ag.algebraic geometry – Construction of holomorphic line bundles on complex torus Answer

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ag.algebraic geometry – Construction of holomorphic line bundles on complex torus

This is an argument for constructing positive line bundles on complex torus. From some knowledge of Abelian varieties, such as Riemann conditions, we know that it is wrong. But I don’t know where this argument goes wrong.

let $\{ v_{s,i} \}_{i=1}^{\infty}$ be sequences of points in $\mathbb{C}^2$where $s=1,2,3,4$. Suppose that $v_{1,i} ,v_{2,i} ,v_{3,i} ,v_{4,i} $ converge to $(1,0),( \sqrt{-1} ,0 ) ,(0,1),( 0,\sqrt{-1} ) $ respectively when $i$ tends to $\infty$. We will denote by $ \Lambda_i $ the lattice generated by $v_{1,i} ,v_{2,i} ,v_{3,i} ,v_{4,i} $. Similarly, we will write the lattice generated by $(1,0),( \sqrt{-1} ,0 ) ,(0,1),( 0,\sqrt{-1} ) $ like $\Lambda_\infty$. Then the quotient spaces $X_i =\mathbb{C}^2 /\Lambda_i $ are $2$-dimensional complex torus, and $X_\infty =\mathbb{C}^2 /\Lambda_\infty $ is an Abelian variety. let $\left( L_\infty , h_\infty \right) $ be a Hermitian holomorphic line bundle on $X_\infty$ look for that $-\sqrt{-1} \partial\bar{\partial} \log h_\infty =2\pi \omega_{Euc} $where $\omega_{Euc} $ is the Euclidean metric on $\mathbb{C}^2$.

let $\pi_i $ denote the projection map $\mathbb{C}^2 \to \mathbb{C}^2 /\Lambda_i =X_i $and $\pi_\infty$ denote the projection map $\mathbb{C}^2 \to \mathbb{C}^2 /\Lambda_\infty =X_\infty $. choosing $N$ points $\{ x_j \}_{j=1}^{N} $ in $\mathbb{C}^2$ look for that $X_{\infty} = \cup_{j=1}^{N} \pi_\infty \left( B_{10^{-3}}\; (x_j) \right) $. Since $ \pi_\infty \left( B_{10^{-1}}\; (x_j) \right) \cong B_{10^{-1}}\; (x_j) $ is a stone manifold, we see that $L_{\infty} |_{ \pi_\infty \left( B_{10^{-1}}\; (x_j) \right)} $ is trivial, and hence we can find holomorphic sections $e_{\infty ,j} \in H^0 \left( \pi_\infty \left( B_{10^{-1}}\; (x_j) \right) , L_\infty \right) $ satisfying that $h_{\infty} \left( e_{\infty, j} ,e_{\infty ,j} \right) =e^{-\pi d^2_{j}} $where $d_j $ is the distance from $\pi_{\infty} ( x_j )$. let $f_{jk ,\infty} $ be the holomorphic function on $\pi_\infty \left( B_{10^{-1}}\; (x_j) \right) \cap \pi_\infty \left( B_{10^{-1}}\; (x_k) \right ) \neq \emptyset $ satisfying $e_{\infty ,j} =f_{jk,\infty} e_{\infty ,k} $. For sufficiently large $i$we have
$$ \pi_i^{-1} \left( \pi_i (B_{10^{-2} }\; (x_k) ) \right) \cap B_{10^{-2}} \;(x_j) \ Subset \pi_\infty^{-1} \left( \pi_\infty (B_{10^{-1} }\; (x_k) ) \right) \cap B_{10^{-1}} \;( x_j) . $$
Consider the following holomorphic functions on each $ \pi_i^{-1} \left( \pi_i (B_{10^{-2} }\; (x_j) ) \right) \cap \pi_i^{-1} \left( \pi_i (B_{10 ^{-2} }\; (x_k) ) \right) \neq \emptyset :$
$$ \psi_{jk,i} = \left( f_{jk,\infty} \circ \pi_\infty \circ \left( \pi_{i} |_{B_{10^{-2}} \; (x_j)} \right)^{-1} \right) \cdot \left( f_{jk,\infty} \circ \pi_\infty \circ \left( \pi_{i} |_{B_{10^ {-2}} \; (x_k)} \right)^{-1} \right) .$$
Then $\{ \psi_{jk,i} \}_{j,k=1}^{N} $ is a family of transition functions on $X_i$ for sufficiently large $i$. It determines a line bundle $L_i$ on $X_i$.

