## 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 $$_X = _Y = _Y = _X = _X =_X$$. 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.