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

3
votes
1answer
Question 1. Given a Laurent polynomial $f(z_1,\cdots,z_n)$, such that the corresponding zero locus $Z$ of $f$ in $(\mathbb{C}^*)^n$ is smooth, can we find a smooth toric variety $\bar{X}$ (together w …
asked Feb 26 '14 by user36931
8
votes
2answers
Given a smooth hypersurface $H$ in $\mathbb{C}^n$, a theorem of Hironaka promises that one can find a strict normal crossings compactification $\bar{H}$ inside of a projective variety $X$. For me, thi …
asked Mar 7 '14 by user36931
5
votes
1answer
Suppose $X$ is a smooth quasi-projective variety over $\mathbb{C}$ and $Z$ a proper subscheme, there is a formal duality isomorphism (here we consider the Zariski topology) due to Hartshorne: $$ tr: …
asked Sep 17 '13 by user36931
10
votes
1answer
Let R be an augmented regular local ring over a field $k$ with maximal ideal m. There is the Grothendieck residue symbol: $$Res: H^n_m(\Omega^n) \to k$$ If $k=\mathbb{C}$ and $R$ is affine space, th …
asked Jul 11 '13 by user36931
2
votes
0answers
The hypersurface $\lbrace (x,y,z) \in \mathbb{C}^3:\frac{(xz+1)^2}{z}-\frac{(yz+1)^3}{z}=1 \rbrace$ is a well-known example of a contractible hypersurface in $\mathbb{C}^3$. See for instance, Example …
asked Aug 24 '17 by user36931
5
votes
0answers
Let $\mathbb{C}P^2$ denote the projective plane. From reading the section of http://homepages.math.uic.edu/~coskun/gokova.pdf which surveys Gieseker stable sheaves, I have understood that there are n …
asked Aug 25 '15 by user36931
10
votes
1answer
Given a smooth quasi-projective variety $X$ over $\mathbb{C}$ and bounded complexes of vector bundles $(P,d)$ and $(P',d')$ with compactly supported cohomology. It is well-known that such complexes sa …
asked Sep 16 '13 by user36931
2
votes
0answers
My question is rather vague and I apologize. Let $X$ be a smooth quasi-projective variety over $\mathbb{C}$. I am interested in whether there are homological properties which distinguish algebraic vec …
asked Dec 10 '13 by user36931
2
votes
1answer
Let $X$ be a simply connected projective manifold of dimension $n$ over $\mathbb{C}$ and $D = \cup D_i$ be a divisor with normal crossings such that its all components $D_i$ are smooth and ample. I …
asked Jul 22 '13 by user36931
3
votes
2answers
On affine toric varieties there is a classical theorem of Danilov which gives some combinatorial ways to describe the global sections of an appropriate sheaf of Kahler differentials as a vector space. …
asked Nov 28 '13 by user36931
1
vote
0answers
This is a more precise version of my previous question. Let $X$ be a smooth variety of dimension $n$ over $\mathbb{C}$ and $Z$ a proper sub-scheme. We denote by $\tilde{X}$ the formal completion of $X …
asked Jul 14 '13 by user36931