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Let $M$ be a compact complex manifold equipped with a line bundle $L$ which has curvature which is non-negative and strictly positive outside of a measure zero set $Z$. In his paper "Holomorphic Morse inequalities" Demailly proved that $M$ is Moishezon (this was a new proof of the Grauert-Riemenschneider conjecture, originally due to Siu).

I want to show that if $M$ is projective, $L$ is semi-ample, that is, base point free. I have a very simple argument, which I guess should be known, but I was unable to find it.

The proof is based on Kawamata BPF theorem, and Demailly's holomorphic Morse inequalities.

Kawamata BPF:
Let $L$ be a nef line bundle on a projective manifold $M$ such that $L^{\otimes a}\otimes K_M^{-1}$ is big and nef for some $a$. Then $L$ is base point free.

Here I was wrong, and I need to apply a correction, thanks to the user YangMills. The bundle $L^{\otimes a}\otimes K_M^{-1}$ should be big and nef. Then the argument I gave becomes invalid, unless $Z$ is very special.

Demailly: $$ h^1(E^k) - h^0(E^k) \leq -\frac{k^n}{n!} \int_U \left(\frac{\sqrt 1}{2\pi} \Theta_E\right)^n + o(k^n), $$ where $E$ is a line bundle, $\Theta_E$ its curvature, and $U\subset M$ the set where $\Theta_E$ has at least $n-1$ positive eigenvalues.

Semipositive line bundles are clearly nef. To apply Kawamata to the semipositive line bundle $L$, we need only to check that $L^{\otimes a}\otimes K_M^{-1}$ is big. However, for very big $a$ the curvature of $L^{\otimes a}\otimes K_M^{-1}$ is strictly positive outside of a small neighbourhood of the zero set $Z$, which contributes very little to the integral $ \int_U (\frac{\sqrt 1}{2\pi} \Theta_E)^n$, hence this integral is positive, and $E=L^{\otimes a}\otimes K_M^{-1}$ is big, because $h^0(E^k)$ grows as $k^n$.

I would be very grateful to anyone who can give me the reference to this, or point out where I did an error (or give a counterexample).

Reference:
Kawamata, Y., Pluricanonical systems on minimal algebraic varieties, Invent. Math., 79, (1985), no. 3, 567-588.

J.-P. Demailly, Holomorphic Morse inequalities, Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), 93--114, Proc. Sympos. Pure Math., 52, Part 2, Amer. Math. Soc., Providence, RI, 1991.

Let $M$ be a compact complex manifold equipped with a line bundle $L$ which has curvature which is non-negative and strictly positive outside of a measure zero set $Z$. In his paper "Holomorphic Morse inequalities" Demailly proved that $M$ is Moishezon (this was a new proof of the Grauert-Riemenschneider conjecture, originally due to Siu).

I want to show that if $M$ is projective, $L$ is semi-ample, that is, base point free. I have a very simple argument, which I guess should be known, but I was unable to find it.

The proof is based on Kawamata BPF theorem, and Demailly's holomorphic Morse inequalities.

Kawamata BPF:
Let $L$ be a nef line bundle on a projective manifold $M$ such that $L^{\otimes a}\otimes K_M^{-1}$ is big and nef for some $a$. Then there exists some $m$ such that $mL$ is base point free.

Here I was wrong, and I need to apply a correction, thanks to the user YangMills. The bundle $L^{\otimes a}\otimes K_M^{-1}$ should be big and nef. Then the argument I gave becomes invalid, unless $Z$ is very special.

Demailly: $$ h^1(E^k) - h^0(E^k) \leq -\frac{k^n}{n!} \int_U \left(\frac{\sqrt 1}{2\pi} \Theta_E\right)^n + o(k^n), $$ where $E$ is a line bundle, $\Theta_E$ its curvature, and $U\subset M$ the set where $\Theta_E$ has at least $n-1$ positive eigenvalues.

Semipositive line bundles are clearly nef. To apply Kawamata to the semipositive line bundle $L$, we need only to check that $L^{\otimes a}\otimes K_M^{-1}$ is big. However, for very big $a$ the curvature of $L^{\otimes a}\otimes K_M^{-1}$ is strictly positive outside of a small neighbourhood of the zero set $Z$, which contributes very little to the integral $ \int_U (\frac{\sqrt 1}{2\pi} \Theta_E)^n$, hence this integral is positive, and $E=L^{\otimes a}\otimes K_M^{-1}$ is big, because $h^0(E^k)$ grows as $k^n$.

I would be very grateful to anyone who can give me the reference to this, or point out where I did an error (or give a counterexample).

Reference:
Kawamata, Y., Pluricanonical systems on minimal algebraic varieties, Invent. Math., 79, (1985), no. 3, 567-588.

