Mixing
Dual abelian variety and the cohomology of elements of dual abelian variety
$begingroup$
Let $X$ be a complex abelian variety. Then the dual abelian variety is $Pic^0(X)$. Is $Pic^0(X)$ is the set of degree zero line bundles on $X$? How do we define degree of a line bundle on a higher dimensional variety? Is $Pic^0(X)$ the set of line bundles on $X$ numerically equivalent to zero?
Another question I have is, if $Pin Pic^0(X)$, what can we say about $H^i(X,P)$? Is $H^0(X,P)=0$? It seems to me that the Euler characteristic is zero.
algebraic-geometry
asked Jan 27 at 18:47
user349424user349424
34317
$endgroup$
add a comment |
$begingroup$
Let $X$ be a complex abelian variety. Then the dual abelian variety is $Pic^0(X)$. Is $Pic^0(X)$ is the set of degree zero line bundles on $X$? How do we define degree of a line bundle on a higher dimensional variety? Is $Pic^0(X)$ the set of line bundles on $X$ numerically equivalent to zero?
Another question I have is, if $Pin Pic^0(X)$, what can we say about $H^i(X,P)$? Is $H^0(X,P)=0$? It seems to me that the Euler characteristic is zero.
algebraic-geometry
asked Jan 27 at 18:47
user349424user349424
34317
$endgroup$
add a comment |
$begingroup$
Let $X$ be a complex abelian variety. Then the dual abelian variety is $Pic^0(X)$. Is $Pic^0(X)$ is the set of degree zero line bundles on $X$? How do we define degree of a line bundle on a higher dimensional variety? Is $Pic^0(X)$ the set of line bundles on $X$ numerically equivalent to zero?
Another question I have is, if $Pin Pic^0(X)$, what can we say about $H^i(X,P)$? Is $H^0(X,P)=0$? It seems to me that the Euler characteristic is zero.
algebraic-geometry
asked Jan 27 at 18:47
user349424user349424
34317
$endgroup$
Let $X$ be a complex abelian variety. Then the dual abelian variety is $Pic^0(X)$. Is $Pic^0(X)$ is the set of degree zero line bundles on $X$? How do we define degree of a line bundle on a higher dimensional variety? Is $Pic^0(X)$ the set of line bundles on $X$ numerically equivalent to zero?
Another question I have is, if $Pin Pic^0(X)$, what can we say about $H^i(X,P)$? Is $H^0(X,P)=0$? It seems to me that the Euler characteristic is zero.
algebraic-geometry
algebraic-geometry
asked Jan 27 at 18:47
user349424user349424
34317
asked Jan 27 at 18:47
user349424user349424
34317
asked Jan 27 at 18:47
user349424user349424
34317
asked Jan 27 at 18:47
user349424user349424
34317
asked Jan 27 at 18:47
user349424user349424
34317
34317
add a comment |
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3089969%2fdual-abelian-variety-and-the-cohomology-of-elements-of-dual-abelian-variety%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3089969%2fdual-abelian-variety-and-the-cohomology-of-elements-of-dual-abelian-variety%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
