We will begin our study about line bundles on abelian varieties. The main result of this note would be following theorem: if $L$ is an ample line bundle on an abelian variety $A$, then the map $\lambda_L$ is surjective. By keeping in mind the circle of ideas we have illustrated in the previous note, it is easier to keep track on the proofs.
1. Kunneth's formula and applications.
Theorem 1.1 (Kunneth's formula). Let $X,Y$ be projective varieties and $F,G$ coherent sheaves on $X,Y$, respectively. Denote $p_1: X\times Y\to X, p_2:X\times Y\to Y$ the projections, and $F\boxtimes G:=p_1^*F\otimes_{\mathscr{O}_{X\times Y}}p_2^*G$. Then
$$H^n(X\times Y, F\boxtimes G)=\bigoplus_{p+q=n}H^p(X,F)\otimes H^q(Y,G)$$
Using Kunneth's formula, one can deduce the vanishing of cohomology group for non-trivial line bundles on $Pic^0(A)$, where $A$ is an abelian variety.
Proposition 1.2. Let $L\in Pic^0(A)$ be a non-trivial line bundle, then $H^n(A,L)=0$, for all $n\ge 0$, and $H^n(A\times A, L)=0$.
Proof. For $n=0$, we assume that $H^0(A,L)\ne 0$, this yields there exists an effective divisor $D\in Div(A)$, such that $L\cong \mathscr{O}_A(D)$. But then, since $L$ is anti-symmetric, we have $(-1)^*L\cong L^{-1}$, and hence $\mathscr{O}_A((-1)^*D)\cong \mathscr{O}_A(-D)$, and from this, $D+(-1)^*D=0$, and $D=0$, since $D$ is effective. This yields a contradiction, since $L\ne \mathscr{O}_A$.
We now look at the higher cohomology groups of $L$, let $i$ be the smallest positive integer such that $H^i(A,L)\ne 0$. Because $m^*L\cong p_1^*L\otimes p_2^*L$, we have by Kunneth's formula
$$H^n(A\times A, m^*L)\cong H^n(A\times A, p_1^*L\otimes p_2^*L)\cong \bigoplus_{p+q=n}H^p(A, L)\otimes H^q(A,L)$$
This yields by the smallest of $i$, and the zero cohomology group of $L$ vanishes, that $H^n(A\times A, m^*L)=0$. The diagram
$$A\xrightarrow{id\times \{0\}}A\times A\xrightarrow{m}A$$
has the composition of two arrows the identity map. This yields $$H^n(A)\to H^n(A\times A)\to H^n(A)$$ has the composition of two arrows again the identity map. From this, one has $H^n(A)=0$, for all $n$. (Q.E.D)
Let us take a look on the case of elliptic curves $E$, $Pic^0(E)$ consists of divisors of the form $D:=[P]-[O]$, and if $P\ne O$, it is of degree zero but not principal, and hence $h^0(E,D)=0$. And by Riemann-Roch, $h^1(E,D)=0$ again. Because $\dim E=1$, $h^n(E,D)=0$ for all $n\ge 2$.
The result of the proposition above is important and will be used very frequently throughout the note.
2. Two important corollaries from Mumford's book.
During these notes, we often use some techniques: pullback line bundles on an abelian variety $A$ to line bundles on $A\times A$ via the projection maps, or multiplication map, as well as restrict line bundles on $A\times A$ to $A\times \{a\}$, for some $a\in A$. In algebraic geometry language, it is the $A\times \{a\}$ is the fiber of $a$ via the projection map $p_2$, and $L|_{A\times \{a\}}$ is again the fiber of $a$ with respect to the invertible sheaf $L$ via the map $p_2$. The two theorems below gives us the relation between the cohomology groups of the fiber and the higher direct image via proper map between projective varieties. Note that we need the information about higher direct image to exact more information about higher cohomology groups via Leray spectral sequence.
Theorem 2.1 (Corollary 2, Page 50, Mumford's book). Let $f:X\to Y$ be a proper morphism between noetherian schemes, and $F$ is a coherent sheaf on $X$, and $Y$ is reduced and connected, then for all $p$, the following are equivalent:
(i) $y\mapsto \dim_{k(y)} H^p(X_y,F_y)$ is a constant function, where $X_y, F_y$ are fibers of $y$ over $f$.
(ii) $R^pf_*F$ is a locally free sheaf on $Y$, and for all $y\in Y$, $R^pf_*F\otimes_{\mathscr{O}_Y}k(y)\cong H^p(X_y,f_y)$.
If these conditions are satisfied, we have further that $R^{p-1}f_*F\otimes_{\mathscr{O}_Y} k(y)\cong H^{p-1}(X_y, f_y)$.
