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edited Apr 9 2012 at 1:10 |
Given a compact Kaehler manifold $M$ of complex dimension $2n$, there are essentially two ways to compute its signature $\sigma(M)$, i.e. the index of the intersection form on $H_{2n}(M,\mathbb{R})$:
1.by Hodge index theorem $\sigma(M)=\sum_{p,q}(-1)^p h^{p,q}$, here $h^{p,q}$ stands for the Hodge numbers.
2.by Hirzebruch signature theorem $\sigma(M)=L[M]$, here $L[M]$ stands for the $L$-genus, i.e. the characteristic number of the top $L$-class.This approach is more general since it works on any $4k$ dimensional real manifolds.
My questions are
1.Since these two approaches rest on different levels of cohomology theory, how are they interrelated?
2.Of course, one possible way to answer Question 1 is to generalize both by the Hirzebruch-Riemann-Roch on Kaehler manifolds, a point already mentioned in Hirzebruch's Neue topologische Methoden. However, I am wondering if someone could relate these two approaches on a more fundamental level.
To be precise,
Is there a formula to express the Chern numbers/Pontryagin numbers out of the Hodge numbers on a compact Kaehler manifold $M$ of complex dimension $2n$? n$? Surely it is the case for $c_n[M]$ interpreted as the Euler characteristic number.
Or, does anyone know such counterexamples that two Kaehler manifolds(notably, Kaehler surfaces, I guess) have the same Hodge numbers but different Chern numbers?
Many thanks!
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edited Apr 9 2012 at 0:33 |
Given a compact Kaehler manifold $M$ of complex dimension $2n$, there are essentially two ways to compute its signature $\sigma(M)$, i.e. the index of the intersection form on $H_{2n}(M,\mathbb{R})$:
1.by Hodge index theorem $\sigma(M)=\sum_{p,q}(-1)^p h^{p,q}$, here $h^{p,q}$ stands for the Hodge numbers.
2.by Hirzebruch signature theorem $\sigma(M)=L[M]$, here $L[M]$ stands for the $L$-genus, i.e. the characteristic number of the top $L$-class.This approach is more general since it works on any $4k$ dimensional real manifolds.
My questions are
1.Since these two approaches rest on different levels of cohomology theory, how are they interrelated?
2.Of course, one possible way to answer Question 1 is to generalize both by the Hirzebruch-Riemann-Roch on Kaehler manifolds, a point already mentioned in Hirzebruch's Neue topologische Methoden. However, I am wondering if someone could relate these two approaches on a more fundamental level.
To be precise,
Is there a formula to express the Chern numbers/Pontryagin numbers out of the Hodge numbers on a compact Kaehler manifold $M$ of complex dimension $2n$? Surely it is the case for $c_n[M]$ interpreted as the Euler characteristic number.
Or, does anyone know such counterexamples that two Kaehler manifolds(notably, Kaehler surfaces, I guess) have the same Hodge numbers but different Chern numbers?
Many thanks!
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edited Apr 8 2012 at 23:44 |
Given a compact Kaehler manifold $M$ of complex dimension $2n$, there are essentially two ways to compute its signature $\sigma(M)$, i.e. the index of the intersection form on $H_{2n}(M,\mathbb{R})$:
1.by Hodge index theorem $\sigma(M)=\sum_{p,q}(-1)^p h^{p,q}$, here $h^{p,q}$ stands for the Hodge numbers.
2.by Hirzebruch signature theorem $\sigma(M)=L[M]$, here $L[M]$ stands for the $L$-genus, i.e. the characteristic number of the top $L$-class.This approach is more general since it works on any $4k$ dimensional real manifolds.
My questions are
1.Since these two approaches rest on different levels of cohomology theory, how are they interrelated?
2.Of course, one possible way to answer Question 1 is to generalize both by the Hirzebruch-Riemann-Roch on Kaehler manifolds, a point already mentioned in Hirzebruch's Neue topologische Methoden. However, I am wondering if someone could relate these two approaches on a more fundamental level.
To be precise,
Is there a formula to express the Chern numbers/Pontryagin numbers out of the Hodge numbers on a compact Kaehler manifold $M$ of complex dimension $2n$?
Or, does anyone know such examples counterexamples that two Kaehler manifolds(notably, Kaehler surfaces, I guess) have same Hodge numbers but different Chern numbers?
