ag.algebraic geometry – Vanishing of intermediate cohomology for a multiple of a divisor

Let $S subset mathbb P^3$ be a smooth projective surface (over complex numbers). Let $C$ be a smooth hyperplane section. Let $Delta$ be a non-zero effective divisor on $S$ such that $h^1(mathcal O_S(nC+Delta))=0, h^1(mathcal O_S(nC-Delta))=0$ for all $n in mathbb Z$. Then my question is the following :

In this situation can we say that: $h^1(mathcal O_S(m Delta))=0$ for $m geq 2$? Can we impose any condition so that this happens?

Any help from anyone is welcome.

dg.differential geometry – Is there a reasonable definition of an Octonionic manifold?

Todorov and Dubous-Violette have recently shown how to understand the structural gauge group of the standard model via octonions.

Q. Is there a octonionic analog of complex manifolds?

There is a sensible quaternionic analog. This is a manifold $M$ with almost quaternionic structure and this in turn means that it has a $GL(n,H).H^times$ G-structure $E$. Here the dot represents the central product which divides out the direct product by their mutual centre. This is equivalently a subbundle of $End TM$ with fibre isomorphic as an algebra to $H$, the quaternionic algebra.

Then a quaternionic manifold is an almost quaternionic manifold that admits a torsion free connection. The bundle $E$ admits a bundle metric and splits into the direct sum of a line bundle $L$ and a vector bundle $E’$. This is akin to the splitting of quaternions into scalar and imaginary parts. Sections of the sphere bundle $S(E’)$ consists of almost complex structures and then choosing a basis $I,J,K$ we have $I^2 = J^2 = K^2 = IJK = -1$.

ag.algebraic geometry – The classifying stack of $PGL(2)$ and the moduli space of genus zero curves

The classifying stack of $PGL(2)$ is the stack quotient $(operatorname{Spec} k/PGL(2))$ where $PGL(2)$ acts trivially on $operatorname{Spec} k$.

Since $(operatorname{Spec} k/PGL(2))$ is a quotient stack, it is an algebraic stack with smooth covering map $q: operatorname{Spec} k to (operatorname{Spec} k/PGL(2))$ defined by the pair $(PGL(2), rho)$ where $rho: PGL(2) times operatorname{Spec} k to operatorname{Spec} k $ is the action map.

On the other hand, $(operatorname{Spec} k/PGL(2))$ is also the algebraic moduli stack of genus-zero curves. Since any map $operatorname{Spec} k to (operatorname{Spec} k/PGL(2))$ is identified by Yoneda’s lemma with an object in the fiber over $(operatorname{Spec} k/PGL(2))(operatorname{Spec} k)$ a smooth covering map $q’:operatorname{Spec} k to (operatorname{Spec} k/PGL(2))$ could just as well have been defined by $mathbb{P}^1$.

How can I recover the map $q$ defined by $(PGL(2), rho)$ from the smooth covering map $q’$ defined by $mathbb{P}^1$?

I know that a map $operatorname{Spec} k to (operatorname{Spec} k/PGL(2))$ should correspond to more than just an object in
$(operatorname{Spec} k/PGL(2))(operatorname{Spec} k)$—maybe an object plus automorphism of that object.

sg.symplectic geometry – Upper triangular similitude for symplectic matrices

It is known that given any matrix $M$ in $Sp(2,mathbb{Z})$ with eigenvalue $+1$, we can find a real symplectic matrix $S$ such that $S^{-1}MS$ is upper triangular with diagonal entries equal to $+1$.

In the case of $Sp(2n,mathbb{Z})$, do we have a similar result ? Any reference or help would be appreciated. Thanks.

riemannian geometry – Does There Exist and Isometry between a Regular Polygon and a Circle

In order to define the question in a meaningful fashion, I am referring to a smooth manifold $mathcal{M}$ within an $epsilon$-neighborhood of a regular polygon $mathcal{P}$ satisfying $max{||x-p||: x in mathcal{M} cap B_{delta}, p in mathcal{P} cap B_{delta}} < epsilon$ for a circular $delta$-neighborhood $B_{delta}$. I have already proved that $mathcal{M}$ is diffeomorphic to the unit circle $S_1$; however, I now wish to examine the possibility of an isometry $I: mathcal{M} to S_1$ where the metric on $mathcal{M}$ is simply taken as a fractional perimeter instead of any ambient distance.

I believe that the existence of $I$ is impossible as even if I were to produce a conformal map, I do not consider any map to preserve the area between $mathcal{M}$ and $S_2$. As local areas were not preserved w.r.t the metric, the notion of magnitude for a vector $v in mathcal{T}_p(mathcal{M})$ would not be preserved. Finally, the first fundamental form would not be preserved, so no isometry could exist.

