ag.algebraic geometry – Smooth projective variety with no second homotopy group

I am looking for an example (if such exist) of a smooth projective variety $X$ whose $mathbb{Q}$-homology is generated by algebraic cycles, and yet does not have a second homotopy group, $pi_2(X)=0.$ Thus, algebraic cycles that span $H_2(X,mathbb{Q})$ are coming from some non-rational curves.

Projective cellular varieties with singular cohomology admitting torsion different from two

Is there a simple enough example of a projective cellular variety $X$ (cellular decomposition by affine spaces) such that $H^*(X(mathbb{R}),mathbb{Z})$ has an element with torsion different from 2? (3,4,5, for example)

Why is every meromorphic function on the complex projective plane a rational function?

You can use the fact that every holomorphic function on the complex plane with a non essential singularity at $infty$ is a polynomial by considering $f(1/z)$ and matching up the taylor series to that and $f$. This gets you most of the way there but what is the last step?

Is quotient of projective scheme over arbitrary base by a finite group also projective

This question probably follows from standard geometric invariant theory. If true I’d to know a reference for it. Given a projective scheme $Xrightarrow S$ over the base $S$. Let’s assume a finite group $G$ is acting on $X$ and its quotient is an $S$-scheme $X//G$. Is the quotient projective or at least proper? (I have seen versions of this over fields but not for arbitrary base.)

gn.general topology – Is there a known topological embedding of the complex projective plane inside $S^5$?

Unless I am mistaken somewhere, I have stumbled on a smooth map from $mathrm{P}^2_{mathbb{C}}$ into $S^5$ which is one-to-one (i.e. injective). Thus it induces a topological embedding of the former into the latter. Is it known that such a topological embedding exists by the way? If so, could someone please point me to the relevant literature? (I am not claiming that my map is an immersion).

ct.category theory – Varieties where every algebra is projective?

Is it possible to classify all varieties (in the sense of universal algebra) where every algebra is projective?

Several years ago I asked a similar question, with “free” in place of “projective”. It turned out it had been answered by Steve Givant in his 1975 thesis, but a variant question I threw in as an afterthought was still open — to classify all varieties where every finitely-generated algebra is free. Keith Kearnes, Emil Kiss, and Agnes Szendrei were able to give a classification. In both cases (with or without the finite-generation condition in the hypothesis) the answer is the same: the only such varieties are sets, pointed sets, vector spaces over a division ring, and affine spaces over a division ring.

So now I’m feeling a little greedier, and want to relax freeness to projectiveness. I think that now the “finitely-generated” and “infinitely-generated” cases will diverge. For instance, every finitely-generated Boolean algebra is projective, but not every Boolean algebra is projective.

There are also at least two notions of projectivity to consider — projectivity means that an algebra lifts against all epimorphisms, whereas regular-projectivity means that it lifts against all regular epimorphisms (i.e. surjections).

Question 1: For which varieties is it the case that every algebra is projective?

Question 1′: For which varieties is it the case that every algebra is regular-projective?

Question 2: For which varieties is it the case that every finitely-generated algebra is projective?

Question 2′: For which varieties is it the case that every finitely-generated algebra is regular-projective?

Question 3: For which rings is the the case that every finitely-generated module is projective?

I’m pretty sure that a ring $R$ has all modules projective if and only if $R$ is a finite product of finite-dimensional matrix algebras over division rings by the Artin-Wedderburn theorem. Projectivity and regular-projectivity coincide in the abelian setting.

So the guess would be that for Question 1, the only varieties are finite products of those varieties where every algebra is free (i.e. of sets, pointed sets, algebras over a division ring, and affine spaces over a division ring) or varieties whose categories of algebras are equivalent to such (although perhaps a syntactic characterization of this condition is nontrivial?). For Question 2 and Question 3 I suspect there might be more interesting examples, which I’d like to hear about even if a classification is out of reach.

dg.differential geometry – Regarding projective manifolds with decomposable real tangent bundle

Let $X$ be a complex projective manifold. Suppose its real tangent bundle $T_{mathbb{R}}X$ splits as a direct sum of two sub-bundles of even rank. Does this give any useful information about the manifold $X$ (e.g. its Euler characteristic, etc.)?

I apologize for being somewhat vague about my question. Any help would be appreciable.

derived categories – Homotopically projective dg $A$-module and perfect $R$-complex

I am stuck on the Lemma 2.12.(2) in this paper.

Let $R$ be a commutative ring, $k$ be a field. Let $A$ be a dg $k$-algebra.

