Over a general field, does $h^1(X,\mathcal{O}_X)=0$ imply $\text{Pic}(X)\simeq \text{NS}(X)$?

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.

• 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. Jun 11 '21 at 15:13
• math.stackexchange.com/questions/1583320/…
– ali
Jun 11 '21 at 20:43
• 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.
– ali
Jun 11 '21 at 20:50

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$^0(X)$