Donu Arapura
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Registered User
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May 16 |
answered | Hodge classes and Leray filtration |
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May 15 |
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Degree of finite morphisms of curves, and of their reduction “modulo p” You're welcome. |
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May 14 |
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Degree of finite morphisms of curves, and of their reduction “modulo p” If you identify the degree with the rank of $f_*\mathcal{O}_X$ as an $\mathcal{O}_Y$-module, then it is not difficult to see that your question(s) has (or have) a positive answer. |
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May 11 |
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Non-compact Riemann surfaces and affine algebraic curves (can't edit) constant -> nonconstant. |
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May 11 |
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Non-compact Riemann surfaces and affine algebraic curves In fact the disk is a simple counterexample. The point is that affine algebraic curves are the same thing as compact Riemann surfaces minus a finite set of points. In particular, they have no constant bounded holomorphic functions, where as the disk has plenty. |
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May 9 |
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Proof that the Hodge-de Rham Rank Equals the Euler Characteristic added 342 characters in body |
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May 9 |
answered | Proof that the Hodge-de Rham Rank Equals the Euler Characteristic |
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May 2 |
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Motivic cohomology and cohomology of Milnor K-theory sheaf There is a strange bug in the software. Sometimes putting backticks "`" around the math will allow it to display correctly |
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May 2 |
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Motivic cohomology and cohomology of Milnor K-theory sheaf added 3 characters in body |
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May 2 |
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$H^1(X,O_X)$, holomorphic $1$-forms, and $b_1(X)/2$ for normal $X$. Georges: this is a somewhat late response. I'd rather not elaborate here, but lack of time is certainly a big factor. |
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Apr 23 |
awarded | ● Enlightened |
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Apr 22 |
awarded | ● Nice Answer |
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Apr 22 |
revised |
$H^1(X,O_X)$, holomorphic $1$-forms, and $b_1(X)/2$ for normal $X$. added 407 characters in body; deleted 1 characters in body |
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Apr 22 |
accepted | $H^1(X,O_X)$, holomorphic $1$-forms, and $b_1(X)/2$ for normal $X$. |
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Apr 22 |
answered | $H^1(X,O_X)$, holomorphic $1$-forms, and $b_1(X)/2$ for normal $X$. |
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Apr 8 |
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surjective submersion and fibrations Yes, if the map is also proper (this goes back to Ehresmann), otherwise there are easy counterexamples. |
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Apr 5 |
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functions of one complex variable: geometric theory I remember looking at Siegel's "Topics in complex function theory" (several vols.) when I was a student, and it seemed very nice. Not sure if it fits all your criteria. |
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Mar 31 |
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cyclic covers over a non-algebraically closed field Yes. Lemma: If $E$ is a spectral sequence of finite dim vector spaces over a field $k$, such that $E_1\otimes L=E_\infty\otimes L$ for some field extension $L$, then $E_1= E_\infty$. Proof: compute dimensions. |
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Mar 27 |
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Another reference request about dualizing sheaves for nodal surfaces To answer Simon's question, algebraic geometry went through very rapid changes in the 1950's, with the advent of sheaf theory etc. Older papers such as Duval's, presumably, were written in an entirely different language. Certainly, the above statement would not even have appeared in the above form. |
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Mar 23 |
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Coboundaries and Gluing in Cech Cohomology - Intuition? Edward, there is no single answer to how to think about higher cohomology. Much of the power of sheaf cohomology stems from the fact that it can be understood in several ways. |
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Mar 22 |
answered | Coboundaries and Gluing in Cech Cohomology - Intuition? |
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Mar 22 |
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is this intersection complex a sheaf? Dear SM, perhaps this will simplify things (?). Since, as you say, it is local, and the local monodromy is diagonalizable, you can reduce to the case where $E$ has rank one with nontrivial monodromy about all components of $D$. Then check $\mathbb{R}j_*E[d] = j_*E[d]= j_!E[d]$ which should yield perversity. Also $j_*E[d]$ should vanish along $D$, which out to imply that this is the minimal extension, i.e. $j_{!*}E[d]$. But perhaps I am overlooking something. |
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Mar 21 |
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Coboundaries and Gluing in Cech Cohomology - Intuition? Edward, as Sándor explains, I was identifying $A(-)$ with subgroup of $B(-)$ etc. If no else gives a more detailed answer, I can in a few days when I have more time. |
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Mar 21 |
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Coboundaries and Gluing in Cech Cohomology - Intuition? Think of it this way: Given an exact sequence $0\to A\to B\to C\to 0$ of sheaves, you may want to a lift a global section $c$ from $C$ to $B$ (for many natural reasons). Lift locally to get $b_i\in B(U_i)$. So $b_i-b_j\in A(U_{ij})$ is the obstruction to patching these. But keep in mind that you might have a made a bad initial choice, whereas $b_i'=b_i+a_i$ might have patched. So that's why you should allow yourself the option of modifying by a coboundary.... |
