ag.algebraic geometry – Construction of derived Quot scheme

I study the construction of derived Quot schemes in paper “ Shifted symplectic structures on derived Quot-stacks ”(arXiv:1908.03021).

Derived quot stacks are constructed from sheaves of non-positively graded dg algebras in section 3 of the paper.

In particular, I have some question about differentials of the dg algebras.

Question

1) The last line of page 14, a differential is constructed by the morphism

$mathcal{V}_j otimes left (bigotimes_{1 leq l leq m+1} mathcal{A}_{i_l} right) otimes (W_i)^vee rightarrow mathcal{V}_j otimes left (bigotimes_{1 leq l leq m} mathcal{A}_{i_l} right) otimes (W_{i+i_{m+1}})^vee$ .

However it seems to me this morphism does not degree 0 morphism and this should be the morphism

$mathcal{V}_j otimes left (bigotimes_{1 leq l leq m+1} mathcal{A}_{i_l} right) otimes (W_{i+i_{m+1}})^vee rightarrow mathcal{V}_j otimes left (bigotimes_{1 leq l leq m} mathcal{A}_{i_l} right) otimes (W_{i})^vee$.

Is this correct ?

2) Does the differential $delta_W$ constructed from the above morphism really become a differential ?
It seems to me that ${delta_W}^2 neq 0$

3) Do we need the differential on $C^{bullet}$ constructed from the multiplication on $oplus mathcal{A_i}$ like that on $B^{bullet}$ on line 10 of page 13.

Thank you !

dg.differential geometry – Smoothness of conformal transformations

Given a smooth pseudo-Riemannian manifold $(M,g)$ one can define the conformal group as the set of smooth diffeomorphisms $varphi:Mto M$ such that there is a positive smooth function $u$ with $varphi^ast g=ug$. One could also define it as the set of all $C^1$-diffeomorphisms $varphi:Mto M$ such that there is a positive continuous function $u$ with $varphi^ast g=ug$. One could also define intermediate cases. I would expect these to be the same but it does not seem obvious, can someone clarify? Do any such answers rely on $M$ being Hausdorff or second-countable/paracompact?

complex geometry – Characterization of a domain of holomorphy

I need to show that the property of being a domain of holomorphy is the same as being a holomorphically convex domain (this result is known as Cartan-Thullen theorem). However, the proofs I found in textbooks (e.g. Shabat) look ugly and are hard to digest.

Is there a reference with a better proof? Can you share your intuition on looking at this result? Thanks in advance.

ag.algebraic geometry – Moduli of rational equivalent classes of 0-cycles

Let $X$ be a smooth variety over a field $k$ and I’d like to think about $CH_0(X)$ the 0-Chow group i.e. the group of rational equivalent classes of 0-cycles. I’m wondering if there is any reasonable formulation to make sense of “family”/”moduli” of rational equivalent classes of 0-cycles, forming an fppf sheaf over $text{Spec} k$? If there is such a sheaf, is there any chance the sheaf is actually representable?

A quick literature search shows some work on certain formulations of “Chow schemes”. But, they seem to parametrize the 0-cycles, instead of the rational equivalent classes.

A known example: if X is in addition one dimensional and proper, we can speak of the Picard scheme which represents the relative Picard functor. I would really like to know what happens when the dimension goes higher.

mathematics – Cylindrical billboarding around an arbitrary axis in geometry shader

I found an answer on this site relating to this question already, but it doesn’t seem applicable in the context of my project.

Basically I’d like to create a method which fits this signature:

float3x3 AxisBillboard(float3 source, float3 target, float3 axis)

That is to say, when given a source point (i.e an object’s position in world space), a target (camera position in world space), and an axis (the object’s up vector, which is not necessarily the global y axis), it produces a 3×3 rotation matrix by which I can multiply the vertices of my point so that it’s properly rotated.

I’ve found many solutions and tutorials online which work nicely assuming I only want to rotate around the y axis.

For example, here’s a solution which billboards around the global y axis:

float3 dir = normalize(target - source);
    
float angleY = atan2(dir.x, dir.z);
c = cos(angleY);
s = sin(angleY);
    
float3x3 rotYMatrix;
rotYMatrix(0).xyz = float3(c, 0, s);
rotYMatrix(1).xyz = float3(0, 1, 0);
rotYMatrix(2).xyz = float3(-s, 0, c);

For context, I’m working on a grass shader, and each individual blade of grass should be billboarded to face the camera while remaining aligned with the normal of the terrain.

mg.metric geometry – When does blending occur?

