Hermitian Metric For Normal Intersection Of Divisors

Mar 11, 2025 by ADMIN 53 views

**Hermitian Metric for Normal Intersection of Divisors**

Introduction

In the realm of complex geometry, the study of divisors and their intersections is a fundamental area of research. A divisor is a formal linear combination of codimension-one submanifolds, and its intersection with another divisor is a crucial aspect of understanding the geometry of the underlying manifold. In this article, we will explore the concept of a Hermitian metric for the normal intersection of divisors, which is a key tool in the study of complex geometry.

Background

Let $X$ be a compact complex manifold, and let $A$ and $B$ be two smooth divisors that intersect transversely. The intersection of $A$ and $B$ is denoted by $P=A\cap B$ . The normal bundles of $P$ , $A$ , and $B$ are denoted by $NP$ , $NA$ , and $NB$ , respectively. The normal bundle of a submanifold is a vector bundle that represents the "directions" in which the submanifold is embedded in the ambient manifold.

Hermitian Metric

A Hermitian metric on a complex vector bundle is a way of assigning a positive definite Hermitian inner product to each fiber of the bundle. In the context of the normal intersection of divisors, a Hermitian metric on $NP$ would provide a way of measuring the "size" of the normal directions at each point of $P$ . This is a crucial tool in understanding the geometry of the intersection.

Existence of a Hermitian Metric

The question of whether a Hermitian metric exists on $NP$ is a non-trivial one. In general, the existence of a Hermitian metric on a complex vector bundle is not guaranteed, and it requires additional conditions on the bundle and the manifold. In the case of the normal intersection of divisors, the existence of a Hermitian metric on $NP$ is closely related to the geometry of the intersection.

Transversality Condition

The transversality condition is a key assumption in the study of the normal intersection of divisors. It states that the intersection of $A$ and $B$ is transverse, meaning that the tangent spaces of $A$ and $B$ at each point of $P$ span the entire tangent space of $X$ at that point. This condition is crucial in ensuring that the normal bundle of $P$ is well-defined and that a Hermitian metric can be constructed on it.

Construction of a Hermitian Metric

Assuming that the transversality condition holds, we can construct a Hermitian metric on $NP$ using the following steps:

Choose a Hermitian metric on $NA$ and $NB$ : We start by choosing a Hermitian metric on the normal bundles of $A$ and $B$ . This is a straightforward step, as we can simply use the given Hermitian metrics on $NA$ and $NB$ .
Define a Hermitian metric on $NP$ : Using the Hermitian metrics on $NA$ and $NB$ , we can define a Hermitian metric on $NP$ by taking the orthogonal complement of the tangent spaces of $A$ and $B$ at each point of $P$ .
Verify the positivity of the Hermitian metric: We need to verify that the Hermitian metric on $NP$ is positive definite, meaning that it assigns a positive value to each non-zero vector in the fiber.

Properties of the Hermitian Metric

Once we have constructed a Hermitian metric on $NP$ , we can study its properties and how it relates to the geometry of the intersection. Some of the key properties of the Hermitian metric include:

Positive definiteness: The Hermitian metric on $NP$ is positive definite, meaning that it assigns a positive value to each non-zero vector in the fiber.
Smoothness: The Hermitian metric on $NP$ is smooth, meaning that it is differentiable with respect to the complex structure of $X$ .
Invariance under holomorphic automorphisms: The Hermitian metric on $NP$ is invariant under holomorphic automorphisms of $X$ , meaning that it is preserved under the action of holomorphic automorphisms.

Applications of the Hermitian Metric

The Hermitian metric on $NP$ has several applications in complex geometry, including:

Study of the geometry of the intersection: The Hermitian metric on $NP$ provides a way of studying the geometry of the intersection of $A$ and $B$ , including the study of the normal directions and the size of the intersection.
Construction of moduli spaces: The Hermitian metric on $NP$ can be used to construct moduli spaces of divisors, which are crucial objects in the study of complex geometry.
Study of the cohomology of the intersection: The Hermitian metric on $NP$ can be used to study the cohomology of the intersection of $A$ and $B$ , including the study of the cohomology groups and the cohomology classes.

Conclusion

Q: What is the significance of the Hermitian metric in complex geometry?

A: The Hermitian metric is a crucial tool in complex geometry, as it provides a way of measuring the "size" of the normal directions at each point of the intersection of two divisors. This is essential in understanding the geometry of the intersection and its properties.

Q: What is the transversality condition, and why is it important?

A: The transversality condition is a key assumption in the study of the normal intersection of divisors. It states that the intersection of two divisors is transverse, meaning that the tangent spaces of the two divisors at each point of the intersection span the entire tangent space of the ambient manifold. This condition is crucial in ensuring that the normal bundle of the intersection is well-defined and that a Hermitian metric can be constructed on it.

Q: How is the Hermitian metric constructed on the normal bundle of the intersection?

A: The Hermitian metric on the normal bundle of the intersection is constructed using the following steps:

Choose a Hermitian metric on the normal bundles of the two divisors.
Define a Hermitian metric on the normal bundle of the intersection by taking the orthogonal complement of the tangent spaces of the two divisors at each point of the intersection.
Verify the positivity of the Hermitian metric.

Q: What are the properties of the Hermitian metric on the normal bundle of the intersection?

A: The Hermitian metric on the normal bundle of the intersection has the following properties:

Positive definiteness: The Hermitian metric assigns a positive value to each non-zero vector in the fiber.
Smoothness: The Hermitian metric is smooth, meaning that it is differentiable with respect to the complex structure of the ambient manifold.
Invariance under holomorphic automorphisms: The Hermitian metric is invariant under holomorphic automorphisms of the ambient manifold.

Q: What are the applications of the Hermitian metric on the normal bundle of the intersection?

A: The Hermitian metric on the normal bundle of the intersection has several applications in complex geometry, including:

Study of the geometry of the intersection: The Hermitian metric provides a way of studying the geometry of the intersection, including the study of the normal directions and the size of the intersection.
Construction of moduli spaces: The Hermitian metric can be used to construct moduli spaces of divisors, which are crucial objects in the study of complex geometry.
Study of the cohomology of the intersection: The Hermitian metric can be used to study the cohomology of the intersection, including the study of the cohomology groups and the cohomology classes.

Q: What are some of the challenges in constructing a Hermitian metric on the normal bundle of the intersection?

A: Some of the challenges in constructing a Hermitian metric on the normal bundle of the intersection include:

Ensuring the positivity of the Hermitian metric.
Verifying the smoothness of the Hermitian metric.
Ensuring the invariance of the Hermitian metric under holomorphic automorphisms.

Q: What are some of the open problems in the study of Hermitian metrics on the normal bundle of the intersection?

A: Some of the open problems in the study of Hermitian metrics on the normal bundle of the intersection include:

Developing a more general theory of Hermitian metrics on the normal bundle of the intersection.
Studying the properties of the Hermitian metric on the normal bundle of the intersection in more detail.
Developing new applications of the Hermitian metric on the normal bundle of the intersection.

Conclusion

In this Q&A article, we have explored some of the key questions and answers related to the Hermitian metric for the normal intersection of divisors. We have discussed the significance of the Hermitian metric, the transversality condition, the construction of the Hermitian metric, and its properties and applications. We have also highlighted some of the challenges and open problems in the study of Hermitian metrics on the normal bundle of the intersection.