J.-P. Demailly, Holomorphic Morse inequalities, Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), 93--114, Proc. Sympos. Pure Math., 52, Part 2, Amer. Math. Soc., Providence, RI, 1991.

Let $M$ be a compact complex manifold equipped with a line bundle $L$ which has curvature which is non-negative and strictly positive outside of a measure zero set $Z$. In his paper "Holomorphic Morse inequalities" Demailly proved that $M$ is Moishezon (this was a new proof of the Grauert-Riemenschneider conjecture, originally due to Siu).

I want to show that if $M$ is projective, $L$ is semi-ample, that is, base point free. I have a very simple argument, which I guess should be known, but I was unable to find it.

The proof is based on Kawamata BPF theorem, and Demailly's holomorphic Morse inequalities.

Kawamata BPF:
Let $L$ be a nef line bundle on a projective manifold $M$ such that $L^{\otimes a}\otimes K_M^{-1}$ is big and nef for some $a$. Then $L$ is semiample.

Here I was wrong, and I need to apply a correction, thanks to the user YangMills. The bundle $L^{\otimes a}\otimes K_M^{-1}$ should be big and nef. Then the argument I gave becomes invalid, unless $Z$ is very special.

Demailly: $$ h^1(E^k) - h^0(E^k) \leq -\frac{k^n}{n!} \int_U \left(\frac{\sqrt 1}{2\pi} \Theta_E\right)^n + o(k^n), $$ where $E$ is a line bundle, $\Theta_E$ its curvature, and $U\subset M$ the set where $\Theta_E$ has at least $n-1$ positive eigenvalues.

Semipositive line bundles are clearly nef. To apply Kawamata to the semipositive line bundle $L$, we need only to check that $L^{\otimes a}\otimes K_M^{-1}$ is big. However, for very big $a$ the curvature of $L^{\otimes a}\otimes K_M^{-1}$ is strictly positive outside of a small neighbourhood of the zero set $Z$, which contributes very little to the integral $ \int_U (\frac{\sqrt 1}{2\pi} \Theta_E)^n$, hence this integral is positive, and $E=L^{\otimes a}\otimes K_M^{-1}$ is big, because $h^0(E^k)$ grows as $k^n$.

I would be very grateful to anyone who can give me the reference to this, or point out where I did an error (or give a counterexample).

Reference:
Kawamata, Y., Pluricanonical systems on minimal algebraic varieties, Invent. Math., 79, (1985), no. 3, 567-588.

J.-P. Demailly, Holomorphic Morse inequalities, Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), 93--114, Proc. Sympos. Pure Math., 52, Part 2, Amer. Math. Soc., Providence, RI, 1991.

Let $M$ be a compact complex manifold equipped with a line bundle $L$ which has curvature which is non-negative and strictly positive outside of a measure zero set $Z$. In his paper "Holomorphic Morse inequalities" Demailly proved that $M$ is Moishezon (this was a new proof of the Grauert-Riemenschneider conjecture, originally due to Siu).

I want to show that if $M$ is projective, $L$ is semi-ample, that is, base point free. I have a very simple argument, which I guess should be known, but I was unable to find it.

The proof is based on Kawamata BPF theorem, and Demailly's holomorphic Morse inequalities.

Kawamata BPF:
Let $L$ be a nef line bundle on a projective manifold $M$ such that $L^{\otimes a}\otimes K_M^{-1}$ is big and nef for some $a$. Then $L$ is base point free.

Here I was wrong, and I need to apply a correction, thanks to the user YangMills. The bundle $L^{\otimes a}\otimes K_M^{-1}$ should be big and nef. Then the argument I gave becomes invalid, unless $Z$ is very special.

Demailly: $$ h^1(E^k) - h^0(E^k) \leq -\frac{k^n}{n!} \int_U \left(\frac{\sqrt 1}{2\pi} \Theta_E\right)^n + o(k^n), $$ where $E$ is a line bundle, $\Theta_E$ its curvature, and $U\subset M$ the set where $\Theta_E$ has at least $n-1$ positive eigenvalues.

Semipositive line bundles are clearly nef. To apply Kawamata to the semipositive line bundle $L$, we need only to check that $L^{\otimes a}\otimes K_M^{-1}$ is big. However, for very big $a$ the curvature of $L^{\otimes a}\otimes K_M^{-1}$ is strictly positive outside of a small neighbourhood of the zero set $Z$, which contributes very little to the integral $ \int_U (\frac{\sqrt 1}{2\pi} \Theta_E)^n$, hence this integral is positive, and $E=L^{\otimes a}\otimes K_M^{-1}$ is big, because $h^0(E^k)$ grows as $k^n$.

I would be very grateful to anyone who can give me the reference to this, or point out where I did an error (or give a counterexample).

Reference:
Kawamata, Y., Pluricanonical systems on minimal algebraic varieties, Invent. Math., 79, (1985), no. 3, 567-588.

