Outer Hodge groups of rationally connected fibrations
I believe the following is true and well known.
Theorem (?). Let $X$ and $Y$ be smooth, irreducible, projective varieties over $mathbb{C}$. Let
$$
fcolon Xrightarrow Y
$$
be a surjective map with generic fibers being irreducible and rationally connected. Then $H^0(X,wedge^kOmega_X)cong H^0(Y,wedge^kOmega_Y)$ for all $kge 0$.
I would appreciate a reference for this result or a simple proof (or, obviously, if this is wrong I would like to know that too).
ag.algebraic-geometry reference-request
asked 7 hours ago
Duck HunterDuck Hunter
232
New contributor
Duck Hunter is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
I believe the following is true and well known.
Theorem (?). Let $X$ and $Y$ be smooth, irreducible, projective varieties over $mathbb{C}$. Let
$$
fcolon Xrightarrow Y
$$
be a surjective map with generic fibers being irreducible and rationally connected. Then $H^0(X,wedge^kOmega_X)cong H^0(Y,wedge^kOmega_Y)$ for all $kge 0$.
I would appreciate a reference for this result or a simple proof (or, obviously, if this is wrong I would like to know that too).
ag.algebraic-geometry reference-request
asked 7 hours ago
Duck HunterDuck Hunter
232
New contributor
Duck Hunter is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
I believe the following is true and well known.
Theorem (?). Let $X$ and $Y$ be smooth, irreducible, projective varieties over $mathbb{C}$. Let
$$
fcolon Xrightarrow Y
$$
be a surjective map with generic fibers being irreducible and rationally connected. Then $H^0(X,wedge^kOmega_X)cong H^0(Y,wedge^kOmega_Y)$ for all $kge 0$.
I would appreciate a reference for this result or a simple proof (or, obviously, if this is wrong I would like to know that too).
ag.algebraic-geometry reference-request
asked 7 hours ago
Duck HunterDuck Hunter
232
New contributor
Duck Hunter is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I believe the following is true and well known.
Theorem (?). Let $X$ and $Y$ be smooth, irreducible, projective varieties over $mathbb{C}$. Let
$$
fcolon Xrightarrow Y
$$
be a surjective map with generic fibers being irreducible and rationally connected. Then $H^0(X,wedge^kOmega_X)cong H^0(Y,wedge^kOmega_Y)$ for all $kge 0$.
I would appreciate a reference for this result or a simple proof (or, obviously, if this is wrong I would like to know that too).
ag.algebraic-geometry reference-request
ag.algebraic-geometry reference-request
asked 7 hours ago
Duck HunterDuck Hunter
232
New contributor
Duck Hunter is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 7 hours ago
Duck HunterDuck Hunter
232
New contributor
Duck Hunter is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 7 hours ago
Duck HunterDuck Hunter
232
New contributor
Duck Hunter is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 7 hours ago
Duck HunterDuck Hunter
232
asked 7 hours ago
Duck HunterDuck Hunter
232
232
New contributor
Duck Hunter is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Duck Hunter is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Duck Hunter is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
By spectral sequence, we just need to prove that $R^if_*mathcal{O}_X=0$ for all $i>0$. This is a special case of Theorem 7.1 in [Kollár, Higher direct images of dualizing sheaves I].
Actually the answer to you question is just Corollary 7.2 in [Kollár, Higher direct images of dualizing sheaves I].
answered 7 hours ago
Chen JiangChen Jiang
934157
1
Thank you! I did not realize Kollár had proved this. What an excellent paper!
– Duck Hunter
6 hours ago
add a comment |
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: "504"
};
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
});
}
});
Duck Hunter is a new contributor. Be nice, and check out our Code of Conduct.
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%2fmathoverflow.net%2fquestions%2f320426%2fouter-hodge-groups-of-rationally-connected-fibrations%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
By spectral sequence, we just need to prove that $R^if_*mathcal{O}_X=0$ for all $i>0$. This is a special case of Theorem 7.1 in [Kollár, Higher direct images of dualizing sheaves I].
Actually the answer to you question is just Corollary 7.2 in [Kollár, Higher direct images of dualizing sheaves I].
answered 7 hours ago
Chen JiangChen Jiang
934157
1
Thank you! I did not realize Kollár had proved this. What an excellent paper!
– Duck Hunter
6 hours ago
add a comment |
By spectral sequence, we just need to prove that $R^if_*mathcal{O}_X=0$ for all $i>0$. This is a special case of Theorem 7.1 in [Kollár, Higher direct images of dualizing sheaves I].
Actually the answer to you question is just Corollary 7.2 in [Kollár, Higher direct images of dualizing sheaves I].
answered 7 hours ago
Chen JiangChen Jiang
934157
1
Thank you! I did not realize Kollár had proved this. What an excellent paper!
– Duck Hunter
6 hours ago
add a comment |
By spectral sequence, we just need to prove that $R^if_*mathcal{O}_X=0$ for all $i>0$. This is a special case of Theorem 7.1 in [Kollár, Higher direct images of dualizing sheaves I].
Actually the answer to you question is just Corollary 7.2 in [Kollár, Higher direct images of dualizing sheaves I].
answered 7 hours ago
Chen JiangChen Jiang
934157
By spectral sequence, we just need to prove that $R^if_*mathcal{O}_X=0$ for all $i>0$. This is a special case of Theorem 7.1 in [Kollár, Higher direct images of dualizing sheaves I].
Actually the answer to you question is just Corollary 7.2 in [Kollár, Higher direct images of dualizing sheaves I].
answered 7 hours ago
Chen JiangChen Jiang
934157
answered 7 hours ago
Chen JiangChen Jiang
934157
answered 7 hours ago
Chen JiangChen Jiang
934157
answered 7 hours ago
Chen JiangChen Jiang
934157
934157
1
Thank you! I did not realize Kollár had proved this. What an excellent paper!
– Duck Hunter
6 hours ago
add a comment |
1
Thank you! I did not realize Kollár had proved this. What an excellent paper!
– Duck Hunter
6 hours ago
1
1
Thank you! I did not realize Kollár had proved this. What an excellent paper!
– Duck Hunter
6 hours ago
Thank you! I did not realize Kollár had proved this. What an excellent paper!
– Duck Hunter
6 hours ago
add a comment |
Duck Hunter is a new contributor. Be nice, and check out our Code of Conduct.
Duck Hunter is a new contributor. Be nice, and check out our Code of Conduct.
Duck Hunter is a new contributor. Be nice, and check out our Code of Conduct.
Duck Hunter is a new contributor. Be nice, and check out our Code of Conduct.
Thanks for contributing an answer to MathOverflow!
- 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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2fmathoverflow.net%2fquestions%2f320426%2fouter-hodge-groups-of-rationally-connected-fibrations%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
