# Correction to: Holomorphic curves in Shimura varieties

- 40 Downloads

## 1 Correction to: Arch. Math. 111 (2018), 379–388 https://doi.org/10.1007/s00013-018-1227-4

This erratum addresses an error in the application of a theorem of Hwang and To in the cited paper.

Erwan Rousseau pointed out that in the proof of [2, Theorem 3.2] the result of [3] about volumes of analytic sets in Hermitian symmetric domains cannot be applied as the set considered in the proof is not in general analytic. To fix this, it is enough to substitute [2, Theorem 3.6] with Proposition 0.7.

We start by recalling some definitions about o-minimal structures and cell decomposition.

Fix an o-minimal expansion \( \tilde{\mathbb {R}} \) of the field of real numbers. By definable, we mean definable in \(\tilde{\mathbb {R}}\).

### Definition 0.1

*cell*is a subset of \( \mathbb {R}^{n} \) defined inductively as follows:

A (0) -cell is a point,

a (1) -cell is a segment,

an \( (i_{1},\ldots ,i_{n-1},0) \)-cell is the graph of a definable function on an \( (i_{1},\ldots ,i_{n-1}) \)-cell,

- an \( (i_{1},\ldots ,i_{n-1},1) \)-cell is defined as the setwhere$$\begin{aligned}&\left\{ (x_{1},\ldots ,x_{n})\in \mathbb {R}^{n} | f(x_{1},\ldots ,x_{n-1})< x_{n} \right. \nonumber \\&\quad \left. < g(x_{0},\ldots ,x_{n-1}) \text { and } (x_{1},\ldots ,x_{n-1})\in C \right\} \end{aligned}$$(0.1)
*C*is an \( (i_{1},\ldots ,i_{n-1}) \)-cell and*f*and*g*are definable functions on*C*.

### Definition 0.2

A cell is said to be *analytic* if the definable functions used in its inductive definition can be chosen analytic.

### Definition 0.3

*decomposition*of \( \mathbb {R}^{n} \) is a partition of \( \mathbb {R}^{n} \) defined inductively as follows:

- A decomposition of \( \mathbb {R} \) is a partition of the type$$\begin{aligned} \left\{ (-\infty ,a_{1}), (a_{1},a_{2}),\ldots , (a_{k},\infty ),\{a_{1}\},\ldots , \left\{ a_{k} \right\} \right\} , \end{aligned}$$(0.2)
a decomposition of \( \mathbb {R}^{n} \) is a partition of \( \mathbb {R}^{n} \) into finitely many cells \( \left\{ A_{i} \right\} _{i\in I} \) such that the set of projections \( \left\{ \pi (A_{i}) \right\} _{i\in I} \) is a decomposition of \( \mathbb {R}^{n-1} \); here \( \pi \) is the projection onto the first \( n-1 \)-coordinates.

### Theorem 0.4

(Cell decomposition). Given definable sets \( A_{1},\ldots ,A_{l} \) in \( \mathbb {R}^{m} \), there is a decomposition of \( \mathbb {R}^{n} \) which partitions each of the \( A_{i} \).

### Definition 0.5

The o-minimal structure \( \tilde{\mathbb {R}} \) is said to admit *analytic cell decomposition* if it satisfies the cell decomposition theorem with the additional requirement that the cells can all be chosen analytic.

A result of van den Dries and Miller in [1, Section 8] implies.

### Theorem 0.6

(Analytic cell decomposition). The o-minimal structures \( \mathbb {R}_{an} \) and \(\mathbb {R}_{an,exp} \) admit analytic cell decomposition.

The proof of the proposition below follows the guideline of [4, Theorem 2.7].

### Proposition 0.7

*U*be a connected \( \tilde{\mathbb {R}} \)-definable subset of \( \mathcal {X} \) of dimension 2 such that \( \dim _{\mathbb {R}}(\bar{U}\cap \partial \mathcal {X}) = 1\). Fix a point \( x_{0}\in \mathcal {X} \). Then there exist real numbers \( c_{1},c_{2} \) such that for any \( R>0 \) sufficiently large,

*R*.

### Proof

\( \Delta _{\alpha ,\beta } = \left\{ r \exp {i \theta } | 0\le r<1 \text { and } \alpha<\theta <\beta \right\} \) where \( 0<\alpha <\beta \) are real numbers,

\( C_{\alpha ,\beta } = \left\{ \exp {i \theta } | \alpha<\theta <\beta \right\} \),

\( \bar{\Delta _{\alpha ,\beta }} = \Delta _{\alpha ,\beta }\cup C_{\alpha ,\beta } \).

\( \psi (\Delta _{\alpha ,\beta }) \) is contained in \( U' \),

\( \psi \) extends to a real analytic function in a neighbourhood of \( \bar{\Delta _{\alpha ,\beta }} \) such that \( \psi (C_{\alpha ,\beta })\subset \bar{U'}\cap \partial { \mathcal {X} } \) is a non-constant real analytic curve.

*R*tends to infinity, the hyperbolic distance of \(I_{ \alpha , \beta }^{R}\) from the origin tends to infinity and its volume is exponential in

*R*. We now use [4, Lemma 2.4 and Lemma 2.8] to see that there exists a constant \( c_{3}>0 \) such that the image \( \psi (I_{ \alpha , \beta }^{R}) \) in \( \mathcal {X} \) is contained in the geodesic ball \( B_{\mathcal {X},h}(x_{0},R) \). We are now ready to calculate the volumes. Let \( R>0 \) be sufficiently large, then

## Notes

### Acknowledgements

Funding was provided by Engineering and Physical Sciences Research Council (Grant No. EP/L015234/1).

## References

- 1.van den Dries, L., Miller, C.: On the real exponential field with restricted analytic functions. Isr. J. Math.
**85**(1–3), 19–56 (1994)MathSciNetCrossRefGoogle Scholar - 2.Giacomini, M.: Holomorphic curves in Shimura varieties. Arch. Math. (Basel)
**111**(4), 379–388 (2018)MathSciNetCrossRefGoogle Scholar - 3.Hwang, J.-M., To, W.-K.: Volumes of complex analytic subvarieties of Hermitian symmetric spaces. Am. J. Math.
**124**(6), 1221–1246 (2002)MathSciNetCrossRefGoogle Scholar - 4.Ullmo, E., Yafaev, A.: Hyperbolic Ax–Lindemann theorem in the cocompact case. Duke Math. J.
**163**(2), 433–463 (2014)MathSciNetCrossRefGoogle Scholar