J.-P. Demailly, Holomorphic Morse inequalities, Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), 93--114, Proc. Sympos. Pure Math., 52, Part 2, Amer. Math. Soc., Providence, RI, 1991.

Let $M$ be a compact complex manifold equipped with a line bundle $L$ which has curvature which is non-negative and strictly positive outside of a measure zero set $Z$. In his paper "Holomorphic Morse inequalities" Demailly proved that $M$ is Moishezon (this was a new proof of the Grauert-Riemenschneider conjecture, originally due to Siu).

I want to show that if $M$ is projective, $L$ is semi-ample, that is, base point free. I have a very simple argument, which I guess should be known, but I was unable to find it.

The proof is based on Kawamata BPF theorem, and Demailly's holomorphic Morse inequalities.

Kawamata BPF:
Let $L$ be a nef line bundle on a projective manifold $M$ such that $L^{\otimes a}\otimes K_M^{-1}$ is big for some $a$. Then $L$ is semiample.

Demailly: $$ h^1(E^k) - h^0(E^k) \leq -\frac{k^n}{n!} \int_U \left(\frac{\sqrt 1}{2\pi} \Theta_E\right)^n + o(k^n), $$ where $E$ is a line bundle, $\Theta_E$ its curvature, and $U\subset M$ the set where $\Theta_E$ has at least $n-1$ positive eigenvalues.

Semipositive line bundles are clearly nef. To apply Kawamata to the semipositive line bundle $L$, we need only to check that $L^{\otimes a}\otimes K_M^{-1}$ is big. However, for very big $a$ the curvature of $L^{\otimes a}\otimes K_M^{-1}$ is strictly positive outside of a small neighbourhood of the zero set $Z$, which contributes very little to the integral $ \int_U (\frac{\sqrt 1}{2\pi} \Theta_E)^n$, hence this integral is positive, and $E=L^{\otimes a}\otimes K_M^{-1}$ is big, because $h^0(E^k)$ grows as $k^n$.

I would be very grateful to anyone who can give me the reference to this, or point out where I did an error (or give a counterexample).

Reference:
Kawamata, Y., Pluricanonical systems on minimal algebraic varieties, Invent. Math., 79, (1985), no. 3, 567-588.

J.-P. Demailly, Holomorphic Morse inequalities, Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), 93--114, Proc. Sympos. Pure Math., 52, Part 2, Amer. Math. Soc., Providence, RI, 1991.

Let $M$ be a compact complex manifold equipped with a line bundle $L$ which has curvature which is non-negative and strictly positive outside of a measure zero set $Z$. In his paper "Holomorphic Morse inequalities" Demailly proved that $M$ is Moishezon (this was a new proof of the Grauert-Riemenschneider conjecture, originally due to Siu).

I want to show that if $M$ is projective, $L$ is semi-ample, that is, base point free. I have a very simple argument, which I guess should be known, but I was unable to find it.

The proof is based on Kawamata BPF theorem, and Demailly's holomorphic Morse inequalities.

Kawamata BPF:
Let $L$ be a nef line bundle on a projective manifold $M$ such that $L^{\otimes a}\otimes K_M^{-1}$ is big and nef for some $a$. Then $L$ is semiample.

Here I was wrong, and I need to apply a correction, thanks to the user YangMills. The bundle $L^{\otimes a}\otimes K_M^{-1}$ should be big and nef. Then the argument I gave becomes invalid, unless $Z$ is very special.

Demailly: $$ h^1(E^k) - h^0(E^k) \leq -\frac{k^n}{n!} \int_U \left(\frac{\sqrt 1}{2\pi} \Theta_E\right)^n + o(k^n), $$ where $E$ is a line bundle, $\Theta_E$ its curvature, and $U\subset M$ the set where $\Theta_E$ has at least $n-1$ positive eigenvalues.

Semipositive line bundles are clearly nef. To apply Kawamata to the semipositive line bundle $L$, we need only to check that $L^{\otimes a}\otimes K_M^{-1}$ is big. However, for very big $a$ the curvature of $L^{\otimes a}\otimes K_M^{-1}$ is strictly positive outside of a small neighbourhood of the zero set $Z$, which contributes very little to the integral $ \int_U (\frac{\sqrt 1}{2\pi} \Theta_E)^n$, hence this integral is positive, and $E=L^{\otimes a}\otimes K_M^{-1}$ is big, because $h^0(E^k)$ grows as $k^n$.

I would be very grateful to anyone who can give me the reference to this, or point out where I did an error (or give a counterexample).

Reference:
Kawamata, Y., Pluricanonical systems on minimal algebraic varieties, Invent. Math., 79, (1985), no. 3, 567-588.

J.-P. Demailly, Holomorphic Morse inequalities, Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), 93--114, Proc. Sympos. Pure Math., 52, Part 2, Amer. Math. Soc., Providence, RI, 1991.