Choosing cut-off functions $\eta_{j} \in C_0^{\infty} (B_{10^{-2}} \; (x_j)) $ for each $j=1,\cdots ,N$satisfying that $0\leq \eta\leq 1$ other $\eta_j |_{B_{10^{-3}} \; (x_j) } =1 $. set $$ \eta_{j,i} = \frac{\eta_{j} \circ \left( \pi_i |_{ \pi_i (B_{10^{-2} }\; (x_j) )} \right) ^{-1} }{ \sum_{l=1}^{N} \eta_{l} \circ \left( \pi_i |_{ \pi_i (B_{10^{-2} }\; (x_l) )} \right)^{-1} } \in C_0^{\infty} (\pi_i \left( B_{10^{-2}} \; (x_j) \right) ) .$$ Then $$h_{i} \big|_{\pi_i (B_{10^{-2} } \; (x_j) )} = e^{- \sum_{l} \eta_{l,i} \left( 2\pi d^2_{l,i} \; – \; 2\log | \psi_{jl,i} \;| \right) } $$ gives a Hermitian metric $h_i$ on $L_i$where $d_{l,i}$ is the distance from $\pi_i (x_l)$. lettering $i\to\infty $we see that $$\left| \psi_{jk,i} \right| \circ \pi_{i} \to \left| f_{jk,\infty} \right|^2 \circ \pi_\infty = e^{\pi d^2_k -\pi d_j^2} \circ \pi_\infty $$ on $ \pi_i^{-1} \left( \pi_i (B_{10^{-2} }\; (x_k) ) \right) \cap B_{10^{-2}} \;(x_j) $ uniformly. Note that the functions $\pi d^2_{j,i} – \log | \psi_{jl,i} | -\pi d^2_{k,i} + \log | \psi_{kl,i} | $ are pluriharmonic. Hence we have
\begin{eqnarray*} -\sqrt{-1} \partial\bar{\partial} \log h_i & = & \sqrt{-1} \partial\bar{\partial} \left( \sum_{l} \ eta_{l,i} \left( 2\pi d^2_{l,i} \; – \; 2\log | \psi_{jl,i} | \right) \right) \\ & = & 4\ pi\omega_{Euc} + \sqrt{-1} \sum_l \left( 2\pi d^2_{l,i} \; – \; 2\log | \psi_{jl,i} | – 2\pi d^2_{j,i} \right) \partial\bar{\partial} \eta_{l,i} \\ & & + \sqrt{-1} \sum_l \partial \eta_{l,i} \wedge \bar{\partial} \left( 2\pi d^2_{l,i} \; – \; 2\log | \psi_{jl,i} | – 2\pi d^2_{j,i} \ right) \\ & & + \sqrt{-1} \sum_l \partial \left( 2\pi d^2_{l,i} \; – \; 2\log | \psi_{jl,i} | – 2 \pi d^2_{j,i} \right) \wedge \bar{\partial} \eta_{l,i} \geq 2\pi \omega_{Euc} , \end{eqnarray*}
on $ \pi_i \left( B_{10^{-2}} \; (x_j) \right) $for sufficiently large $i$. It follows that for sufficiently large $i$, $(L_i ,h_i)$ is a positive line bundle on $X_i$.

But the set of Abelian varieties cannot contain an open subset of the moduli space of complex torus, contradiction.

So, where is this proof wrong?

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