All Questions
8,187 questions with no upvoted or accepted answers
12
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0
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716
views
Lifting abelian varieties in (the closed fiber of) a fixed Neron model
Suppose that $R$ is a dvr with field of fractions $K$ and residue field $k$ and that $A_K$ is an abelian variety over $K$ with Neron model $A$ over $R$. Then the closed fiber $A_k$ is a smooth ...
- 1,609
12
votes
0
answers
851
views
Compact Symplectic Fano (strongly monotone) manfiolds
What are known examples of compact symplectic Fano manifolds, apart from those that come from algebraic geometry?
We define symplectic Fano manifold as a symplectic manifold $(M,w)$, such that
$[c_1(...
- 28.9k
12
votes
0
answers
582
views
Pencils with many completely decomposable fibers
Let $F= \frac{G}{H} : \mathbb P^n \to \mathbb P^1$ be a non-constant rational function ($G$ and $H$ homogenous polynomials of the same degree
in $\mathbb C^{n+1})$.
The fiber over $(\lambda:\mu) \in ...
- 8,828
11
votes
0
answers
183
views
Are algebras with rational structures dense in varieties of real Lie nilpotent algebras?
One says that a real nilpotent Lie algebra has $\mathbb Q$-structure if it has a basis with rational structure constants. It is well known that there are nilpotent Lie algebras without $\mathbb Q$-...
- 723
11
votes
0
answers
437
views
A rather strange algebra
Let $k$ be an algebraic closed field of zero characteristic and $X$ an affine smooth variety, with $A=\mathcal{O}(X)$ the algebra of regular functions and $\mathcal{V}$ the Lie algebra of vector ...
- 3,318
11
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0
answers
339
views
Interpretation of $H^3(\mathrm{Gal}(L/K),L^\times)$
During my work I came across the group $H^3(\mathrm{Gal}(L/K),L^\times)=H^3(L/K,L^\times)$ for certain (infinite) Galois extensions $L/K$ (for an arbitrary field $K$) and I wondered if there is an ...
- 61
11
votes
0
answers
374
views
Example of abelian variety over finite field which doesn't lift
What is an example of an abelian variety over a finite field $\mathbb{F}_p$ which doesn't lift to $\mathbb{Z}_p$? This question seems to hint that they should exist, but no example is given.
Note that ...
- 21.4k
11
votes
0
answers
375
views
Quasi-separated rigid-analytic space without a formal model?
Well, my question is slightly embarrassing. When learning rigid geometry (mostly from Bosch's book) I realized that I don't know the answer to the following basic question.
Question. Is there an ...
- 16.2k
11
votes
0
answers
885
views
Infinite-dimensional affine space in algebraic geometry and algebraic topology
In topology, there are of course many different infinite-dimensional topological vector spaces over $\mathbb R$ or $\mathbb C$. Luckily, in algebraic topology, one rarely needs to worry too much about ...
- 64k
11
votes
0
answers
188
views
Can geometric point counting detect prime powers?
Let's call a polynomial $P(x)\in \mathbb Z[x]$ realizable if there exists some polynomial-count scheme $X$ over $\mathbb Z$ that satisfies $|X(\mathbb F_q)|=P(q)$, for all prime powers $q$.
Suppose $n$...
- 85.6k
11
votes
0
answers
364
views
Is there yet an example of a non-negative convex polynomial that cannot be written as a sum-of-squares?
I have read that it remains an open question, whether an example can be constructed of a non-negative convex polynomial that cannot be written as a sum-of-squares. My reading includes the following ...
- 173
11
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0
answers
234
views
When is cohomology of a finitely presented dg-algebra computable?
Given a smooth affine variety $X$ defined over $\mathbb{Q}$, its singular cohomology is isomorphic to the algebraic de Rham cohomology, which is the cohomology of the complex $\Omega_X^0\to\Omega_X^1\...
- 3,772
11
votes
0
answers
875
views
Why diamonds are only defined in characteristic $p$?
