Pushforward of Chern character

Definition

The pushforward of the Chern character is a fundamental concept in algebraic geometry that describes how the Chern character of a vector bundle behaves under the operation of pushing forward along a morphism between varieties. This process encapsulates the idea of transferring geometric and topological information from one space to another, which is essential for understanding invariants in the context of the Riemann-Roch theorem for curves and surfaces. By relating properties of bundles on different spaces, it becomes possible to derive important results about their dimensions and associated cohomology classes.

5 Must Know Facts For Your Next Test

1. The pushforward of the Chern character is denoted as $f_* c_1(E)$ where $f: X \to Y$ is a morphism and $E$ is a vector bundle over $X$.
2. This pushforward operation respects the additive property of Chern characters, meaning that if you have a direct sum of bundles, the pushforward distributes over them.
3. In the context of the Riemann-Roch theorem, the pushforward of the Chern character helps in relating the cohomological dimensions of vector bundles on curves to those on surfaces.
4. The pushforward can be computed using integration along fibers, which involves taking an integral over the fibers of the morphism in question.
5. Understanding the pushforward of the Chern character is crucial for establishing isomorphisms between various cohomology groups in algebraic geometry.

Review Questions

• How does the pushforward of the Chern character relate to the Riemann-Roch theorem and what implications does this have for understanding vector bundles?
  • The pushforward of the Chern character is deeply connected to the Riemann-Roch theorem as it enables us to transfer information about vector bundles from one space to another via morphisms. By using this operation, we can relate dimensions of spaces associated with bundles on curves to those on surfaces, allowing us to draw conclusions about their geometric properties. This relationship is essential for applying Riemann-Roch in practical scenarios involving algebraic curves and their higher-dimensional counterparts.
• Explain how the properties of the pushforward of Chern characters facilitate calculations involving direct sums of vector bundles.
  • The properties of the pushforward ensure that it distributes over direct sums, meaning that if you have two vector bundles $E_1$ and $E_2$, then $f_* (c_1(E_1 \oplus E_2)) = f_* c_1(E_1) + f_* c_1(E_2)$. This additive nature simplifies calculations significantly because it allows us to consider individual contributions from each bundle rather than having to treat them as a single entity. Consequently, this property plays a crucial role in computations within cohomology theories where multiple bundles are involved.
• Assess how understanding the pushforward of the Chern character influences broader concepts in algebraic geometry, particularly in relation to cohomology groups.
  • Understanding the pushforward of the Chern character directly influences our ability to establish relationships between different cohomology groups in algebraic geometry. By transferring Chern classes through morphisms, we gain insight into how these classes interact across varieties, leading to deeper connections between their cohomological dimensions. This comprehension not only enriches our theoretical framework but also aids in solving problems related to invariants and geometric properties across varying dimensions and structures.

Related terms

Chern character: A characteristic class that associates a vector bundle with a cohomology class in a topological space, encapsulating information about its curvature and topology.

Riemann-Roch theorem: A fundamental theorem in algebraic geometry that provides a formula for calculating dimensions of spaces of meromorphic functions and differentials on algebraic curves.

pushforward sheaf: A construction that allows the transfer of sheaves from one variety to another through a continuous map, preserving certain properties of the original sheaf.