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Nevanlinna theory and the Bott-Chern classes

Nevanlinna theory deals with the following type of questions: Let $ f:\mathbb{C}
\to \mathbb{C}P^1$ be a holomorphic map. Given $ a$ in $ \mathbb{C}P^1$ , what is the inverse image $ f^{-1}(a)$ ? Since $ \mathbb{C}$ is noncompact, there may be infinitely many points in the pre-image $ f^{-1}(a)$ . Sometimes $ f^{-1}(a)$ will be empty, meaning that $ f$ misses the point $ a$ in $ \mathbb{C}P^1$ .

The exponential map $ \exp : \mathbb{C}\to \mathbb{C}P^1$ misses exactly two points, 0 and $ \infty$ , in $ \mathbb{C}P^1$ . According to a classical theorem of Picard, a nonconstant holomorphic map $ f:\mathbb{C}
\to \mathbb{C}P^1$ cannot miss more than two points.

Nevanlinna theory refines Picard's theorem in a beautiful way. To each $ a \in \mathbb{C}P^1$ , it attaches a real number $ \delta (a)$ between 0 and $ 1$ inclusive, the deficiency index of $ a$ . The deficiency index is a normalized way of counting the number of points in the inverse image. If $ f^{-1}(a)$ is empty, then the deficiency index is $ 1$ .

In this context the first main theorem of Nevanlinna theory says that a nonconstant holomorphic map $ f:\mathbb{C}
\to \mathbb{C}P^1$ has deficiency index 0 almost everywhere. The second main theorem yields the stronger inequality:

$\displaystyle \sum_{a\in \mathbb{C}P^1} \delta (a) \le 2.

Ahlfors generalized these two theorems to holomorphic maps with values in a complex projective space $ \mathbb{C}P^n$ .

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 $ M$ the exterior derivative $ d$ decomposes into a sum $ d= \partial +
\bar{\partial}$ , and the smooth $ k$ -forms decompose into a direct sum of $ (p,q)$ -forms. Let $ A^{p,p}$ be the space of smooth $ (p,p)$ -forms on $ M$ . Then the operator $ \partial \bar{\partial}$ makes $ \oplus A^{p,p}$ into a differential complex. Thus, the cohomology $ H^*\{ A^{p,p},
\partial \bar{\partial} \}$ is defined.

A Hermitian structure on a holomorphic rank $ n$ vector bundle $ E$ on $ M$ determines a unique connection and hence a unique curvature tensor. If $ K$ and $ K'$ are the curvature forms determined by two Hermitian structures on $ E$ and $ \phi$ is a $ \operatorname{GL}(n,\mathbb{C})$ -invariant polynomial on $ \mathfrak{gl}(n,\mathbb{C})$ , then it is well known that $ \phi (K)$ and $ \phi(K')$ are global closed forms on $ M$ whose difference is exact:

$\displaystyle \phi (K) - \phi(K') = d \alpha

for a differential form $ \alpha$ on $ M$ . This allows one to define the characteristic classes of $ E$ as cohomology classes in $ H^*(M)$ .

In the holomorphic case, $ \phi (K)$ and $ \phi(K')$ are $ (p,p)$ -forms closed under $ \partial \bar{\partial}$ . Bott and Chern found that in fact,

$\displaystyle \phi (K) - \phi(K') = \partial \bar{\partial} \beta

for some $ (p-1,p-1)$ -form $ \beta$ . For a holomorphic vector bundle $ E$ , the Bott-Chern class of $ E$ associated to an invariant polynomial $ \phi$ is the cohomology class of $ \phi (E)$ , not in the usual cohomology, but in the cohomology of the complex $ \{ A^{p,p},
\partial \bar{\partial} \}$ .

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