $4k$-manifolds is the following statement : If two manifolds are glued by an orientation-preserving diffeomorphism of their bou …

Let $M$ be a compact manifold with an $\mathbb S^1$-action that fixes a point on $M$. Is it correct that $\pi_1(M/S^1)=\pi_1(M)$? I believe this is correct and is a corollary of some well-known state …

I am aware that the following result is a classical one (by now). But I am not able to understand who proved it. What should be a proper reference to this statement? Theorem. Let $M^4$ be a compact s …

Let $X$ be an affine variety over $\mathbb C$ and let $G$ be a reductive group (over $\mathbb C$) acting on $X$. It is well known then that the ring of invariants $\mathbb C[X]^G$ is finitely generate …