I'm trying to read Scholze's article "Etale cohomology of diamonds" (arXiv link) and both in this article and in Berkeley notes, the diamonds are defined as sheaves on the category of ...
- 1,093
11
votes
1
answer
592
views
Degree inequality of a polynomial map distinguishing hyperplanes
Let $H_1, \ldots, H_n$ be $n$ linearly independent hyperplanes in $k^n$, for some arbitrary field $k$. Let $X = H_1 \cup H_2 \cup \cdots \cup H_n$. Is it true that if $F=(f_1, \ldots, f_n)$ is a ...
- 539
11
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0
answers
623
views
Does Merkurjev's argument help Voevodsky's program?
In the talk
Unimath - its present and its future, July 10, 2017. Video and slides of a talk, Isaac Newton Institute for Mathematical Sciences, Cambridge. (abstract)
Voevodsky mentioned that he was ...
- 35.5k
11
votes
0
answers
420
views
Good reduction of finite etale covers of abelian varieties
Let $R$ be a dvr (whose residue characteristic is zero if it helps) with fraction field $K$.
Let $A$ be an abelian variety over $K$ with good reduction over $R$. Let $X\to A$ be a finite etale ...
- 513
11
votes
0
answers
270
views
Non-isomorphic curves with isomorphic Jacobians
There are many examples of pairs of non-isomorphic curves of genus 2 or 3 whose Jacobians are isomorphic as (unpolarized) Abelian varieties, see e.g. this post and its answers. This is relatively easy ...
- 38k
11
votes
0
answers
745
views
When is fppf better than fpqc (and vice versa)?
Depending on a geometer's needs, they may use the Zariski/etale/syntomic/etc. topology on the spaces they consider. I know some settings where etale topology is better suited for the task than the ...
user145520
11
votes
0
answers
607
views
The virtual fundamental class as derived intersection
Say $X$ is a smooth projective variety and $\beta\in H_2(X)$ is a class. Then there is a finite-type proper scheme (or in general, stack) $SM : = \overline{\mathcal{M}}_{g,n}(X,\beta)$ of stable maps ...
- 7,962
11
votes
0
answers
1k
views
What are the uses of topoi in algebraic geometry today?
Since Grothendieck introduced them in the 1960s, topoi have found many applications in algebraic geometry, category theory, and logic. For instance, they appear in the development of étale and ...
- 11.8k
11
votes
0
answers
749
views
What is known at $\ell = p$ about realizing Jacquet-Langlands & local Langlands as the cohomology of Lubin-Tate space with level structure?
Background:
(Mostly my paraphrased interpretation of the introduction of Strauch's Deformation spaces of one-dimensional formal
modules and their cohomology, with additional details from Carayol's ...
- 3,539
11
votes
0
answers
344
views
A purely algebraic argument for existence of a section of a smooth projective morphism to the projective line
If I am reading this post correctly, any smooth projective $\mathbb{C}$-morphism of schemes $X\rightarrow \mathbb{P}^1$ admits a section. I am afraid of the topological argument presented there. Is ...
user142965
11
votes
0
answers
325
views
Examples and non-examples of Tannakian $\infty$-categories
My definition of a Tannakian $\infty$-category is taken from this paper ("Tannaka duality over ring spectra" by James Wallbridge).
(p. $53$, definition $7.9$) Let $R$ be an $E_{\infty}$-...
- 439
11
votes
0
answers
241
views
Grothendieck-Teichmüller conjecture and tropicalization of moduli of curves
Abramovich, Caporaso and Payne (2014) have constructed functorial tropicalization maps from the Berkovich analytification of the moduli spaces of stable curves, $\overline{M}_{g,n}$, to the moduli ...
- 111
11
votes
0
answers
201
views
Holomorphically convex manifolds and Bergman complete manifolds
Suppose $X$ is a complex manifold which admits the Bergman metric (for definitions, see for instance Kobayashi's book "Hyperbolic Complex Spaces"). Suppose moreover that the Bergman metric of $X$ is ...
- 7,902
11
votes
0
answers
252
views
Can the orbit with respect to a rotation on the torus hit an algebraic variety infinitely often?
