## 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: 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?