# On sufficient conditions on an analytic map to be algebraic(=regular)

Let $X$ and $Y$ be smooth quasi-projective varieties defined over $\mathbf{C}$ and let $$f:X(\mathbf{C})\rightarrow Y(\mathbf{C})$$ be a holomorphic map (not necessarily regular=algebraic). Then it is natural to ask what are additional conditions that one can impose on the data $(f,X,Y)$ in order to force $f$ to be algebraic. Let me give 3 examples of such conditions:

1) Assume that $f$ is finite, unramified and that $X(\mathbf{C})$ has only one algebraic structure. Then a combination of Grauert-Remmert and GAGA implies that $f$ is algebraic. Note that (a postiori) the finiteness assumption on $f$ is essential since one has for example the exponential map $exp:\mathbf{C}\rightarrow\mathbf{C}^{\times}$ which is not algebraic but satisfy all the other assumptions (except the finiteness). Moreover, in general, it is also essential to assume that $X(\mathbf{C})$ has only one algebraic structure since there are examples of complex manifolds with at least 2 non-equivalent algebraic structures.

2) If $X$ is compact then from GAGA we ge automatically that $f$ is algebraic

3) Say that $X$ is a curve and $Y=\mathbb{P}^1(\mathbf{C})-\{0,1,\infty\}$. Then Picard's theorem (+removable singularity result) imply that $f$ is meromorphic on the compactification of $X$ and therefore $f$ is algebraic. (If I remember correctly, I think that there is some kind of generalization of Picard's result to higher dimension from the work of Kwack).

So with these 3 examples in mind, here is my question:

Q: what is known in the litterature about additional conditions that one may impose on the data $(f,X,Y)$ in order to force $f$ to be algebraic?

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1) is not quite correct since there are non-isomorphic algebraic varieties which are isomorphic as complex manifolds. –  ulrich Aug 16 '11 at 15:41
@Ulrich, Could you give me such an example? –  Hugo Chapdelaine Aug 16 '11 at 15:52
More examples of non-isomormphic algebraic varieties which are analytically isomorphic are given here: mathoverflow.net/questions/68421/… –  Sam Gunningham Aug 16 '11 at 17:00
@Hugo: There will always be an algebraic structure on $X$ which will make $f$ algebraic (as you say, by Grauert-Remmert and GAGA) but there could be algebraic structures on $X$ which are not the restriction of the algebraic structure from $\bar{X}$ (if $\bar{X}$ is fixed). –  ulrich Aug 16 '11 at 17:26
@Hugo: map a fiber of $L$ into $C\times\mathbb{C}$ through the alg.var.isomorphism, then project down to $C$; it must be constant (genus is high), so any isomorphism between the two bundles sends fiber to fiber (possibly on a different point). Now you can produce a bundle isomorphism by composing with the appropriate automorphism of the basis and adding on each fiber the right vector to fix the origin. –  Samuele Aug 16 '11 at 20:33

Borel (1972, J. Diffl. Geometry) proved that $f$ is always algebraic if $Y$ is the quotient of a bounded symmetric domain by a torsion-free arithmetic subgroup. This is a super-generalization of your example 3 (the quotient of the complex upper half plane by $\Gamma(2)$ is isomorphic to the projective line minus three points). The proof uses a generalization of work of Kwack plus the resolution of singularities.

Added: Kwack (1969) generalized the big Picard theorem by proving that any holomorphic map from the punctured unit disk into a hyperbolic complex space can be extended holomorphically to the whole unit disk. [A reduced complex space is said to be hyperbolic if the Kobayashi pseudodistance is a distance (Kobayashi 1967).]

Borel 1972 replaced the punctured disk in Kwack's theorem with a product of punctured disks and disks.

Resolution of singularities allows you to realize a smooth algebraic variety as an open subvariety of a smooth projective variety in such a way that the boundary is a divisor with normal crossings (hence analytically a product of punctured disks and disks).

These statements sometimes allow you to extend your map to an analytic map of projective varieties, where you can apply Chow's theorem to prove that it is regular.

References:

Borel, Armand. Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem. J. Differential Geometry 6 (1972), 543--560.

Kwack, Myung H., Generalization of the big Picard theorem. Ann. of Math. (2) 90 1969 9--22.

Kobayashi, Shoshichi, Invariant distances on complex manifolds and holomorphic mappings. J. Math. Soc. Japan 19 1967 460--480.

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Thanks Anon. Can you give me a precise state? What is the analogue $\mathbb{P}^1-\\{0,1,\infty\}}$ in higher dimension? –  Hugo Chapdelaine Aug 17 '11 at 16:16
Thanks a lot Anon for the precisions and the references –  Hugo Chapdelaine Aug 23 '11 at 16:28