Question: Does there exist a $d$-tuple $\alpha = (\alpha_1,\dots,\alpha_d) \in \mathbb{R}^d$ (with $1,\alpha_1, \dots,\alpha_d$ linearly independent over $\mathbb{Q}$) and an algebraic variety $V \...
- 1,914
11
votes
0
answers
605
views
Fourier transforms and nontrivial vector bundles
We know that in arithmetic, geometry and analysis, Fourier transforms of various forms show up. For example, we have the classical Fourier transform, Fourier-Mukai transforms in the setting of ...
- 281
11
votes
0
answers
600
views
The Grassmannian Gr(2,8) and an E7 surprise
Are there any mathematical explanations for the following surprising facts?
$$\int_{Gr(2,8)} c_{\text{top}}(TX(-2)) = 6556 = \frac{1}{2} \deg(E_7/P(\alpha_7)) + 1,$$
and
$$\int_{Gr(2,6)} c_{\text{top}}...
- 1,298
11
votes
0
answers
227
views
Matrices that admit a power that is symmetric
We fix an integer $n\geq 2$. Let $S_n$ be the set of real symmetric matrices in $M_n(\mathbb{R})$. We consider the algebraic sets $Y_k=\{A\in M_n(\mathbb{R});A^k\in S_n\},k\geq 2$ and the sequence $...
- 3,741
11
votes
0
answers
474
views
Is the $\hat{A}$-genus invariant under crepant birational maps between smooth algebraic varieties?
The degree of the Todd class of the tangent bundle of a variety $Y$ gives the holomorphic genus of $Y$, which is a birational invariant. We also know that the $\hat{A}$-roof of a smooth variety $Y$ ...
- 3,022
11
votes
0
answers
480
views
Sheaf-theoretic Grothendieck groups
Let $S$ be a scheme, $M\to S$ a commutative monoid object in algebraic $S$-spaces, ie. an algebraic $S$-space such that, functorially on $S$-schemes $T$, $M(T)$ is a commutative monoid with neutral ...
user113393
11
votes
1
answer
900
views
Cohomological bounds for scalar curvature of an extremal Kähler metric
There is an interesting trick used in Chen-LeBrun-Weber's paper on the extremal Kähler metrics of $\mathbb{CP}^2\#2\overline{\mathbb{CP}^2}$, and I would like to know whether it can be (has been?) ...
- 3,212
11
votes
0
answers
246
views
Reference for $E_{\infty}$ algebra structure on cohomology of a site
Let $A$ be a sheaf of commutative rings on a site $X$. (In my applications, $X$ comes from one of the standard Grothendieck topologies on algebraic varieties.) It should be true that $R\Gamma (X, A)$ ...
- 2,809
11
votes
0
answers
407
views
Algebraizability of formal schemes
Let $R$ be a complete DVR with quotient field $K$ and let $f:\mathfrak{X} \to \mathrm{Spf}(R)$ be a smooth proper formal scheme.
If the (rigid analytic) generic fibre of $f$ is (the analytification ...
- 10.5k
11
votes
0
answers
656
views
What is known about mapping class groups of 4-manifolds?
I am mostly interested in the case when you have a smooth degree $d$ algebraic surface $X$ over $\mathbb C$ and we can define three distinct groups: $\pi_0(\mathrm{Diff}^+(X))$, $\pi_0(\mathrm{Homeo}^+...
- 765
11
votes
0
answers
450
views
K-stability is invariant under D-equivalency
Kawamata conjectured that
Let $X$ and $Y$ be birationally equivalent smooth
projective varieties. Then the following are equivalent. We denote by
$D^b(Coh(X))$ the derived category of bounded ...
user21574
11
votes
0
answers
295
views
Computing $h^1$ of dual of graph of central fibre of the degeneration of Kaehler-Einstein manifolds
Consider a Kaehler degeneration $\mathcal X\to \Delta$ of smooth manifolds: Here $\Delta$ is the unit disc, $\pi$ a proper flat map, smooth over $\Delta^∗=\Delta−\{0\}$. The general fibres are $X_t=\...
