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Results tagged with complex-geometry
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user 4333
Complex geometry is the study of complex manifolds, complex algebraic varieties, complex analytic spaces, and, by extension, of almost complex structures. It is a part of differential geometry, algebraic geometry and analytic geometry.
6
votes
3
answers
1k
views
Is there an obvious way for showing singularities are quotient?
I'm stuck on a technicality concerning singularities.
Basically, I have to show that the singularities of a $\mathbf{certain}$ normal projective variety over $\mathbf{C}$ are rational. (I won't both …
- 9,497
11
votes
1
answer
1k
views
How does $f_* O_X$ measure ramification and Grothendieck-Riemann-Roch
Let $f:X\longrightarrow Y$ be a finite morphism of smooth projective varieties over a field $k$ of characteristic zero, where $\dim X=\dim Y$. Then $f$ is flat. Hence $f_\ast \mathcal{O}_X$ is a coher …
- 9,497
22
votes
1
answer
874
views
Is being of general type stable under generization
This question is about how varieties of general type over an algebraically closed field of characteristic zero $k$ behave under generization in families.
Definition. An integral projective s …
- 9,497
8
votes
2
answers
970
views
Covers of the projective line over Z and arithmetic Grauert-Remmert
This question is the two-dimensional analogue of Etale coverings of certain open subschemes in Spec O_K
There I asked if one could characterize in a way certain covers of $\textrm{Spec} \ O_K$. As Ca …
- 9,497
14
votes
1
answer
362
views
Are any of these complex surfaces ever projective?
Let $C$ and $T$ be compact connected Riemann surfaces (or: smooth projective connected curves over $\mathbb{C}$) of genus at least two and let $X:=C\times T$. Let $(c,t)$ be a point of $X$, and let $ …
- 9,497
16
votes
1
answer
1k
views
Is the set of surfaces over Spec Z with ample canonical sheaf empty
Main question. Does there exist a smooth projective morphism $X\to$ Spec $\mathbf Z$ of relative dimension two such that the canonical sheaf $\omega_{X_{\mathbf Q}}$ of the generic fibre $X_{\mathbf Q …
- 9,497
23
votes
1
answer
4k
views
GAGA and Chern classes
My question is as follows.
Do the Chern classes as defined by Grothendieck for smooth projective varieties coincide with the Chern classes as defined with the aid of invariant polynomials and connect …
- 9,497
9
votes
2
answers
1k
views
Finite etale atlas for Deligne-Mumford stacks
Let $X$ be a smooth finite type separated connected Deligne-Mumford stack over $\mathbb C$.
Does there exist a finite etale morphism $Y\to X$ with $Y$ a scheme?
What if $X$ is an algebraic space (i. …
- 9,497
19
votes
0
answers
607
views
Coarse moduli spaces of stacks for which every atlas is a scheme
Let $X = [P/G]$ be a smooth finite type separated DM-stack over $\mathbb C$ given as the quotient of a smooth quasi-projective scheme $P$ by the action of a smooth (finite type separated) reductive gr …
- 9,497
10
votes
1
answer
772
views
Can one bound the Todd class of a 3-dimensional variety polynomially in c_3
This question is on bounding the degree of the Todd class on a complex threefold.
Let $X$ be a smooth compact connected complex surface. Let $c_i=c_i(TX)$ be its $i$-th Chern class. Recall the foll …
- 9,497
11
votes
2
answers
740
views
Families of Fano varieties over non-hyperbolic curves
Let $C$ be a non-hyperbolic (smooth quasi-projective connected complex algebraic) curve. That is, $C$ is isomorphic to $\mathbb P^1, \mathbb A^1, \mathbb G_m$, or an elliptic curve.
Let $f:X\to C$ be …
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