Many thanks!
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edited Apr 8 2012 at 23:39 |
Given a compact Kaehler manifold $M$ of complex dimension $2n$, there are essentially two ways to compute its signature $\sigma(M)$, i.e. the index of the intersection form on $H_{2n}(M,\mathbb{R})$:
1.by Hodge index theorem $\sigma(M)=\sum_{p,q}(-1)^p h^{p,q}$, here $h^{p,q}$ stands for the Hodge numbers.
2.by Hirzebruch signature theorem $\sigma(M)=L[M]$, here $L[M]$ stands for the $L$-genus, i.e. the characteristic number of the top $L$-class.This approach is more general since it works on any $4k$ dimensional real manifolds.
My questions are
1.Since these two approaches rest on different levels of cohomology theory, how are they interrelated?
2.Of course, one possible way to answer Question 1 is to generalize both by the Hirzebruch-Riemann-Roch on Kaehler manifolds, a point already mentioned in Hirzebruch's Neue topologische Methoden. However, I am wondering if someone could relate these two approaches on a more fundamental level.
To be precise,
Is there a formula to express the Chern numbers/Pontryagin numbers out of the Hodge numbers on a compact Kaehler manifold $M$ of complex dimension $2n$?
Or, does anyone know such examples that two Kaehler manifolds(notably, Kaehler surfaces, I guess) have same Hodge numbers but different Chern numbers?
Many thanks!
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edited Apr 8 2012 at 23:32 |
Given a compact Kaehler manifold $M$ of complex dimension $2n$, there are essentially two ways to compute its signature $\sigma(M)$, i.e. the index of the intersection form on $H_{2n}(M,\mathbb{R})$:
1.by Hodge index theorem $\sigma(M)=\sum_{p,q}(-1)^p h^{p,q}$, here $h^{p,q}$ stands for the Hodge numbers.
2.by Hirzebruch signature theorem $\sigma(M)=L[M]$, here $L[M]$ stands for the $L$-genus, i.e. the characteristic number of the top $L$-class.This approach is more general since it works on any $4k$ dimensional real manifolds.
My questions are
1.Since these two approaches rest on different levels of cohomology theory, how are they interrelated?
2.Of course, one possible way to answer Question 1 is to generalize both by the Hirzebruch-Riemann-Roch on Kaehler manifolds, a point already mentioned in Hirzebruch's Neue topologische Methoden. However, I am wondering if someone could relate these two approaches on a more fundamental level.
To be precise,
Is there a formula to express the Chern numbers out of the Hodge numbers ?
3.Examples on a compact Kaehler manifold $M$ of the complex dimension $2n$?
Or, does anyone know such examples that two Kaehler manifolds(notably, Kaehler surfaces, I guess) have same nature are highly welcomed.Hodge numbers but different Chern numbers?
Many thanks!
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edited Apr 8 2012 at 22:49 |
Given a compact Kaehler manifold $M$ of complex dimension $2n$, there are essentially two ways to compute its signature $\sigma(M)$, i.e. the index of the intersection form on $H^{2n}(M,\mathbb{R})$:H_{2n}(M,\mathbb{R})$:
1.by Hodge index theorem $\sigma(M)=\sum_{p,q}(-1)^p h^{p,q}$, here $h^{p,q}$ stands for the Hodge numbers.
2.by Hirzebruch signature theorem $\sigma(M)=L[M]$, here $L[M]$ stands for the $L$-genus, i.e. the characteristic number of the top $L$-class.This approach is more general since it works on any $4k$ dimensional real manifolds.
My questions are
1.Since these two approaches rest on different levels of cohomology theory, how are they interrelated?
2.Of course, one possible way to answer Question 1 is to generalize both by the Hirzebruch-Riemann-Roch on Kaehler manifolds, a point already mentioned in Hirzebruch's Neue topologische Methoden. However, I am wondering if someone could relate these two approaches on a more fundamental level. Is there a formula to express the Chern numbers out of the Hodge numbers?
3.Examples of the same nature are highly welcomed.
Many thanks!
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edited Apr 8 2012 at 22:42 |
Given a Kaehler manifold $M$ of complex dimension $2n$, there are essentially two ways to compute its signature $\sigma(M)$, i.e. the index of the intersection form on $H^{2n}(M,\mathbb{R})$:
1.by Hodge index theorem $\sigma(M)=\sum_{p,q}(-1)^p h^{p,q}$, here $h^{p,q}$ stands for the Hodge numbers.