Now, we will use this fact to deduce some useful information about line bundles on abelian varieties.
Corollary/Lemma 2.2. Let $L$ be a line bundle on $A\times A$, such that $L_{A\times \{a\}}Pic^0(A)$ is non-trivial on $A\times A$, then $H^n(A\times A, L)$ vanishes for all $n$.
Proof. Due to Proposition 1.2, we have $H^n(A\times \{a\}, L|_{A\times \{a\}})=0$, for all $n$, this yields the map $a\mapsto h^p(A\times \{a\}, L|_{A\times \{a\}})$ is a constant function. Hence, by Theorem 2.1, we have $R^qp_{2,*}L$ is trivial. This yields $H^p(A, R^qp_{2,*}L)$ is trivial for all $n$. By Leray spectral sequence, there exists a spectral sequence such that
$$E_2^{p,q}=H^p(A,R^qp_{2,*}L)\Rightarrow H^{p+q}(A\times A, L)$$
And this yields $H^{n}(A\times A, L)$ is zero. (Q.E.D)
Now, if $L$ is an ample line bundle on $A$, and $M$ is any line bundle on $A$, let $N:=\Lambda(L)\otimes p_2^*M=m^*L\otimes p_1^*L^{-1}\otimes p_2^*L^{-1}\otimes p_2^*M$, where $\Lambda(L)$ is the Mumford line bundle on $A\times A$. Once we restrict $N$ to $A\times \{a\}$, we can see $N|_{A\times \{a\}}=\lambda_L(a)\in Pic^0(A)$, and $N|_{A\times \{a\}}$ is trivial iff $a\in K(L)$, which is finite. This yields by Proposition 1.2 and Theorem 2.1 that for any open subset $U\subset A\setminus K(L)$, $R^qp_{2,*}N|_U$ vanishes. Hence $Supp(R^qp_{2,*}N)\subseteq K(L)$, and it has dimension zero. This implies $H^p(A,R^np_{2,*}N)$ vanishes whenever $p\ge 1$. Now, making use of Leray's spectral sequence, we have
$$E_2^{p,q}=H^p(A,R^qp_{2,*}N)\Rightarrow H^{p+q}(A\times A, N)$$
When $p=0$, $E_2^{0,q}=H^0(A,R^qp_{2,*}N)\Rightarrow H^{q}(A\times A, N)$. Using complex sequence in the second page of the spectral sequence, we have
$$E^{1,q-2}=H^1(A,R^{q-2}p_{2,*}N)=0\to E_2^{0,q}\to 0$$
And hence, $E_{\infty}^{0,q}$ is actually stable at the second page. This yields $R^qp_{2,*}N\cong H^{q}(A\times A, N)$. And we have proved
Corollary/Lemma 2.3. Let $L$ be an ample line bundle on $A$, and $M$ is any line bundle on $A$. Let $N:=\Lambda(L)\otimes p_2^*M$ be a line bundle on $A\times A$, then $R^qp_{2,*}N\cong H^q(A\times A, N)$ for all $q$.
The following theorem also reflects the relation between higher direct image sheaves and the cohomology groups of fibers.
Theorem 2.4 (Corollary 4, page 53, Mumford's book). Let $X,Y,F$ be defined as in Theorem 2.1, if $R^kf_*F=0$, for $k\ge k_0$, then $H^k(X_y, F_y)=0$ for all $y\in Y$.
Combining all together, we are now ready for the proof of the main theorem.
Theorem 2.5. Let $L$ be an ample line bundle on $A$, then $\lambda(L)$ is a surjective map.
Proof. Assume that $\lambda(L)$ is not surjective, i.e. there exists $M\in Pic^0(A)$, such that $\lambda_L(a)\ne M$ for all $a\in A$, this yields $K_a:=\lambda_L(a)\otimes M^{-1}$ is a non-trivial line bundle on $Pic^0(A)$.
Let $N=\Lambda(L)\otimes p_2^*M^{-1}$. Once we restrict $N$ on $\{a\}\times A$, we get $K_a$, which is nontrivial in $Pic^0(A)$. By Corollary 2.2, $H^n(A\times A, N)$ vanishes. Again, if we restrict $N$ on $A\times \{a\}$, we get $\lambda_L(a)$, by Corollary 2.3, we have $R^qp_{2,*}N\cong H^q(A\times A, N)=0$, for all $q$. Using Theorem 2.4, we have $H^0(A\times \{0\}, N|_{A\times \{0\}})=0$. But it is a contradiction, since $\mathscr{O}_A(A)\ne 0$. Hence, $\lambda(L)$ is surjective. (Q.E.D)
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