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

**3**

votes

**1**answer

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

**2**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**2**answers

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

**0**answers

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