## ag.algebraic geometry – Extending a holomorphic vector bundle: a reference request

Let $$Y$$ be a complex manifold, $$Xsubset Y$$ a compact submanifold, and $$Eto X$$ a holomorphic vector bundle. Can $$E$$ be extended
to a bundle over an open neighborhood of $$X$$ in $$Y$$? (Four years ago I have asked this question on MO Extending the tangent bundle of a submanifold for the case $$E=T_X$$.)

After tinkering with this problem for a while I found a necessary condition, there is an invariant in $$H^2(X, mathcal{N}_{X/Y}^*otimes End(E))$$ which must be zero for an extension to be possible. So far, so good.
Now I think about writing it down and submitting somewhere (assuming it is a new result). But, in a decent paper there are supposed to be references to known results in the same direction, right? And this is what the real problem is: damned if I have a clue where to look! It is all miles away from areas I am familiar with (mostly differential geometry), and this far I could not find anything remotely relevant. So, it would be nice if someone helps me with this.

## ag.algebraic geometry – “Universal coefficent theorem” for pro-étale cohomology

In algebraic topology, for any space with finite homology type, the universal coefficient theorem states that for any abelian group $$G$$, we have
$$H^n(X,G)cong left( H^n(X,mathbb{Z})otimes Gright)oplus text{Tor}_1(H^{n+1}(X,mathbb{Z}),G).$$
My question is whether the analogous statement is true for the pro-étale cohomology, namely if $$R$$ is a $$mathbb{Z}_ell$$-algebra, do we have
$$H^n_{proét}(X,underline{R})cong left(H^n_{proét}(X,underline{mathbb{Z}_ell})otimes Rright)oplus text{Tor}_1(H_{proét}^{n+1}(X,underline{mathbb{Z}_ell}),R)$$
for a sufficiently nice scheme? I’m mostly interested in the case of a smooth, projective scheme over some algebraically closed field (possibly of positive characteristic).
Also, would this decomposition respect that Galois action on the cohomology?

## ag.algebraic geometry – Affine scheme as Algebraic Space

We working in the following with Knutson’s definition of an algebraic space
(ie via equivalence relation; there is also another equivalent def via
sheaves but let us work here with the following one):

An algebraic space $$X$$ comprises a scheme $$U$$ and a closed subscheme
$$R subset U times U$$ satisfying the following two conditions:

1. $$R$$ is an equivalence relation as a subset $$U times U$$;
2. the two projections $$p_i: R to U$$ onto each factor are étale.

Knutson adds an extra condition that the diagonal map is quasi-compact.

A couple of notes on used notations: the equivalence realtion $$R subset U times R$$ is considered as
categoretical equivalence relation (also called “internal relation”),
that means that for all
$$T in (Sch)$$ the set $$Hom(T,R) subset Hom(T, U times U)= Hom(T,U) times Hom(T,U)$$ is the equivalence relation in usual sense.

Question: How one can see that an “usual” scheme $$U$$ is an
algebraic space in the sense above? Assume wlog $$U$$ affine. The
crucial task is to find an equivalence relation $$R subset U times U$$
corresponding to $$U$$ such that projections $$p_i: R to U$$ are etale.

The most natural choice seems to me the image with respect the
diagonal map $$Delta: U to U times U$$, ie $$R:= Delta(U)$$.
$$Delta$$ is always an immersion and thus
$$Delta(U)$$ is always a locally closed subscheme of $$U times U$$.

If we take this choice for $$R$$, why $$p_i: R to U$$ are etale?
Or is it conventional to take another choice for $$R$$? eg the
closure of the image? if yes, why?

## ag.algebraic geometry – Closure of the product of subfunctors

Background:

• Let $$X: textbf{CRing} to textbf{Set}$$ be a preheaf on the category of affine schemes and $$Z subseteq X$$ a subfunctor. One defines $$Z$$ to be closed if for every ring $$A$$ and every morphism $$f: text{Hom}(A , -) to X$$ the inverse image $$f^{-1}(Z)$$ is of the form $$R mapsto { varphi : A to R | varphi(I) = 0 }$$ for some ideal $$I subseteq A$$.
• The intersection of subfunctors is defined naively, as is the closure (denoted by $$overline{Z}$$) of a subfunctor $$Z subseteq X$$ (it is the intersection of all closed subfunctors of $$X$$ containing $$Z$$).
• If $$Y$$ is another presheaf, the product of $$X$$ and $$Y$$ is also defined naively.

