Iitaka subadditivity, variation, and logarithmic additivity

The explanation

Take a curve shaped like a doughnut's surface, a torus. It carries exactly one special form of each degree, up to scale. The counts run 1, 1, 1, 1 and never grow. Now take a two-holed doughnut surface, which mathematicians call a curve of genus 2. In degrees 2, 3, 4 and 5 it carries 3, 5, 7 and 9 independent forms. The count grows like the degree itself. A sphere carries none at all.

These forms are called pluricanonical forms. The Kodaira dimension records how fast their count grows. Growth like the degree to the power 1 gives Kodaira dimension 1, as for the genus-2 curve. Constant counts give 0, as for the torus. No forms at all give minus infinity, as for the sphere. A space whose count grows as fast as its own dimension allows is called of general type.

Now multiply two genus-2 curves into a product. Its counts multiply too: 9, 25, 49 and 81 in degrees 2 to 5. That is 3², 5², 7² and 9². Growth like the degree squared gives Kodaira dimension 2, which is 1 plus 1. A product is the simplest fibration. A fibration is a space laid over a base, with one copy of a fiber above each base point. In a general fibration the fibers can twist and change from point to point.

Iitaka asked whether twisting can ever destroy forms. His subadditivity question asks for this rule. The total space's Kodaira dimension is at least the fiber's plus the base's. A product meets it with equality. Iitaka set up the language in 1971. An explicit version for open spaces appeared in his 1977 note.

Many special cases followed. Kawamata treated curve bases in 1982 and fibers with good minimal models in 1985. Viehweg handled bases of general type in 1983. Kollár handled fibers of general type in 1987. Birkar settled total dimension at most six in 2009. Each case drew its positivity from a different part of the fibration. No single method covered all of them. Earlier preprints by Tsuji and Maehara state general inequalities, but these papers do not use themSource 1.

This family of five papers claims to prove the full rule and several stronger forms. The main paper covers Campana's orbifold versionSource 2. It counts multiple fibers, where the fiber above a point is repeatedSource 3. It also allows open spaces, where forms may have poles along a removed boundarySource 4. The open version uses the logarithmic Kodaira dimension. The complex plane with one point removed shows why that matters. It has logarithmic Kodaira dimension 0. The sphere that completes it, by adding two points, has minus infinitySource 5. A second paper claims Popa's stronger inequalitySource 6. It counts the directions in which the fibers truly vary. When that number beats the base's term, it can replace it. That paper needs a base with logarithmic Kodaira dimension at least 0Source 6. A third claims the reverse inequality, and so exact equality, when the family is smooth over the open baseSource 7.

The single obstacle was finding enough global forms. They must keep the growth of the fibers and of the base at onceSource 8. The papers split each fiber into a part of general type and a part with Kodaira dimension 0Source 9. A part with Kodaira dimension 0 carries just one form in the relevant degrees. A line of Hodge classes records how that one form changes along the base. These are measurements of a fiber's shape, built from its forms. The papers claim this line carries enough positivitySource 10. It comes through a second map, to a space that tracks how fibers changeSource 11. A fifth paper supplies a related positivity result for compact Kähler fibrationsSource 12.

The rules give lower bounds, not values. They do not compute the Kodaira dimension of any particular space. The fiber in each statement is a very general one, chosen away from a countable set of bad pointsSource 13. The equality result needs smooth boundary pieces over the open baseSource 14. The same holds wherever those pieces cross.

How sure is this? A computer proof checker called Lean has checked one branch of the equality theorem. It covers a very general base point whose fiber has logarithmic Kodaira dimension minus infinity. Then the total space has minus infinity too, and its logarithmic pluricanonical forms of positive degree all vanish. The other branches of that theorem are outside the check. So are the subadditivity, orbifold and variation theorems. No specialist review has been published.

What it doesn't show. This does not compute the Kodaira dimension of any particular space; it only bounds it.