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All of the following is from Beauville's wonderful, short, dense book: "Complex Algebraic Surfaces." I really recommend it if you want to learn about the classification or even general techniques in s …

An explicit realization of degree $2$ and degree $g+1$ maps that separate points can be provided. Suppose the equation of a hyperelliptic curve is $$C:y^2=f(x)$$ with $\deg(f)=2g+2$. "Complete the squ …

Are there examples of algebraic singularities which may be smoothed analytically but not algebraically? It certainly seems possible, but if not, why? Are there conditions under which this becomes true …

Let $\mathcal{X}\rightarrow \textrm{Spec}\,\mathbb{C}[[t]]$ be a flat family of K3 surfaces such that the central fiber $X_0$ has a $-2$ curve $C$. Then one can perform a flop along $C$ inside the thr …

Every divisor class $D$ on a surface is the difference of two smooth, connected curves. Choose a very ample divisor $A$ and an $n$ so that $D+nA$ is also very ample. Then $(D+nA)-nA=D$ so $D$ is the …

I think yes. Let $J$ be the complex structure on $\mathbb{R}^{2g}$. Let $L$ be the lattice. Then there is a complex subtorus if and only if $L$ intersects some complex-linear subspace in a full sub-la …

It is easy to show using birational factorization that the only curves on a surface which can be contracted to get an algebraic, smooth surface are smooth $(-1)$-curves. Furthermore, I know of example …

Let $\mathcal{X}\to S$ be a smooth family of projective varieties over a smooth curve $S$. Let $\beta$ be a curve class supported in some fiber and consider the relative moduli space of stable maps $\ …

I won't completely answer your question, but will try to just rephrase it in a certain way. You are asking when two given moduli spaces of lattice-polarized K3 surfaces $M_L$ and $M_{L'}$ intersect. T …

Is the quotient of a smooth complex projective surface by an involution projective? Suppose the quotient happens to be smooth; does that change the situation?

When is a complete toric variety a Poincare duality space? Is there an "if and only if" condition? And is this condition local? Given an analytically-locally-toric compactification of a smooth variet …

Suppose you have a non-compact complex manifold $X$ with a Hodge metric, whose associated Kahler form has integral cohomology class. In the compact case, one would be able to conclude that $X$ is proj …

Let $M$ be a compact oriented manifold of dimension $n$. It is well-known that there is a perfect intersection pairing $$H_k(M;\mathbb{Z})_{torsion\,\,free}\otimes H_{n-k}(M;\mathbb{Z})_{torsion\,\,fr …

naf's comment is correct. This example can be seen as an instance of Mumford's construction of degenerations of abelian varieties into unions of toric varieties. Here is a "purely toric" construction …

I am slightly confused by Francesco's comment and answer, because the Frolicher spectral sequence does degenerate for compact complex surfaces. So why isn't a non-Kahler surface an example? Perhaps th …