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S. Li
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I was wondering in the world of non-Kähler compact complex manifold, what is the relationship between $h^1(\mathcal{O}_X)$, $h^0(\Omega_X)$, and first betti number?

The only thing I know is that for surfaces, we have $h^1(\mathcal{O}_X) \geq \frac{b_1}{2} \geq h^0(\Omega_X)$, the reason being that every holomorphic 1-form on a compact surface must be closed. Does this inequality hold true for general compact complex manifold? I'm mainly interested in the relation between $h^1(O_X)$ and $h^0(\Omega_X)$.

Another somewhat related question is: does there exist a compact complex manifold with non-closed holomorphic 1-forms?

Thanks to an anonymous friend's comment. On page 444 of Griffiths and Harris' book Principles of Algebraic Geometry, they gave an example (Iwasawa manifold) of a non-closed holomorphic 1-form.

I was wondering in the world of non-Kähler compact complex manifold, what is the relationship between $h^1(\mathcal{O}_X)$, $h^0(\Omega_X)$, and first betti number?

The only thing I know is that for surfaces, we have $h^1(\mathcal{O}_X) \geq \frac{b_1}{2} \geq h^0(\Omega_X)$, the reason being that every holomorphic 1-form on a compact surface must be closed. Does this inequality hold true for general compact complex manifold?

Another somewhat related question is: does there exist a compact complex manifold with non-closed holomorphic 1-forms?

Thanks to an anonymous friend's comment. On page 444 of Griffiths and Harris' book Principles of Algebraic Geometry, they gave an example (Iwasawa manifold) of a non-closed holomorphic 1-form.

I was wondering in the world of non-Kähler compact complex manifold, what is the relationship between $h^1(\mathcal{O}_X)$, $h^0(\Omega_X)$, and first betti number?

The only thing I know is that for surfaces, we have $h^1(\mathcal{O}_X) \geq \frac{b_1}{2} \geq h^0(\Omega_X)$, the reason being that every holomorphic 1-form on a compact surface must be closed. Does this inequality hold true for general compact complex manifold? I'm mainly interested in the relation between $h^1(O_X)$ and $h^0(\Omega_X)$.

Another somewhat related question is: does there exist a compact complex manifold with non-closed holomorphic 1-forms?

Thanks to an anonymous friend's comment. On page 444 of Griffiths and Harris' book Principles of Algebraic Geometry, they gave an example (Iwasawa manifold) of a non-closed holomorphic 1-form.

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Michael Albanese
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Relation between $h^1(O_X\mathcal{O}_X)$, $h^0(\Omega_X)$, and first betti number for general complex manifold?

I was wondering in the world of non-Kähler compact complex manifold, what is the relationship between $h^1(O_X)$$h^1(\mathcal{O}_X)$, $h^0(\Omega_X)$, and first betti number.?

The only thing I know is that for surfaces, we have $h^1(O_X) \geq \frac{b_1}{2} \geq h^0(\Omega_X)$$h^1(\mathcal{O}_X) \geq \frac{b_1}{2} \geq h^0(\Omega_X)$, the reason being that every holomorphic 1-form on a compact surface must be closed. Does this inequality hold true for general compact complex manifold?

Another somewhat related question is: does there exist a compact complex manifold with non-closed holomorphic 1-forms?

Thanks to an anonymous friend's comment. On page 444 of Griffiths and Harris' book Principles of Algebraic Geometry, they gave an example (Iwasawa manifold) of a non-closed holomorphic 1-form.

Relation between $h^1(O_X)$, $h^0(\Omega_X)$, and first betti number for general complex manifold?

I was wondering in the world of non-Kähler compact complex manifold, what is the relationship between $h^1(O_X)$, $h^0(\Omega_X)$, and first betti number.

The only thing I know is that for surfaces, we have $h^1(O_X) \geq \frac{b_1}{2} \geq h^0(\Omega_X)$, the reason being that every holomorphic 1-form on a compact surface must be closed. Does this inequality hold true for general compact complex manifold?

Another somewhat related question is: does there exist a compact complex manifold with non-closed holomorphic 1-forms?

Thanks to an anonymous friend's comment. On page 444 of Griffiths and Harris' book , they gave an example (Iwasawa manifold) of a non-closed holomorphic 1-form.

Relation between $h^1(\mathcal{O}_X)$, $h^0(\Omega_X)$, and first betti number for general complex manifold?

I was wondering in the world of non-Kähler compact complex manifold, what is the relationship between $h^1(\mathcal{O}_X)$, $h^0(\Omega_X)$, and first betti number?

The only thing I know is that for surfaces, we have $h^1(\mathcal{O}_X) \geq \frac{b_1}{2} \geq h^0(\Omega_X)$, the reason being that every holomorphic 1-form on a compact surface must be closed. Does this inequality hold true for general compact complex manifold?

Another somewhat related question is: does there exist a compact complex manifold with non-closed holomorphic 1-forms?

Thanks to an anonymous friend's comment. On page 444 of Griffiths and Harris' book Principles of Algebraic Geometry, they gave an example (Iwasawa manifold) of a non-closed holomorphic 1-form.

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S. Li
  • 619
  • 4
  • 17

I was wondering in the world of non-Kähler compact complex manifold, what is the relationship between $h^1(O_X)$, $h^0(\Omega_X)$, and first betti number.

The only thing I know is that for surfaces, we have $h^1(O_X) \geq \frac{b_1}{2} \geq h^0(\Omega_X)$, the reason being that every holomorphic 1-form on a compact surface must be closed. Does this inequality holdshold true for general compact complex manifold? 

Another somewhat related question is: does there exist a compact complex manifold with non-closed holomorphic 1-forms?

Thanks to an anonymous friend's comment. On page 444 of Griffiths and Harris' book , they gave an example (Iwasawa manifold) of a non-closed holomorphic 1-form.

I was wondering in the world of non-Kähler compact complex manifold, what is the relationship between $h^1(O_X)$, $h^0(\Omega_X)$, and first betti number.

The only thing I know is that for surfaces, we have $h^1(O_X) \geq \frac{b_1}{2} \geq h^0(\Omega_X)$, the reason being that every holomorphic 1-form on a compact surface must be closed. Does this inequality holds true for general compact complex manifold? Another somewhat related question is: does there exist a compact complex manifold with non-closed holomorphic 1-forms?

I was wondering in the world of non-Kähler compact complex manifold, what is the relationship between $h^1(O_X)$, $h^0(\Omega_X)$, and first betti number.

The only thing I know is that for surfaces, we have $h^1(O_X) \geq \frac{b_1}{2} \geq h^0(\Omega_X)$, the reason being that every holomorphic 1-form on a compact surface must be closed. Does this inequality hold true for general compact complex manifold? 

Another somewhat related question is: does there exist a compact complex manifold with non-closed holomorphic 1-forms?

Thanks to an anonymous friend's comment. On page 444 of Griffiths and Harris' book , they gave an example (Iwasawa manifold) of a non-closed holomorphic 1-form.

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S. Li
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