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Definition. I say a topological space $Y$ has finite-type if $Y$ is a finite union of open sets $Y=U_1\cup \dotsb \cup U_N$ such that each possible intersection of the $U_i$ is either empty or is cont …

Lefschetz hyper-plane theorem for smooth projective varieties, $X\subset \mathbb{P}^{n+1}$ says: For smooth hyperplane section $Y= X\cap H$, the restriction map $H^i(X) \rightarrow H^i(Y)$ is an isomo …

Is it true that every smooth rational variety X is simply connected? How is the proof? Would it be still true if X has mild (for example orbifold) singularities?

Let $L\to V$ be complex line bundle and $F_{t}:V\to V$, $t\in [0,1]$, be a loop of diffeomorphisms, $F_0=F_1=$ identity. For every $x\in V$, we get a loop $\gamma_x(t)=\{F_t(x)\}$ whose class in $\pi …

An orbifold structure on some topological space $X$ is a covering of $X$ with local quotient charts $V/G$, where $V$ is some connected manifold and $G$ effectively acts on $V$ via a finite group of di …

Let $\Sigma_g$ be a Riemann surface of genus $g\geq 2$ and $G=\pi_1(\Sigma_g)$. Let $\pi\colon \mathbb{H}\to \Sigma_g$ be the universal covering map. What kind of surface is $\mathbb{H}/[G,G]$? More …

Let $B$ be a compact manifold, and $\hat{B}\to B$ be the maximal abelian covering of $B$; i.e. $\hat{B}$ is the quotient of the universal cover with respect to the commutator subgroup of $\pi_1(B)$. G …

As pointed out to me by Guangbo Xu, if the $S^1$-action gives $V$ the structure of an $S^1$-bundle $\pi:V\to B$, then such connection exists if and only if $\pi_*(c_1(L))=0 \in H_1(B)$, where $\pi_*$ …

Let $X$ be a compact Kahler manifold and $A\subset H_1(X,\mathbb{Z})$ be a subgroup. Corresponding to $A$ there is an abelian covering $X_A \to X$ with $Deck(X_A)=H_1(X,\mathbb{Z})/A$. For example if …

Let X be a K3 surface and $Y=X/\mathbb{Z}_2$, an Enrique surface. Long exact sequence of homotopy groups corresponding to fiberaion $\pi:X\to Y$, says that $\pi_2(X)=\pi_2(Y)$, while we know $H_2(X)$ …

There is a paper "on the homology of double branched covers" by Lee, which is kind of related to your question.

Let $C$ be the set of points $(a,b,c,d) \in \mathbb{C}^4$ which satisfy 1) $ \left|a\right|^2+\left|c\right|^2=\left|b\right|^2+\left|d\right|^2 =1 $. 2) $ a\bar{b}+c\bar{d}=0 $ There is a (compone …

For an isolated plane curve singularity, given by homogeneous equation $F=0 \subset \mathbb{C}^2$, one consider the curve $(F=0) \cap S^3 \subset S^3$, and we call it the link of singularity. some pro …

Also there are C.Y 3-folds with this property constructed via toric geometry (I think due to Batyrev)