Context: In section 1.14 of Jens Jantzen’s great book “Representations of Algebraic Groups”, the following is stated: If $$X$$ and $$Y$$ are presheaves which are schemes over a noetherian ring $$k$$ and $$Z subseteq X$$ is a subscheme, and if $$Z, X$$ are algebraic and $$Y$$ is flat, then $$overline{Z times Y} = overline{Z} times Y$$. For the proof, he references Demazure-Gabriel I, section 2, 4.14 (although in my copy of Bell’s translation this reference unfortunately doesn’t exist).

Actual Question:
Is this true for general presheaves? I.e. if $$X$$ and $$Y$$ are presheaves and $$Z subseteq X$$ is a subfunctor, is it true that $$overline{Z times Y} = overline{Z} times Y$$? I worry that it isn’t because of the conditions in Jantzen stated above, but I haven’t been able to decide either way. (Also side question: does anyone know the correct reference in the translation?)

## ag.algebraic geometry – GAGA for vector bundles over Riemann surfaces

Serre’s GAGA theorem gives an equivalence of categories between algebraic and analytic coherent sheaves over a complex projective variety. The proof relies on the finiteness of the cohomologies of coherent analytic sheaves over compact complex manifolds, which is a non-trivial analytic result.

I was wondering that maybe GAGA theorem could be proven quite more easily in the case of vector bundles over a Riemann surface $$X$$ by using the existence of meromorphic functions as the underlying analytic result and “Grothendieck” theorem about the classification of vector bundles on the Riemann sphere.

The idea is to consider a meromorphic function $$f:X rightarrow mathbb{P}^1$$ and, for any vector bundle $$Erightarrow X$$, consider the pushforward $$f_*E$$, which is a vector bundle over $$mathbb{P}^1$$, which by the Grothendieck theorem is algebraic.

My question is if this idea can be used to show that, since $$f_*E$$ is algebraic, then $$E$$ is also algebraic.

## ag.algebraic geometry – Algebraic Space: Two equivalent constructions

According to Wikipedia
there are two common ways to define algebraic spaces:
they can be defined as either quotients of schemes by étale
equivalence relations,
or as sheaves on a big étale site that are locally isomorphic
to schemes.
Namely:

I) a la Knutson:

An algebraic space $$X$$ comprises a scheme $$U$$ and a closed subscheme
$$R subset U times U$$ satisfying the following two conditions:

1. $$R$$ is an equivalence relation as a subset $$U times U$$;
2. the two projections $$P_i: R to U$$ onto each factor are étale.

Knutson adds an extra condition that the diagonal map is quasi-compact.

II) as a sheaf:

An algebraic space $$mathfrak {X}$$ can be defined as a sheaf of sets
$$mathfrak {X}:(operatorname{Sch}/S)^{text{op}}_{text{ét}} to operatorname{Sets}$$
such that

1. There is a surjective étale morphism $$h_X to mathfrak {X}$$;
2. the diagonal morphism $$Delta _{{mathfrak {X}}/S}: mathfrak {X} to mathfrak {X} times mathfrak {X}$$
is representable and quasicompact (thanks to David’s careful remark).

(Rmk: in II)1. we identified a scheme $$X$$ with its image $$h_X$$ wrt the Yoneda
embedding $$X to operatorname{Hom}(X,{-})$$.)

Two questions:

1. About construction I). Wikipedia moreover says that if $$R$$
is the trivial equivalence over each connected conponent of $$U$$
(i.e. for all $$x,y in U$$ lying in same component then
$$xRy$$ iff $$x=y$$) then the so defined algebraic space is a scheme
in the usual sense. Why?

2. Where I can find a proof/ reason that the constructions
I) and II) are indeed equivalent?

## ag.algebraic geometry – Galois representations and pro-étale Site

On a scheme, we can define the pro-étale site. This is an improvement over the étale site in that we can define the $$ell$$-adic cohomology as the sheaf cohomology of the constant sheaf $$underline{mathbb{Z}_ell }$$ instead of as a limit. We can also define $$mathbb{Z}_ell(1):=lim mu_{ell^n}in text{Ab}(X_{text{proét}})$$ (see Remark 5.2.4 in https://arxiv.org/pdf/1309.1198.pdf). Assume that $$X$$ is defined over some number field $$k$$. My question is if
$$H^n_{text{proét}}(X,mathbb{Z}_ell(i))cong H^i_{text{ét}}(X,mathbb{Z}_ell(i))$$
as Galois modules.

## ag.algebraic geometry – Is the determinantal ideal of the span of a linearly independent set of rank-one matrices radical?