Do you all have suggestions as to how I may prove or disprove the existence of such an isometry $I: mathcal{M} to S_1$.

dg.differential geometry – Existence parallel vector fields and its effect on the topology of manifolds (Karp’s Thesis)

It seems that there is no digital copy of Leon Karp’s Ph.D. thesis

L. Karp, Vector fields on manifolds, Thesis, New York Univ., 1976.

on internet and his paper excerpted from his thesis is very brief and without any detailed proof. (I wonder that peers read the thesis or they trust to the advisory committee).

Karp, Leon, Parallel vector fields and the topology of manifolds, Bull. Am. Math. Soc. 83, 1051-1053 (1977). ZBL0376.53024.

There he generalized a theorem of S. S. Chern and proved that

Theorem. If $M^n$
admits a vector field that is parallel with respect to
some Riemannian metric then the Betti numbers of $M$ satisfy $b_{k+1}geq b_k-b_{k-1}$ and $b_1geq 1$.

There is also no review on zbMATH. I want to know about its sketched proof; and similar results on bivectors if exist any.

dg.differential geometry – Uniform convergence of Eigenfunction decomposition on Riemannian sphere?

Let ${u_k}_{k=1}^infty$ be a sequence of ($L^2$ normalized) mutually orthogonal eigenfunctions of the operator $-Delta$ on the sphere $mathbb{S}^n$ (here $Delta$ is the Laplace Beltrami operator). Let $u$ be a smooth (real valued) function on the sphere. It is a well-known result that we can write $u=sum_{k=1}^infty c_k u_k$ for some (real) constants $c_k$. My question is: Is the convergence of this sum uniform?

I am trying to prove that the optimal constant in the Poincare inequality is $lambda_1=n$. That is to say, I am trying to prove the inequlity $int_{mathbb{S}^n} |nabla u|^2 geq n int_{mathbb{S}^n} |u|^2$. Here is what I have done so far:

First, integrate by parts on the LHS so that it suffices to prove $int_{mathbb{S}^n} -uDelta u geq n int_{mathbb{S}^n} |u|^2$. Then use $u=sum_{k=1}^infty c_k u_k$ and assume that the convergence is uniform. Then we can switch the order of the sum with the derivative and integral (and use the fact that ${u_k}$ are orthonomal) so that
begin{align*}
int_{mathbb{S}^n} -left(sum_{k=1}^infty c_k u_kright)Delta left(sum_{j=1}^infty c_j u_jright)&=
int_{mathbb{S}^n} -left(sum_{k=1}^infty c_k u_kright)left(sum_{j=1}^infty c_j Delta u_jright)=
int_{mathbb{S}^n} -left(sum_{k=1}^infty c_k u_kright)left(sum_{j=1}^infty lambda_j c_j u_jright)
\&= sum_{j,k}c_k c_j lambda_jint_{mathbb{S}^n} u_k u_j= sum_{j,k}c_k c_j lambda_j delta_{jk}=sum_{j}c_j^2 lambda_jgeq lambda_1 sum_j c_j^2.
end{align*}

By the same logic, the last sum is equal to $int |u|^2$.

Now obviously, this proof requires some argument showing that the sum commutes with $Delta$ and the integral but I have not been able to find a reference that the sum converges uniformly. My thought is that this would follow from some basic facts in Harmonic analysis though I am no expert in that field. Would anyone be able to provide a reference for this?

dg.differential geometry – Adjunction formula for non compact surfaces

Let M be a non compact complex surface and S an embedded compact Riemann surface in M.

I already know how to show the following equality of fiber bundle:

$$Omega^2_{M}|S =Omega^1_S otimes N^*_S$$
where $Omega^2_{M}|S$ is the canonical vector bundle of $M$ restricted to $S$ and $Omega^1_S$ is the canonical vector bundle of $S$ and $N^*_S$

is the dual of the normal vector bundle of $S$ over $M$.

Question : is it possible to deduce from this formula the following :
$$K_M.S + S^2 = 2g-2$$
in term of intersection multiplicity of divisors and 𝑔

denotes the genus of S.

i konwI know this formula is correct when M𝑀

is compact and it is known as the adjunction formula but iI don’t know for the non compact-compact case.

thanksThanks for your help.

algebraic geometry – Zariski site is subcanonical?

I want to show that the zariski site is subcanonical as an exersice of the book “Sheaves in geometry and logic”
and I need help with it…
To be honest I didnot really understand the definition of what a zariski site exactly is…
I tried to take the structure sheaf $O=Hom(A,-)$(where $A$ is an $k-algebra$) which is equivalent to $Hom(-,V(A))$ as an arbitrary pre-sheaf and and claim it is actually a sheaf…
I took $f-1,…f_nin A$ such that $A=<f_1,…,f_n>$
And then I defined a matching family $alpha _i : V_A_f_i rightarrow V_A$ and I draw the pull back diagram consisting of $V(A_f_i)rightarrow V(A)$ , $V(A_f_j)rightarrow V(A)$,$V(A_{f_i,f_j})rightarrow V(A_f_i)$ and $V(A_{f_i,f_j})rightarrow V(A_f_j)$ ……

Please help….