Denote by $Mod$$A$ the dg category of right dg $A$-modules.

$hproj$$A$ is defined as the full dg subcategory of $Mod$$A$ consists of h-projective dg modules.

Triangulated category $mathrm{Perf}(A)$ is defined as the full thick triangulated subcategory of $D(A)$ generated by the dg $A$-module $A$.

$Perf(A)$ is defined as the full dg category of $hproj$$A$ consisting of objects that belongs to $mathrm{Perf}(A)$.

An $R$-complex $F in D(R)$ is perfect if it is locally on Spec $R$ quasi-isomorphic to a finite complex of free $mathcal{O}_{mathrm{Spec} R}$-modules of finite rank.

A $R$-h-projective dg $R$-algebra is called compact if it is perfect as an $R$-complex.

Lemma 2.12. (1) Let $A$ be an $R$-h-projective dg $R$-algebra, then any object in $hproj$$A$ is also $R$-h-projective. Similarly, any object in $hproj$$left(A^{mathrm{op}} otimes_{R} Aright)$ is $A$-h-projective, $A^{text {op}}$-h-projective and $R$-h-projective.

(2) Assume in addition that $A$ is compact. Then $color{red}{②}$ any object in $Perf(A)$ is $R$-h-projective and $R$-perfect, hence is $color{red}{③}$ homotopy equivalent, as an $R$-complex, to a strict perfect complex. The same is true about $R$-complexes $operatorname{Hom}_{A}(M, N), operatorname{Hom}_{R}(M, R)$ and $M otimes_{A} P$ for $M, N in Perf(A)$ and $P in Perfleft(A^{text{op}}right)$.

Proof. (2) This is clear, $color{red}{①}$ as every object in $Perf(A)$ is homotopy equivalent to a direct summand
of a dg $A$-module that has a finite filtration with subquotients being shifts of $A$.

How to show $color{red}{①}$ and $color{red}{①} implies color{red}{②} implies color{red}{③}$?

Actually, $color{red}{②} implies color{red}{③}$ is covered by the following statement, but I can’t figure it out.

(paper, p.5) It is known that a perfect $R$-complex is quasi-isomorphic to a strict perfect $R$-complex, i.e. to a finite complex of finitely generated projective $R$-modules (TT90, Theorem 2.4.3). It follows that a perfect $R$-complex which is h-projective, is homotopy equivalent to a strict perfect $R$-complex.

Thank you very much.

rt.representation theory – length of projective cover/injective hull of a simple module is finite

Let $A$ be an Artinian algebra and let $l$ denote the length of a module. Let $S$ be a simple $A-$module.

Consider the projective Cover $P(S) rightarrow S$ and the injective hull $S rightarrow I(S)$. I would like to prove the following:

P(S) and I(S) are modules of finite length.

I unfortunately have no idea how to tackle this. Does anybody have a clue?

algebraic geometry – Equivalent Definitions of Projective Morphisms with Line Bundles

This is part b) of Vakil 17.3 A, self-study. We say a morphism $pi: X to Y$ is projective if $X$ is isomorphic as $Y$-schemes to the relative $operatorname{Proj}$ of some quasicoherent sheaf of algebras $mathcal L$ on $Y$ (such that $mathcal L$ is finitely generated in degree $1$).

In part a) of this question, we are asked to show that $pi$ is projective iff there is a finite type quasicoherent sheaf $mathcal L_1$ on $Y$ such that there exists a closed embedding of $Y$-schemes $X hookrightarrow mathbb P mathcal L_1$. I believe I have done this.

Part b) takes it a step further: given an invertible sheaf $mathcal L$ on $X$, we want to show $pi$ is projective with $mathcal O(1) simeq mathcal L$ iff there is a finite type quasicoherent sheaf $mathcal L_1$ on $Y$ such that there exists a closed embedding of $Y$-schemes $i: X hookrightarrow mathbb P mathcal L_1$ with $i^* mathcal O_{mathbb P mathcal L_1}(1) simeq mathcal L$.

I am only unable to show that given the latter, we get the isomorphism $mathcal O(1) simeq mathcal L$ of the former. The only observation I have been able to make is that if we could show $i^* mathcal O_{mathbb P mathcal L_1}(1) simeq mathcal O(1)$, we would be done. For example, we could show that $mathcal L$ is very ample and that $X$ is a proper $Y$ scheme, which we have not shown, and, in fact, not even make precise outside the context in which $Y$ is an affine scheme, so I don’t believe this is the intended route.