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# Nevanlinna theory and the Bott-Chern classes

Nevanlinna theory deals with the following type of questions: Let be a holomorphic map. Given in , what is the inverse image ? Since is noncompact, there may be infinitely many points in the pre-image . Sometimes will be empty, meaning that misses the point in .

The exponential map misses exactly two points, 0 and , in . According to a classical theorem of Picard, a nonconstant holomorphic map cannot miss more than two points.

Nevanlinna theory refines Picard's theorem in a beautiful way. To each , it attaches a real number between 0 and inclusive, the deficiency index of . The deficiency index is a normalized way of counting the number of points in the inverse image. If is empty, then the deficiency index is .

In this context the first main theorem of Nevanlinna theory says that a nonconstant holomorphic map has deficiency index 0 almost everywhere. The second main theorem yields the stronger inequality:

Ahlfors generalized these two theorems to holomorphic maps with values in a complex projective space .

In [38] Bott and Chern souped up Nevanlinna's hard analysis to give a more conceptual proof of the first main theorem.

A by-product of Bott and Chern's excursion in Nevanlinna theory is the notion of a refined Chern class, now called the Bott-Chern class, that has since been transformed into a powerful tool in Arakelov geometry and other aspects of modern number theory.

Briefly, the Bott-Chern classes arise as follows. On a complex manifold the exterior derivative decomposes into a sum , and the smooth -forms decompose into a direct sum of -forms. Let be the space of smooth -forms on . Then the operator makes into a differential complex. Thus, the cohomology is defined.

A Hermitian structure on a holomorphic rank vector bundle on determines a unique connection and hence a unique curvature tensor. If and are the curvature forms determined by two Hermitian structures on and is a -invariant polynomial on , then it is well known that and are global closed forms on whose difference is exact:

for a differential form on . This allows one to define the characteristic classes of as cohomology classes in .

In the holomorphic case, and are -forms closed under . Bott and Chern found that in fact,

for some -form . For a holomorphic vector bundle , the Bott-Chern class of associated to an invariant polynomial is the cohomology class of , not in the usual cohomology, but in the cohomology of the complex .

Next: Characteristic numbers and the Up: The life and works Previous: The index theorem for
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