- 415
11
votes
0
answers
324
views
Why is the CM-type preserved after base changing from char 0 to char p?
There is a transition in the theory of complex multiplication which seems to be glossed over in all expositions I can find. I would like to explicitly find a theorem that allows me to do this.
...
- 3,539
11
votes
0
answers
532
views
Third cohomology of symplectic $6$-manifolds
Suppose that $A$ is a finitely generated abelian group with even rank and without $2$-torsion. Does there exist a compact symplectic $6$-manifold $(M,\omega)$ such that $A$ Is isormorphic to $H^{3}(M,\...
- 6,995
11
votes
0
answers
288
views
Generalization of Dwork's Derivation of the Picard-Fuchs equation
Background:
Let $V_\lambda$ be the elliptic curve $x^3+y^3+z^3 - \lambda xyz=0$. Then, when we consider $\omega_\lambda \in H^1_{dR}(V_\lambda)$, since $H^1_{dR}(V_\lambda)$ is only 2-dimensional, ...
- 3,539
11
votes
0
answers
358
views
3264 rational conics
It's a classical fact that there are 3264 plane conics tangent to 5 general conics, over $\mathbb{C}$ [1]. It was also shown that the 3264 can be defined over the reals [2] or [3]. More precisely, ...
- 271
11
votes
0
answers
552
views
The intrinsic meaning of abelian sheaf cohomology of a category
Basically my question is:
Suppose I meet an alien mathematician which understands everything through category theory and category theory alone. How would I convince said mathematician that certain ...
- 7,799
11
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0
answers
451
views
Semistability of tensor products under automorphisms of tensored vector spaces
Let $A,B,C,D,E,F$ be vector spaces over a field.
Let $x\in A \otimes B \otimes C$ and $y \in D \otimes E \otimes F$ be tensors that are semistable with respect to the natural actions of $\text{SL}(A) ...
- 149k
11
votes
0
answers
411
views
Lazard's theorem and Hopf structures on the polynomial algebra
Let $k$ be an algebraically closed field of characteristic $0$. A well-known result of Lazard's states that an algebraic group which is isomorphic as a variety to an affine space is unipotent (M. ...
- 426
11
votes
0
answers
372
views
MU and the integral Hodge Conjecture
Totaro proved that the cycle class map for an algebraic variety factors through complex cobordism. In other words, we have a factorization
$$CH^i(X) \rightarrow MU^{2i}(X) \rightarrow H^{2i}(X; \...
- 2,809
11
votes
0
answers
726
views
Are Kähler differentials the same as one-forms for compact manifolds?
Let $M$ be a manifold and let $A = \mathcal{C}^\infty(M)$ be the ring of smooth real-valued functions.
An old posting asks about the relationship of Kähler differentials and ordinary differential ...
user13113
11
votes
0
answers
310
views
Surfaces with $q=2$ and generically finite Albanese map
I have a family of surfaces of general type $S$ with $q(S)=2$, and such that the Albanese map $$\alpha \colon S \longrightarrow A:=\mathrm{Alb}(S)$$ is generically finite of degree $n$. By a result of ...
- 66.3k
11
votes
0
answers
524
views
Singular curve on an abelian surface
Let $C_2$ be a smooth genus $2$ curve and $J(C_2)$ its Jacobian. It is well known that the blow-up of $J(C_2)$ at the origin $o$ is isomorphic to the second symmetric product $\textrm{Sym}^2(C_2)$, ...
- 66.3k
11
votes
0
answers
2k
views
Is it worth the efforts to read books/papers written in Weil's algebraic geometry language
There is much important work written in Weil's language of algebraic geometry rather than schemes (besides Weil himself, I can think of Shimura, Neron immediately).
My question is: is it worth the ...
- 111
11
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0
answers
606
views
Letters of a bi-rationalist
V.V. Shokurov has written several papers over the course of about 10 years which are called "Letters of a bi-rationalist". Here are the ones that I could find:
Letters of a bi-rationalist. I. A ...
- 6,871