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Apr
30
comment Families of abelian varieties on the line (or more generally simply connected varieties)
NB I heard you like Serre-Tate, so you might appreciate this little fact as well: Nori and Srinivas in "Varieties in positive characteristic with trivial tangent bundle" prove that in characteristic p, every family of ordinary abelian varieties over a smooth projective curve becomes trivial after a finite etale cover. In particular, every family of ordinary abelian varieties over a proper variety (no assumptions on $\pi_1$) is isotrivial.
Apr
29
comment Families of abelian varieties on the line (or more generally simply connected varieties)
J. Kollár, "Shafarevich maps and automorphic forms", MR1341589 ams.org/mathscinet-getitem?mr=1341589
Apr
29
comment Families of abelian varieties on the line (or more generally simply connected varieties)
There is a notion, introduced I think by Kollar, of "large fundamental group". A variety has large $\pi_1$ if the image of the $\pi_1$ of every subvariety is infinite. He states a conjecture (attributed to Shafarevich) that a variety has large (topological) fundamental group iff the universal cover is a Stein space. I think that this conjecture should imply (modulo stacky issues) that the answer to Q1 (and maybe Q2, by looking at the corresponding period domain) is affirmative.
Apr
23
comment Number of rational points in a non-smooth variety
You can assume $X$ reduced, and then $X$ has a smooth dense open $U$. Let $Z=X\setminus U$, which has dimension $\leq n-1$. If you accept the asymptotic for smooth varieties, by induction on $n$ you get that $\# X(\mathbb{F}_{q^k}) = \# U(\mathbb{F}_{q^k}) + \# Z(\mathbb{F}_{q^k}) = (1+o(1))q^{nk} + O(q^{(n-1)k}) = (1+o(1)) q^{nk}$.
Apr
19
comment Complete the following sequence: point, triangle, octahedron, . . . in a dg-category
The one which is the convex hull of the midpoints of the edges of the $(n+1)$-simplex (cf. Gelfand–Manin, reference 1 above). I guess that should be $\Delta_{n+1, 2}$?
Apr
19
answered Complete the following sequence: point, triangle, octahedron, . . . in a dg-category
Apr
8
comment Canonical metric on moduli space of singular Calabi-Yau varieties
Could you please clarify what you mean by 'degeneration of Calabi-Yau varieties with singular fibers'?
Mar
16
reviewed Approve Where can square roots come from when they are not distances?
Mar
14
answered How to stop worrying about enriched categories?
Mar
12
revised Lifting of Frobenius on semi-abelian varieties
fixed formatting
Mar
12
answered Lifting of Frobenius on semi-abelian varieties
Mar
12
comment Lifting of Frobenius on semi-abelian varieties
What do you mean by `$A$ lifts to $W(k)$'? Do you want the lift to be a scheme or a formal scheme?
Mar
12
revised Lifting of Frobenius on torsors over abelian varieties
added a diagram
Mar
12
revised Lifting of Frobenius on torsors over abelian varieties
added 37 characters in body
Mar
12
answered Lifting of Frobenius on torsors over abelian varieties
Mar
6
reviewed Approve Asymptotic behavior of Sturm-Liouville eigenvalues
Mar
5
comment When does an $E_\infty$ algebra come from a commutative differential graded algebra?
@DenisNardin Above I assume that the $K^i$ are flat, so $H^*(K\otimes \mathbb{F}_p) = H^*(K)\otimes \mathbb{F}_p$.
Mar
5
comment Exterior power of a torsion-free sheaf on a DVR
@Ron: I still don't see why it shouldn't be. Do you have an example where it isn't?
Mar
5
comment Is flatness preserved under exterior power
@Vinteuil you are right, I misunderstood the question. Perhaps it is still true that an $A$-flat $B$-module is a colimit of $A$-free $B$-modules, but it is unclear whether $\bigwedge^m_B$ of an $A$-free $B$-module is $A$-free...
Mar
5
comment Exterior power of a torsion-free sheaf on a DVR
Maybe I'm missing something. Isn't it always flat?