2.by Hirzebruch signature theorem $\sigma(M)=L[M]$, here $L[M]$ stands for the L genus, $L$-genus, i.e. the characteristic number of the top $L$-class.This approach is more general since it works on any $4k$ dimensional real manifolds.
My questions are
1.Since these two approaches rest on different levels of cohomology theory, how are they interrelated?
2.Of course, one possible way to answer Question 1 is to generalize both by the Hirzebruch-Riemann-Roch on Kaehler manifolds, a point already mentioned in Hirzebruch's Neue topologische Methoden. However, I am wondering if someone could relate these two approaches on a more fundamental level. Is there a formula to express the Chern numbers out of the Hodge numbers?
3.Examples of the same nature are highly welcomed.
Many thanks!
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edited Apr 8 2012 at 22:20 |
Given a Kaehler manifold $M$ of complex dimension $2n$, there are essentially two ways to compute its signature $\sigma(M)$, i.e. the index of the intersection form on $H^{2n}(M,\mathbb{R})$:
1.by Hodge index theorem $\sigma(M)=\sum_{p,q}(-1)^p h^{p,q}$, here $h^{p,q}$ stands for the Hodge numbers.
2.by Hirzebruch signature theorem $\sigma(M)=L[M]$, here $L[M]$ stands for the L genus, i.e. the characteristic number of the top $L$-class.This approach is more general since it works on any $4k$ dimensional real manifolds.
My questions are
1.Since these two approaches rest on different levels of cohomology theory, how are they interrelated?
2.Of course, one possible way to answer Question 1 is to generalize both by the Hirzebruch-Riemann-Roch on Kaehler manifolds, a point already mentioned in Hirzebruch's Neue topologische Methoden. However, I am wondering if someone could relate these two approaches on a more fundamental level. Is there a formular formula to express the Chern numbers by out of the Hodge numbers?
3.Examples of the same nature are highly welcomed.
Many thanks!
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edited Apr 8 2012 at 22:15 |
Given a Kaehler manifold $M$ of complex dimension $2n$, there are essentially two ways to compute its signature $\sigma(M)$, i.e. the index of the intersection form on $H^{2n}(M,\mathbb{R})$:
1.by Hodge index theorem $\sigma(M)=\sum_{p,q}(-1)^p h^{p,q}$, here $h^{p,q}$ stands for the Hodge numbers.
2.by Hirzebruch signature theorem $\sigma(M)=L[M]$, here $L[M]$ stands for the L genus, i.e. the characteristic number of the top $L$-class.This approach is more general since it works on any $4k$ dimensional real manifolds.
My questions are
1.Since these two approaches rest on different levels of cohomology theory, how are they interrelated?
2.Of course, one possible way to answer Question 1 is to generalize both by the Hirzebruch-Riemann-Roch on Kaehler manifolds, a point already mentioned in Hirzebruch's Neue topologische Methoden. However, I am wondering if someone could relate these two approaches on a more fundamental level. Is there a formular to express the Chern numbers by the Hodge numbers?
3.Examples of the same nature are also highly welcomed.
Many thanks!
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asked Apr 8 2012 at 22:06 |
Two approaches to compute the signature of a Kaehler manifold
Given a Kaehler manifold $M$ of complex dimension $2n$, there are essentially two ways to compute its signature $\sigma(M)$, i.e. the index of the intersection form on $H^{2n}(M,\mathbb{R})$:
1.by Hodge index theorem $\sigma(M)=\sum_{p,q}(-1)^p h^{p,q}$, here $h^{p,q}$ stands for the Hodge numbers.
2.by Hirzebruch signature theorem $\sigma(M)=L[M]$, here $L[M]$ stands for the L genus, i.e. the characteristic number of the top $L$-class.This approach is more general since it works on any $4k$ dimensional real manifolds.
My questions are
1.Since these two approaches rest on different levels of cohomology theory, how are they interrelated?
2.Of course, one possible way to answer Question 1 is to generalize both by the Hirzebruch-Riemann-Roch on Kaehler manifolds, a point already mentioned in Hirzebruch's Neue topologische Methoden. However, I am wondering if someone could relate these two approaches on a more fundamental level.
3.Examples of the same nature are also welcomed.
Many thanks!
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