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Mar 19 |
awarded | ● Enlightened |
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Mar 18 |
accepted | what is Deligne’s cohomological descent (and what are some examples) |
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Mar 17 |
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Another fibration with a given singular fiber class. Correction: I guess I was thinking of $Y$ as a surface. In general, you would need to blow up along a smooth codim 2 subset of $X$ supported on $D$. |
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Mar 17 |
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Another fibration with a given singular fiber class. Without further hypothesis, I don't see how you would prove something like this. Consider the following example. Suppose $g:Y\to C$ is a surjective map with general fibre $D$, blow up $Y$ along a point on $D$ to get $X$. Now $D$ appears as a component of singular fibre of $X$, yet all the other smooth fibres are deformation equivalent to $D$. |
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Mar 17 |
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The “interplay” between additive and multiplicative structure in a field Peter: perhaps none of the above, maybe a computer scientist? Joel: I'm sorry to nitpick, and I'm certainly no expert, but didn't Ax prove decidability of the theory of finite fields? |
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Mar 17 |
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Is the modification a rational map? Yes, they are. $f$ is given by a sequence of blow ups, so in fact, it is a regular map (or morphism if you prefer). The inverse is, however, usually only rational. |
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Mar 16 |
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cohomological criterion of triviality Georges, yes you're right, $1\gg 0$ in this case! |
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Mar 15 |
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cohomological criterion of triviality Jikki, you're welcome. If you really need such a result, you should probably require $E$ to be semistable. |
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Mar 15 |
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cohomological criterion of triviality No, consider $E=\mathcal{O}(n)\oplus \mathcal{O}(-m)$ on $\mathbb{P}^1$ with $n,m\gg 0$. |
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Mar 12 |
awarded | ● Nice Answer |
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Mar 11 |
awarded | ● Nice Answer |
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Mar 10 |
revised |
what is Deligne’s cohomological descent (and what are some examples) deleted 3 characters in body |
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Mar 10 |
revised |
what is Deligne’s cohomological descent (and what are some examples) added 551 characters in body |
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Mar 9 |
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what is Deligne’s cohomological descent (and what are some examples) Right, the MHS is independent of the choice. The idea is that any two simplicial resolutions are dominated by a third, so the resulting MHS's are comparable... |
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Mar 9 |
answered | what is Deligne’s cohomological descent (and what are some examples) |
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Mar 8 |
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Decomposing Semisimple Perverse Sheaves Pick a presentation $\mathcal{A}^n\to \mathcal{A}^m\to E\to 0$. Now you can realize $K_{E}$ in your category as the kernel of $K^m\to K^n$ where the map is the transpose of the original matrix. |
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Mar 8 |
answered | a local system on an open of the projective line |
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Mar 1 |
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exactness on pullback of sheaves OK, up to a point, but there is no reason to conclude that $Tor_1=0$ at $p\in Y$ from what you said previously. Consider $X=Spec k[x]$, $Y$ the closed point defined by $(x)$ and $A,B,C$ the sheaves corresponding to $xk[x]$, $k[x]$ and $k$, then your assumptions hold but $Tor_1(k,k)\not=0$. |
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Feb 26 |
accepted | Does the Čech cohomology always yield long exact sequences from short ones? |
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Feb 26 |
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Does the Čech cohomology always yield long exact sequences from short ones? I added some details. (Don't know much about Kan extensions, sorry.) |
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Feb 26 |
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Does the Čech cohomology always yield long exact sequences from short ones? added 966 characters in body; deleted 5 characters in body |
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Feb 25 |
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Does the Čech cohomology always yield long exact sequences from short ones? Sure, I can try to do this in few days, when I finish some other things. |
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Feb 25 |
answered | Does the Čech cohomology always yield long exact sequences from short ones? |
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Feb 24 |
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Kahler Metric Fundamental Forms and Cohomology Ring Generators Serge, you're right! The higher Chern classes of the universal bundle won't be in the ring generated by $H^2$ of the Grassmanian in general. I wasn't really thinking when I made the comment. |
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Feb 23 |
accepted | Kahler Metric Fundamental Forms and Cohomology Ring Generators |
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Feb 23 |
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Kahler Metric Fundamental Forms and Cohomology Ring Generators Yes, Grassmanians should work. |
