ag.algebraic geometry – Purely inseparable extension of curves is universally homeomorphism

Let $f: Xrightarrow Y$ be a morphism of integral scheme such that the corresponding extension of function fields is purely inseparable, then $f$ is universally homeomorphism.

Could someone give a reference or proof of the above fact?

In fact, I am only concern smooth projective curves, I want to prove a non-constant map of curves is bijective if the extension of function fields in purely inseparable. I also want to ask if someone could give me a proof without using the theory of scheme, I have no idea with this.

Thanks very much.

ag.algebraic geometry – A problem about deformation of schemes

Let $X$ be a scheme over a field $k$ and let $Z$ be a closed subscheme of $X$. Then the morphism of cotangent complexes $L_{Z/k} to L_{Z/X}$ in $D(mathcal O_Z)$ induces a homomorphism
$$mathop{mathrm{Hom}}nolimits_{mathcal O_Z}(mathcal I_Z/mathcal I_Z^2, mathcal O_Z) = mathop{mathrm{Ext}}nolimits_{mathcal O_Z}^1(L_{Z/X}, mathcal O_Z) to mathop{mathrm{Ext}}nolimits_{mathcal O_Z}^1(L_{Z/k}, mathcal O_Z),$$
which can be identified with the induced homomorphism of tangent spaces of deformation categories $mathscr{Hilb}_{Z/X/k} to mathscr{Def}_Z$ over $(mathsf{Art}/k)^mathrm{op}$.

There is a natural homomorphism
$$mathop{mathrm{Ext}}nolimits_{mathcal O_Z}^1(L_{Z/k}, mathcal O_Z) to mathop{mathrm{Hom}}nolimits_{mathcal O_Z}(H^{-1}(L_{Z/k}), mathcal O_Z). $$
What is the group $mathop{mathrm{Hom}}nolimits_{mathcal O_Z}(H^{-1}(L_{Z/k}), mathcal O_Z)$, I heard that if $Z$ is a curve, then it characterize the deformation of the closed subscheme of singular points $D$ of $Z$, so is it true that
$$mathop{mathrm{Hom}}nolimits_{mathcal O_Z}(H^{-1}(L_{Z/k}), mathcal O_Z) cong mathop{mathrm{Ext}}nolimits_{mathcal O_D}^1(L_{D/k}, mathcal O_D)?$$
And what if $Z$ is a general scheme.

ag.algebraic geometry – Metric of negative holomorphic sectional curvature

Let $X$ be a Kähler manifold which admits a Hermitian metric of negative holomorphic sectional curvature. Does $X$ admit a Kähler metric with negative holomorphic sectional curvature?

This question is motivated by an old paper of Klembeck’s (Geodesic convexity and plurisubharmonicity). In this paper, he shows that the existence of a Hermitian metric on an open Hermitian manifold $X$ with negative Riemannian sectional curvature does not guarantee $X$ is Stein. This issue is circumvented if the metric is Kähler.

ag.algebraic geometry – Picard group of connected linear algebraic group

Here’s a statement:

Suppose $G$ is a connected linear algebraic group over a field $k$, then $Pic(G)$ is a finite group.

I know this is true when $k=mathbb{C}$. My question is does this true for abitrary field $k$? If not, how about furthermore when $G$ is smooth or even reductive? Is there any reference?

Thanks for any help.

ag.algebraic geometry – What is $f^*TX$ for a general morphism $fcolonmathbb{P}^1to X$?

Let $X$ be a projective homogeneous space over $mathbb{C}$, i.e. $G/P$ where $G$ is a simple, simply connected linear algebraic group and $P$ is a parabolic subgroup. Let $fcolonmathbb{P}^1to X$ be a general morphism.

Q. What does $f^*TX$ look like?

By Grothendieck’s theorem, $f^*TX$ splits as a sum $bigoplus_imathcal{O}(a_i)$. How to determine the numbers $a_i$?

A general morphism $f$ as above has obviously image an $mathrm{SL}_2(x)$-orbit, where $x$ is the sum of some root vectors $sum_ix_{theta_i}$. It definitely helps to know this, but I don’t see the solution clearly.

Presumably, the solution is as follows: Let $d$ be the degree of $f$. Then, $f^*TX=bigoplus_{alphain R^-setminus R_P^-}mathcal{O}(d,alpha)$. How to prove this?

Here, $R^-setminus R_P^-$ denotes the roots in $mathfrak{g}/mathfrak{p}$.

ag.algebraic geometry – Zero schemes of sections of locally free sheaves

Let $X$ be a scheme. Let $mathcal E$ be a locally free sheaf of rank $r$ on $X$ and let $s$ be a section of $mathcal E$. Then the zero scheme of $s$ is defined as follows: Consider the homomorphism $mathcal O_X to mathcal E$ induced by $s$, taking duals, we obtain $mathcal E^vee to mathcal O_X$. Then $Z(s)$ is defined to be the scheme associated to the sheaf of ideals $mathop{mathrm{im}}(mathcal E^vee to mathcal O_X)$.

First question: is there some natural conditions of regularity defined on $s$. For example, when $mathcal E = mathcal L$ is an invertible $mathcal O_X$-module, then $s$ is said to be regular if and only if the induced homomorphism $mathcal O_X to mathcal L$ is injective, and in this case $Z(s)$ is an effective Cartier divisor on $X$. So I think the condition of regularity should satisfy that if $s$ is regular, then any generic point of irreducible components of $Z(s)$ has codimension $r$ in $X$.

