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There are Calabi-Yau threefolds that do not have Calabi-Yau threefolds as their mirror partners like rigid ones ($h^{1,2} =0$). I am wondering if there are known any examples of non-rigid Calabi-Yau …

By a kummer surface, I mean a quotient of two-dimensional complex torus by multiplication by $-1$. It is a K3 surface and known to be simply connected. But it is not clear to me why it is simply-conne …

Let $c_1, c_2$ be smooth curves on a plane in $\mathbb P^3$ that intersect at a point $p$ with multiplicity $m \ge 1$ and $H$ be an ample divisor on $\mathbb P^3$. Let $\pi_1:X_1 \rightarrow \mathbb P …

If $X$ is a smooth projective variety, then the line bundle is trivial if and only if $D$ is the zero divisor. That's because $L$ is trivial only if $H^{n-1} \cdot D =0$, where $H$ is an ample diviso …

For a given Hilbert polynomial $P$, consider the Hilbert scheme $\text{Hilb}_P (\mathbb P^n)$. If it is not empty, then does it contain a smooth variety? Here $\text{Hilb}_P (\mathbb P^n)$ is regarde …

In the Wikipedia page for 'Kodaira dimension', there is 'the classification table of algebraic three-folds' in the section 'Any dimension'. What are those 'others' in the very bottom of this table? A …

I am considering smooth projective rational threefolds $X$'s with base-point-free anticanonical pencil, i.e., the anticanonical linear system $|-K_X|$ is a base-point-free pencil. It is a generalizati …

I am looking for an example of a smooth projective threefold $X$ with fibration $ \pi : X \rightarrow \mathbb P^1$ such that a generic fiber $F$ of $\pi$ is a smooth $K3$ surface, $K_X$ is linearly …

Take a look at arXiv:math/0703315. It gives an explicit pair of birational Calabi-Yau threefolds which are cohomologically non-isomorphic.

Let $\pi:X \rightarrow Y$ be a double cover between compact manifolds $X$, $Y$ and $\theta$ be the deck transformation. Let $H^2(X, \mathbb Z)^\theta$ be a group of $\theta^*$-invariant elements in $H …

Let me explain more concretely what's happening. Choose a smooth quintic hypersurface $W$ in $\mathbb C \mathbb P^4$ that intersect transversely with $X \cap Y$. Then $W$ and $X+Y$ are linearly equiv …

Let $X$ be a smooth projective threefold with $h^{0,1}(X) = h^{0,2}(X)=0$ that has a smooth anticanonical section $D$. Then $D$ is necessarily a $K3$ surface. Consider a subgroup $$Pic_X(D) = i^*(Pic …

By a Calabi-Yau manifold, I mean a compact complex manifold whose canonical bundle is trivial and $$H^i (X, O_X) = H^0(X, \Omega_X^i) = 0$$ for $0 < i < \dim X$. Infinitely many topological types of …

Let $G$ be a finite subgroup of the group of automorphsims of a $K3$ surface $S$. Consider the quotient $S/G$. I am interested in the collection of such qutients: $$\{ S/G \mid S\text{ is a K3 surf …

I am looking for examples of non-Kähler compact complex manifolds of dimension four with trivial canonical class and $H^i(M, {\mathcal O}_M) = H^0(M, \Omega^i_M) =0$ for $0< i < \dim M$. In dimensio …