Let $$k$$ be an algebraically closed field, and let $$X_1,dots, X_n in M_m(k)$$ be rank-one matrices that are linearly independent over $$k$$. For a fixed integer $$1 leq r leq m$$, consider the ideal $$I subseteq k(t_1,dots, t_n)$$ generated by the $$r times r$$ minors of $$t_1 X_1+dots+t_n X_n.$$ Is $$I=sqrt{I}$$?

If we drop the assumption that $$X_1,dots, X_n$$ are rank one, then this is false. Indeed, let $$n=m=r=2$$, $$X_1=begin{bmatrix} 1 & 1\ 0 & 1 end{bmatrix}$$, and $$X_2=begin{bmatrix} 1 & 0\ 0 & 1 end{bmatrix}$$. Then $$I=langle det(t_1 X_1+t_2 X_2)rangle=langle(t_1+t_2)^2rangle$$ is not radical.

## ag.algebraic geometry – Algebraic stack of fixed points and higher stacks

In this paper, M. Romagny defines for an action of a group scheme $$G$$ on a stack $$X$$ the fixed point stacks $$X^G$$ associated
to the group action on a stack and in Theorem 3.3 he proves that if

1. the group $$G$$ is proper and flat of finite representation
2. $$X$$ is a Deligne-Mumford stack

then $$X^G$$ is algebraic. Later in this note, he proves that condition 2. can be relaxed to $$X$$ being algebraic with the diagonal being locally of finite presentation.

I am mostly interested in actions by complex tori on algebraic stacks locally of finite type. In this case, one doesn’t have the properness from condition 1. Is it still true that the stack of fixed points $$X^G$$ is algebraic?

Another question that I am interested in, is what happens if one considers higher stacks and torus actions on them. Are there some results regarding this?

## ag.algebraic geometry – Monodromy of a non flat connection

I’m asking the following question (i couldn’t find a similar question in the litterature or on the web). Let $$mathcal{E}$$ be a vector bundle of rank $$n$$ over a complex manifold $$M$$ and $$D$$ a divisor. Let $$nabla$$ be a (non flat) meromorphic connection on it, with pole over $$D$$.

1. Although the connection is not flat, we can consider the locally constant sheaf $$mathcal{E}^nabla$$ over $$Mbackslash D$$ of flat sections, which is a subsheaf of $$mathcal{E}|_{Msetminus D}$$. Suppose the sections of $$mathcal{E}^nabla$$ have moderate growth around $$D$$ (for local basis of $$mathcal{E}$$). Then let $$mathcal{L}$$ the $$mathcal{O}_M$$-module defined by : $$mathcal{L}=(mathcal{O}_{Msetminus D} underset{mathcal{O}_M}{otimes} mathcal{E}^nabla) cap mathcal{E}$$
Can we say that it is equivalent to a lattice for $$nabla$$ in the sense that $$z_1 nabla_{frac{partial}{partial z_1}}(mathcal{L})subset mathcal{L}$$ ?
2. If 1., is true, suppose the following : for every $$x in D$$, there exist a local basis $$Y^x_1,ldots,Y^x_n$$ of $$mathcal{E}$$ in a neighbhorood of $$x$$ such that : $$nabla_{(Y^x_i)_i} = d + A$$ where the connection form $$A$$ has no logarithmic terms.

Can we say that $$nabla$$ has no monodromy in the sense that $$mathcal{E}^nabla$$ is a locally constant sheaf over $$M$$ ?

Thanks very much for trying to help me, best regards

Edit : In fact, i’m pretty sure that we can find a $$mathcal{O}_M$$-module $$mathcal{L}$$ which is loccally free, and is a submodule of $$mathcal{E}$$ such that $$nabla$$ is regular on $$mathcal{L}$$ : we know that a such module always exists by Riemann Hilbert on punctured disks , but since for $$(X,U) in mathcal{E}^nabla(U)$$, and for $$(Z,V) in mathcal{L}(V)$$ we have $$Z|_V = f X$$ with $$fin mathcal{O}_{Msetminus D}(U)$$ which has a moderate growth, and $$X$$ has moderate growth relatively to $$mathcal{E}$$, then $$Z$$ must have moderate growth relatively to $$mathcal{E}$$ so that $$mathcal{L}hookrightarrow mathcal{E}otimes mathcal{O}_M(-D)$$ Then we can consider $$mathcal{L}’=z_1^N mathcal{L}$$ for $$N$$ large enough for $$mathcal{L}’$$ to be in $$mathcal{E}$$ and it remains a lattice for $$nabla$$.

I think my problem is to know wether $$mathcal{E}/mathcal{L}$$ is locally free. If so, then i think it’s immediate that the monodromy can be read on the matrix form $$A$$ relatively to a (local) basis of $$mathcal{E}$$.