First, a definition. Blending is the operation of taking two or more polytopes, arranging them in a compound so that some elements coincide completely, and removing those coincident pairs.

Last year, I discovered that a $4_{21}$ polytope could be vertex-inscribed in a $2_{41}$ polytope in 270 ways, and that it is possible to blend all 270 of them, giving a polytope with less than 2160*270 7-orthoplex facets. Later on, I learned that the $4_{21}$ could in fact be vertex-inscribed in any uniform polytope with $E_8$ symmetry. Therefore my question is: which other $E_8$-symmetric polytopes lead to blending when $4_{21}$s are vertex-inscribed in them?

ag.algebraic geometry – Associated point of Coherent Sheaf

That’s a question about a proof I found in E. Sernesi’s
Deformations of Algebraic Schemes on page 188:

enter image description here

The sheaf $F$ is assumes to be coherent on $X= mathbb{P}^r$. The question is
why the hyperplace $H subset mathbb{P}^r$ is assumed not to contain
any point of $Ass(F)$, the associated points
(https://stacks.math.columbia.edu/tag/02OI)

then the exact sequence $0 to O(-H) to O_{P^r} to O_H to 0$ stays
exact after beeing tensored by $- otimes F(k)$?

Obviously the problem is of local nature, so we can study what happens at affine
piece of $mathbb{P}^r$. Let $R$ be a ring, $M$ a coherent $R$-module
and $f in R$ a nonzero element which defines the hyperplace. The
assumption that $H$ not contains point of $Ass(M)$, translates to the
ring theoretic assumpion that for every associated point of $M$, that
is a prime $p subset R$ which annihalates some nonzero $m in M$, ie
$p= Ann(m)$, the element $f$ is not contaned in $p$.

Therefore the claim is that the sequence
$0 to R xrightarrow{cdot f} R to R/(f)$ stays exact after tensor
by $M$. Why that’s true?

ag.algebraic geometry – “Eigenvalues” for semi-linear actions

Suppose I consider a free module $M$ of rank $r$ over the ring $R = mathbb Z_ell((t))$. Let $sigma$ be a $mathbb Z_ell$ linear operator on $M$ so that $sigma(f(t)m) = f((1+t)^q-1)sigma(m)$ for $m in M$. I think this is often called a semi-linear operator. ($mathbb Z_ell$ refers to the $ell$-adics, not the finite field).

Can we always find a basis $v_1,dots,v_r$ of $M$ over $R$ such that there exists some fppf extension $R to R’$, an extension of $sigma: R’ to R’$ and $alpha_1,dots,alpha_r in R’$ so that $sigma(v_i) = alpha_iv_i$? This notion is supposed to be a replacement for an eigenvalue/eigenvector.

Note that unlike eigenvalues, the $alpha_i$ are not unique! If i replace $v_i to p(t)v_i$ for some unit $p(t) in R^times$, then $alpha_i to p((1+t)^q-1)alpha_i/p(t)$.

dg.differential geometry – Intuition behind Nakano positivity

I am learning about Nakano positivity of hermitian vector bundles, which is the strongest notion of positivity we can ask. I don’t understand what is the geometric meaning of it. Let me briefly summarize the situation:

$overline E=(E,h)$ is a (holomorphic) hermitian vector bundle on a complex manifold $X$. Then we have a natural notion of curvature $Theta(E)$ which is a global (1,1)-form of real type on $X$ with values in $Gamma(E)otimes Gamma(E)^vee$.

At this point, the curvature $Theta(E)$ induces a hermitian form $theta_E$ on the vector bundle $T_Xotimes E$. Such a form is defined locally just by taking the local coefficients appearing in $Theta(E)$ (we are fixing also an orthonormal frame).

We say that $E$ is Nakano positive if $theta_E$ is actually a hermitian metric for $T_Xotimes E$, i.e. if $theta_E$ is defined positive.

What is the meaning of $theta_E$? What does it “measure” when it is a metric? Do we have an intrinsic construction of $theta_E$ without
appealing to local cohordinates?

complex geometry – Fundamental $1$-form for a Riemannian manifold?

Take a Hermitian manifold $(M,I,g)$ where $I$ is the complex structure and $g$ is the Hermitian metric. The associated fundamental $2$-form
$
g(cdot,I(cdot))
$

captures a lot of the information about the Hermitian geometry of the manifold.

For quaternionic manifolds an analogous situation giving a fundamental $4$-form.

By analogy there “should be” a fundamental $1$-form for a Riemannian manifold. Is this true? If not then why not?