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Let $X$ be a smooth algebraic surface over $\Bbb{C}$.

From the exponential sheaf sequence $1\to 2\pi i\,\Bbb{Z}\to \mathcal{O}_X\to\mathcal{O}_X^*\to 1$ we get a long exact sequence on cohomology, from which we obtain the maps $$H^1(X,\mathcal{O}_X)\to\text{Pic(X)}\xrightarrow{\delta} H^2(X,\Bbb{Z})$$

The Néron-Severi group is given by $\text{NS}(X):=\text{im }g$ and, by definition, divisors $D_1,D_2\in\text{Pic}(X)$ are algebraically equivalent if and only if $\delta(D_1)=\delta(D_2)$.

Consequently, if $h^1(X,\mathcal{O}_X)=0$ then by exactness $\delta$ is injective and induces an isomorphism $\text{Pic}(X)\simeq \text{NS}(X)$, i.e. linear equivalence and algebraic equivalence coincide in $X$.

(if there is something wrong in the above, please let me know)

Now my question is: in a more general setting, do linear and algebraic equivalencies coincide when $X$ is over an algebraically closed field $k$ and $h^1(X,\mathcal{O}_X)=0$ holds?

I hope the answer is yes, but I don't know a general version of the exponential sheaf sequence, so I don't know how to get the sequence on cohomology.

P.S.: if additional hypothesis on $X$ are necessary, I'd happy to know which.

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  • $\begingroup$ I'm not sure what you are asking. The exponential sequence works for any (analytic) non-singular variety over $\mathbb C$. By the GAGA principle, this result extends to projective, algebraic non-singular varieties of arbitrary dimension (not just surfaces). If you want to chane the base field though, I don't know. $\endgroup$ Jun 11 '21 at 15:13
  • $\begingroup$ math.stackexchange.com/questions/1583320/… $\endgroup$
    – ali
    Jun 11 '21 at 20:43
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    $\begingroup$ as said in the answer in the above question, if $X$ is projective and smooth,$Pic^0(X)$ is a connected (abelian) variety of dimension $dim H^1(X,O_X)$and so vanish iff $H^1(X,O_X)$ vanishes. $\endgroup$
    – ali
    Jun 11 '21 at 20:50
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Question: "I hope the answer is yes, but I don't know a general version of the exponential sheaf sequence, so I don't know how to get the sequence on cohomology. P.S.: if additional hypothesis on X are necessary, I'd happy to know which."

Answer: In general there is a well defined map

$$dlog: \mathcal{O}_X^* \rightarrow \Omega^1_{X/k}$$

defined locally by $dlog(a):=\frac{da}{a}$. The map $dlog$ is a map of sheaves of abelian groups and induce a well defined map

$$c:H^1(X,\mathcal{O}_X^*)\cong Pic(X) \rightarrow H^1(X, \Omega^1_{X/k})$$

for any field $k$ (or base scheme $S$). There is no exponential map $exp: \mathcal{O}_X \rightarrow \mathcal{O}_X^*$ in general, and there is no definition of $H^2(X, \mathbb{Z})$ in general. If $k$ is the complex number field and if $\Gamma \subseteq k^2$ is a lattice, you may realize all elliptic curves $E(\Gamma)$ over $k$ as quotients $E(\Gamma):=k^2/\Gamma$. Similar constructions exist for abelian varieties over $k$. Similar constructions do not exist for other fields such as number fields $\mathbb{Q} \subseteq K$.

Example: Abelian varieties over a number field $K$ "usually" arise as the Jacobian variety/Picard variety $J(C)$ of some curve $C$ over $K$. The "Jacobian variety" is defined using the "language of representable functors". For an elliptic curve $E(\Gamma)$ over the complex numbers we may similarly define the Jacobian variety $J(E(\Gamma))$ and there is an isomorphism $J(E(\Gamma)) \cong E$ by Hartshorne Thm.IV.4.11.

The relationship between $J(X)$ and Pic$^0(X)$

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