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

26 votes
Accepted

Algebraic surface of constant width?

There exist many algebraic surfaces of constant breadth with no continuous symmetries, even ones with no symmetries at all. To see this, consider the properties of the support parametrization: The su …
Martin Sleziak's user avatar
13 votes
Accepted

A quadratic $O(N)$ invariant equation for 4-index tensors

Well, this is not actually an answer to either of the OP's questions; at most, it provides an easier way to classify the solutions for the $n=3$ case, and that might point a way towards an analysis fo …
Robert Bryant's user avatar
8 votes

Are the quaternionic Grassmannians quaternionic Kaehler manifolds?

Perhaps the OP really wants to know why quaternionic Grassmannians other than the quaternionic projective spaces are not considered to be 'quaternion-Kähler'. The reason goes back to Berger's classifi …
Robert Bryant's user avatar
15 votes
Accepted

Algebraic atlas on smooth manifolds

The answers to your question in the case $n=1$ are well-known. In higher dimensions, the answers are less complete, but something is known. For example, in the real case when $n=1$, there is only one …
Robert Bryant's user avatar
53 votes
Accepted

When is a singular point of a variety ($\mathcal{C}^\infty$-) smooth?

NB: This answer is directed to the questions about the real case, not the complex case, which was already treated by Francesco. Added 5 July 2021: Because of some questions I have received over the …
Robert Bryant's user avatar
5 votes

Invariant theory over $\mathbb R$

As YCor commented, the main point is to show that the invariant polynomials separate orbits. This follows from the compactness of $\mathrm{SO}(n)$. The point is this: Because $\mathrm{SO}(n)$ is co …
Community's user avatar
  • 1
18 votes

Is every minimal hypersurface in $S^n$ algebraic?

The answer to this question is 'no' for most minimal surfaces of revolution in the $3$-sphere. Consider surfaces in $S^3 = \{\,(z,w)\in\mathbb{C}^2\,|\,|z|^2+|w|^2=1\,\}$ that are invariant under the …
Robert Bryant's user avatar
7 votes

Subspaces of $ A_{n}(\mathbb {Q})$ in which all nonzero matrices are invertible

For a field $\mathbb{F}$, let $\mu_\mathbb{F}(n)$ denote the maximal dimension of a subspace $N\subset A_n(\mathbb{F})$ such that all the nonzero elements of $N$ are invertible. For simplicity, I wil …
Robert Bryant's user avatar
21 votes
Accepted

Problems concerning subspaces of $M_{n}(\mathbb{Q}) $

Let's call this maximal dimension function $\rho_{\mathbb{Q}}:\mathbb{N}\to\mathbb{N}$, i.e., $\rho_{\mathbb{Q}}(n)$ is the largest possible dimension of a subspace $N\subset M_n(\mathbb{Q})$ such tha …
Robert Bryant's user avatar
4 votes
Accepted

Existence of complex function?

The answer is 'yes' there do exist such functions that are non-constant with singularities only along surfaces $\Sigma\subset\mathbb{C}^2$, and here is how one can understand them: First, it helps to …
Robert Bryant's user avatar
2 votes
Accepted

Product of subgroups of $SU(8)$ algebraic set?

Yes, $G_1G_2\subset\mathrm{SU}(8)$ is an algebraic set. Here is the argument: Let $G_1{\times}G_2$ act on $\mathrm{SU}(8)\subset\mathrm{End}(\mathbb{C}^8)\simeq\mathbb{C}^{64}$ by the rule $(g_1,g_2) …
Robert Bryant's user avatar
12 votes
Accepted

To describe an invariant trivector in dimension 8 geometrically

Here's another very nice (but still algebraic) interpretation that explains some of the geometry: Recall that $\operatorname{SL}(2,\mathbb{C})$ has a $2$-to-$1$ representation into $\operatorname{SL} …
LSpice's user avatar
  • 12.9k
1 vote
Accepted

Algebraic geometric conditions on the variety $V(F)$ such that the manifold defined by $F$ h...

As we know, the projective hypersurface in $\mathbb{P}^n$ defined by a homogeneous polynomial equation $$ F(x^0,\ldots,x^n)=0 $$ of degree $m$ is nonsingular if $x=0$ is the only solution to the equat …
Robert Bryant's user avatar
2 votes
Accepted

Question about the implicit function theorem. an example of a homogeneous form for which its...

Here is a simple example: Take $F = w^3 +3 w u^2 -v^3$ on $\mathbb{R}^3$ with coordinates $(u,v,w)$. At the point $p=(u,v,w)=(1,0,0)$, we have that $F=0$ can be solved for $w$ as a function of $(u,v …
Robert Bryant's user avatar
14 votes

Unique almost complex structure up to diffeomorphism

Dmitri's answer is fine, but there is a different argument that is purely local that is worth bearing in mind as well: On a $2n$-manifold $M$, the set of almost complex structures on $M$ are the secti …
Robert Bryant's user avatar

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