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Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

4 votes
1 answer
243 views

“Logarithmic” form of Kodaira Embedding

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 …
1 vote

Toroidal embedding

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 …
Philip Engel's user avatar
  • 1,493
4 votes
0 answers
221 views

When is a toric variety a Poincare duality space?

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 …
6 votes
0 answers
282 views

Algebraic deformation invariance of Gromov-Witten invariants

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 $\ …
5 votes
1 answer
337 views

Quotient of a smooth projective surface by an involution

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?
8 votes
1 answer
2k views

Contracting a curve of negative self-intersection on a surface

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 …
3 votes

Isomorphic schemes over DVR

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 …
Philip Engel's user avatar
  • 1,493
4 votes
0 answers
246 views

Is there an analogue of the linking pairing in etale, crystalline, etc cohomology theories?

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 …
6 votes

Examples of non-Kähler compact complex manifolds which satisfy the Dolbeault isomorphism

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 …
Philip Engel's user avatar
  • 1,493
5 votes
Accepted

A very general complex torus is simple

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 …
Philip Engel's user avatar
  • 1,493
2 votes
0 answers
147 views

Open Period Integrals of Elliptically Fibered K3 surfaces

Let M be the period domain for elliptic K3 surfaces $(X,\Omega)$ with a holomorphic two-form. Denote the fiber class $f$. Then $$M=\{\Omega\in f^\perp\otimes \mathbb{C}\,:\, \Omega\cdot \Omega=0, \,\O …
10 votes

If $A$ is the ring of continuous functions on a genus $g$ surface, can the genus of $X$ be s...

The torus has two functions $f$ and $g$ which are (1) relatively prime, (2) each have two square roots, and (3) whose product has $4$ square roots. For instance take two functions which vanish on disj …
Philip Engel's user avatar
  • 1,493
3 votes
Accepted

Loci in the moduli space of K3 surfaces associated to lattices

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 …
Philip Engel's user avatar
  • 1,493
3 votes

Existence of a morphism between two toric varieties

I don't think this is possible. Let $F_1$ and $F_2$ be two disjoint fibers of the projection $\mathbb{P}^1\times\mathbb{P}^1\rightarrow \mathbb{P}^1$. Suppose that there were a surjective morphism $$m …
Philip Engel's user avatar
  • 1,493
3 votes
1 answer
346 views

Analytically but not algebraically smoothable singularity

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 …

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