Now suppose we have already defined some conditions of regularity. Assume now that $X$ is a smooth projective variety and let $s$ be a “regular” section. Consider the $r$-cycle associated to $Z(s)$. Prove the following statement: The class of the $r$-cycle associated to $Z(s)$ in the $r$-th Chow group $mathop{mathrm{CH}}^r(X)$ equals the $r$-th Chern class $c_r(mathcal E)$ of $mathcal E$.

Hence the linear equivalence class of $Z(s)$ is independent of the choice of $s$ and we get a well-defined map
$${text{locally free sheaves of rank } r text{ on } X} / {text{isomorphisms}} to mathop{mathrm{CH}}nolimits^r(X),$$
is this map bijective? (When $r = 1$, we obtain an isomorphism $mathop{mathrm{Pic}}(X) to mathop{mathrm{Cl}}(X)$.)

If the map above is bijective, then the group structure on $mathop{mathrm{CH}}nolimits^r(X)$ should induces a group structure on ${text{locally free sheaves of rank } r text{ on } X} / {text{isomorphisms}}$, and what is it? (When $r = 1$, it is tensor products of invertible $mathcal O_X$-modules. However, we cannot simply take the tensor product of two locally free sheaves of rank $r$).

ag.algebraic geometry – $L^r_M = i_* circ hat{L}^{r-1}_M circ i^*$ by the projection formula and the Poincare duality

This is a question arising when I am reading

M. A. A. de Cataldo, L. Migliorini – The Hard Lefschetz Theorem and the topology of semismall maps, Ann. sci. École Norm. Sup., Serie 4 35 (2002) 759-772.

I give a reduction of the semantics. Let $M$ be a line bundle (in fact it is a lef line bundle), $Y$ is the zero locus of a non zero section of $M$. Assume the Chern class of $M_{vert Y}$ satisfies the condition of the Hard Lefschetz. What we want to prove is that
$$L^r_M = i_* circ hat{L}^{r-1}_M circ i^*,$$
where $L$ means wedge product with a Chern class and $hat{L}_M = L_{M_{vert Y}}$.

I could give a proof my self. Let $Y = {f_1 = 0}$ and $f_2$ is another section of $M$ traversing $Y$. Note that $c_M = (sqrt{-1}partial bar{partial} f_1 ) = (sqrt{-1}partial bar{partial} f_2 )$ and $c_{M_{vert Y}} = (sqrt{-1}partial bar{partial} f_2circ i )= i^*(sqrt{-1}partial bar{partial} f_2 ) = i*c_M$. So
$$i_* circ hat{L}^{r-1}_M circ i^* omega = i_* i^*({c_M^{r-1} wedge omega}).$$
It remains to show that $i_*i^* = L_M$. This can be seen by $<i_*i^* alpha , beta >_X = <i^* alpha , i^* beta>_Y = <i^* (alpha wedge bar{ beta}) , 1>_Y = <alpha wedge bar{ beta}, c_M>_X = <alpha, beta wedge c_M>_X =<alpha wedge c_M, beta>_X$. So $i_*i^* = L_M$.

However, in the article it is said that this is proved by the projection formula and the Poincaré duality. I don’t understand how it works and some explanation could help a lot, not only on this question but also the understanding of the cohomology theory. It’s more appreciated if you can work on derived categories.

ag.algebraic geometry – Todd polynomials

Let $T_k(x_1,ldots,x_n)$ be the Todd polynomials, $e_k(x_1,ldots,x_n)$ the elementary symmetric polynomials and $p_k(x_1,ldots, x_n)$ the power sums of degree $k$.

We have the following generating formulas
begin{align*}
sum_{kgeq 0}T_k(x_1,ldots,x_n)t^k = prod_{i=1}^nfrac{tx_i}{1-e^{-tx^i}},,\
sum_{kgeq 0}e_k(x_1,ldots,x_n)t^k = prod_{i=1}^n(1+tx_i),,\
sum_{kgeq 0}frac{1}{k!}p_k(x_1,ldots,x_n)t^k = sum_{i=1}^ne^{tx_i},.
end{align*}

There is an explicit relation between $(1/k!)p_k$ and $e_k$ in terms of Newton’s identities which can be expressed using generating series (see e.g. wikipedia). Is there some similar expression for $T_k$ in terms of $e_k$ or $p_k$?

For example if $X$ is a hyperkähler complex manifold. Replacing $x_i$ with $alpha_i$ the roots of $c(TX)$, one can show that (see (3.13)):
$$
td(X) = text{exp}Big(-2sum_{ngeq0} b_{2n}text{ch}_{2n}(TX)Big),,
$$

where $b_{2n}$ are the modified Bernoulli numbers. Is there possibly even a less explicit formula without any assumptions on the geometry?

ag.algebraic geometry – Tangent bundle of Lagrangian Grassmannians

Consider the Lagrangian Grassmannian $LG(r,2r)$ of $r$-dimensional isotropic subspaces of a vector space of dimesnion $2r$ and let $T_{LG(r,2r)}$ be its tangent bundle.

Does there exist a closed formula for the Chern classes of $T_{LG(r,2r)}$ in terms of $r$ and the Schubert cycles generating the cohomology of $LG(r,2r)$?

ag.algebraic geometry – Do connected algebraic stacks have a smooth cover by a connected scheme?

An algebraic stack $X$ has an induced topological space $|X|$ given by equivalence classes of fields mapping to $X$ as outlined in the stacks project. If $|X|$ is connected, does that imply there exists a smooth map $V to X$ from a connected scheme $V$?

My naive guess is to take an atlas $V to X$ and try to glue it along its intersection, piecing together disjoint components of $V$, or perhaps take a colimit along inclusions over connected, smooth $V to X$.

A positive answer to this question would simplify a technical argument.