Shafarevich counterexamples in dimension two and with large fundamental group

The explanation

Draw the unit circle in the complex plane. Ask which points z satisfy |g(z)| ≤ the largest value of |g| on the circle, for every polynomial g. Take g(z) = z³. On the circle |g| is always 1, so any point with |z| > 1 fails. Inside the circle, every point passes, by the maximum principle. The set of passing points is called the holomorphic hull of the circle. Here it is the closed unit disc, a bounded closed set. A complex space is holomorphically convex when every such hull of a compact set is compactSource 1. The complex plane passes this test.

Every curved space has a universal cover. It unrolls every loop that cannot be shrunk. An elliptic curve, the surface of a doughnut, unrolls to the whole complex plane. A curve with two or more holes unrolls to the open disc. Both covers are holomorphically convex. Shafarevich asked whether this always happens, for every smooth projective varietySource 2. A projective variety is a shape cut out by polynomial equations. This is the Shafarevich conjecture on holomorphic convexity.

The conjecture held in many cases. Gurjar and Shastri proved it for elliptic surfaces in 1985, and Katzarkov for virtually nilpotent fundamental groups, an especially tame kindSource 3. In 2012, Eyssidieux, Katzarkov, Pantev and Ramachandran proved the linear caseSource 4. There the loop group of the variety, its fundamental group, can be faithfully written with complex matrices.

A known trap shows how it could fail. Take a connected chain of compact curves, where the loops through the whole chain cannot be shrunk. Lift it to the universal cover. It becomes an endless chain of compact pieces. Every holomorphic function is constant on each compact pieceSource 5. Neighbours share a point, so the constants all agree. The hull of one point then contains the whole endless chain, which is not compactSource 6. In 1998, Bogomolov and Katzarkov suggested surfaces built this way. Their key step, that a certain group stays infinite, remained a conjectureSource 7.

This family of two papers claims two counterexamples. The surface paper claims a smooth projective surface whose universal cover is not holomorphically convexSource 8. Its chain is made of spheres, glued at single points. It starts from a family of spheres with 48 moving puncturesSource 9. The hard part is keeping the group infinite after all missing fibers are filled inSource 10. The paper uses a counting method for groups, in the style of Golod and Shafarevich. By the linear case above, the surface's fundamental group then admits no faithful matrix pictureSource 11.

The fourfold paper targets a stronger form. Say a variety has large fundamental group when its universal cover contains no compact complex subvariety of positive dimension. The stronger form asks that this cover then be SteinSource 12. Stein spaces are the most flexible kind, with enough holomorphic functions to embed in some ℂᴺ. The paper claims a smooth projective fourfold with large fundamental group whose universal cover is not SteinSource 13. A fourfold has four complex dimensions. Inside it sits a surface A whose own loop group is ℤ⁴. Only a copy of ℤ survives in the fourfold's group, so 4 − 1 = 3 loop directions die there. In the universal cover, one direction of A unrolls into a line and three stay closed loops. The lifted surface is a line times a three-dimensional torus, with a three-dimensional holeSource 14. A Stein surface can have holes of real dimension at most twoSource 15. The method needs no matrix picture of the groupSource 16.

How sure is this? No Lean formalization is listed for either paper. Each rests on delicate claims that certain groups stay infinite. No specialist review has been published.

What it doesn't show. This does not affect the linear case, where the fundamental group has a faithful matrix picture.