Torelli Theorem: Understanding Polarization And Riemann Surfaces

The Torelli theorem stands as a cornerstone in the study of Riemann surfaces, Hodge theory, and moduli spaces. It essentially provides a bridge between the geometry of a compact Riemann surface and its Hodge structure. But what happens when we delve deeper into the nuances of this theorem, particularly concerning polarization? Are there instances where two distinct Riemann surfaces, despite not being isomorphic, exhibit identical integral Hodge structures on their first cohomology group? Let's unravel this fascinating question.

Understanding the Torelli Theorem

The Torelli theorem, in its basic form, states that a compact Riemann surface is uniquely determined by its polarized Hodge structure. More formally, if you have two compact Riemann surfaces, X and Y, and an isomorphism between their first cohomology groups, $H^1(X, \mathbb{Z})$ and $H^1(Y, \mathbb{Z})$, that preserves the Hodge structure and the polarization, then X and Y are isomorphic. This is a powerful statement because it connects the complex-analytic structure of a Riemann surface to the algebraic structure of its cohomology.

To fully appreciate the theorem, we need to break down its key components. A Riemann surface is a one-dimensional complex manifold, essentially a surface that locally looks like the complex plane. These surfaces are fundamental in complex analysis and algebraic geometry. The first cohomology group, $H^1(X, \mathbb{Z})$, captures essential topological information about the Riemann surface, specifically how many independent 'holes' it has. The Hodge structure provides a way to decompose the complexified cohomology group, $H^1(X, \mathbb{C})$, into subspaces that reflect the complex structure of the Riemann surface. Finally, polarization is a crucial piece; it’s a bilinear form on the cohomology group that satisfies certain positivity conditions, ensuring that the Hodge structure behaves nicely.

The Role of Hodge Structure

The Hodge structure on $H^1(X, \mathbb{Z})$ encodes a wealth of information about the Riemann surface X. It arises from the decomposition of the complexified cohomology $H^1(X, \mathbb{C})$ into two subspaces, $H^{1,0}(X)$ and $H^{0,1}(X)$, where $H^{1,0}(X)$ represents the space of holomorphic 1-forms and $H^{0,1}(X)$ is its complex conjugate. This decomposition allows us to study the complex structure of X through algebraic means. An isomorphism between the Hodge structures of two Riemann surfaces implies that their holomorphic and anti-holomorphic forms behave in a similar manner. However, this alone is not sufficient to guarantee that the surfaces themselves are isomorphic. The key is that the isomorphism must also preserve the polarization.

The Importance of Polarization

Polarization adds an extra layer of structure that is critical for the Torelli theorem to hold. It's a bilinear form $Q: H^1(X, \mathbb{Z}) \times H^1(X, \mathbb{Z}) \rightarrow \mathbb{Z}$ that satisfies certain properties. For Riemann surfaces, this polarization is often given by the cup product, which measures the intersection of cohomology classes. The polarization essentially tells us how the different 'cycles' on the Riemann surface intersect each other. This intersection information is crucial because it is deeply tied to the geometry of the surface.

The polarization condition ensures that the Hodge structure is compatible with the underlying geometry. Without it, you could potentially have two non-isomorphic Riemann surfaces with isomorphic Hodge structures, but different intersection behaviors of their cycles. This is where the Torelli theorem shows its strength: it asserts that if the Hodge structures are isomorphic and the polarizations match, then the Riemann surfaces must be the same.

Counterexamples and the Significance of Polarization

Now, let's address the core question: Are there two non-isomorphic compact Riemann surfaces with isomorphic integral Hodge structures on $H^1(-, \mathbb{Z})$? The answer is a resounding no, if the isomorphism respects the polarization. However, if we drop the requirement that the isomorphism preserves the polarization, then the answer becomes yes. This subtle distinction underscores the importance of polarization in the Torelli theorem.

Cases Without Polarization

Consider a scenario where we have two Riemann surfaces, X and Y, that are not isomorphic. Suppose there exists an isomorphism $\phi: H^1(X, \mathbb{Z}) \rightarrow H^1(Y, \mathbb{Z})$ that preserves the Hodge structure but does not preserve the polarization. In other words, $Q_Y(\phi(a), \phi(b)) \neq Q_X(a, b)$ for some $a, b \in H^1(X, \mathbb{Z})$, where $Q_X$ and $Q_Y$ are the polarization forms on X and Y, respectively. In this case, X and Y can indeed be non-isomorphic. The non-preservation of polarization indicates that the intersection behaviors of cycles on X and Y are fundamentally different, even though their Hodge structures are similar.

To visualize this, think of a Riemann surface as a donut. The first cohomology group tells you about the two fundamental cycles that go around the hole and through the hole. The polarization tells you how these cycles intersect. If you have two donuts with different shapes, their Hodge structures might be isomorphic, but the way the fundamental cycles intersect could be different. This difference is captured by the polarization, and if the isomorphism doesn't respect this difference, then the donuts are not isomorphic.

Why Polarization Matters

Polarization is essential because it ties the algebraic structure of the cohomology to the geometric structure of the Riemann surface. Without it, we lose the ability to uniquely determine the Riemann surface from its Hodge structure. The Torelli theorem relies on the fact that the polarization encodes information about the intersection of cycles, which in turn reflects the geometry of the surface. If we ignore the polarization, we are essentially ignoring a crucial piece of the puzzle.

Implications for Moduli Spaces

The Torelli theorem has profound implications for the study of moduli spaces of Riemann surfaces. The moduli space, denoted by $\mathcal{M}_g$, is the space that parameterizes all Riemann surfaces of genus g. The Torelli theorem allows us to embed $\mathcal{M}_g$ into a larger space called the Siegel upper half-space, which parameterizes polarized abelian varieties. This embedding is known as the Torelli map.

By understanding the Torelli theorem and the role of polarization, we can gain insights into the structure of the moduli space. The Torelli map provides a way to study the geometry of $\mathcal{M}_g$ using the tools of algebraic geometry and representation theory. The polarization condition ensures that this map is well-defined and that the embedding accurately reflects the geometry of the Riemann surfaces.

Examples and Further Considerations

While constructing explicit examples of non-isomorphic Riemann surfaces with isomorphic but non-polarized Hodge structures can be technically challenging, the underlying principle is clear. The polarization provides a critical constraint that ensures the uniqueness of the Riemann surface. Without this constraint, the Hodge structure alone is insufficient to determine the surface uniquely.

The Schottky Locus

Another interesting aspect related to the Torelli theorem is the Schottky locus. The Schottky problem asks: Which polarized abelian varieties are Jacobians of Riemann surfaces? In other words, which points in the Siegel upper half-space actually come from the Torelli map? This is a difficult question that has been studied extensively. The solution to the Schottky problem would give us a deeper understanding of the relationship between Riemann surfaces and abelian varieties.

The Torelli theorem provides a powerful tool for studying the Schottky locus. By understanding the properties of the Torelli map and the polarization condition, we can gain insights into the geometry of the Schottky locus and its relationship to the moduli space of Riemann surfaces.

Conclusion

In summary, the polarization in the Torelli theorem is not just a technical detail; it's a fundamental requirement that ensures the unique determination of a Riemann surface from its Hodge structure. While it is possible to find non-isomorphic Riemann surfaces with isomorphic Hodge structures if we ignore the polarization, the Torelli theorem holds true when the polarization is taken into account. This theorem has deep implications for the study of Riemann surfaces, Hodge theory, and moduli spaces, providing a powerful connection between geometry and algebra.

For further reading on related topics, consider exploring resources on Hodge theory and complex algebraic geometry.