14.2 Riemann-Roch theorems in algebraic geometry

Riemann-Roch theorems are powerful tools in algebraic geometry, connecting sheaf cohomology to geometric invariants. They extend from curves to higher-dimensional varieties, providing a way to calculate Euler characteristics and study moduli spaces.

These theorems showcase the interplay between K-theory and algebraic geometry. By expressing cohomological information in terms of Chern characters and Todd classes, they reveal deep connections between topology and algebraic structure in geometry.

Grothendieck-Riemann-Roch Theorem

Generalization of Classical Riemann-Roch Theorem

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• The Grothendieck-Riemann-Roch theorem extends the classical Riemann-Roch theorem to higher dimensions and arbitrary coherent sheaves
• Applies to smooth projective varieties over a field $k$
• Expresses the Euler characteristic $\chi(X,F)$ of a coherent sheaf $F$ on $X$ in terms of Chern characters and Todd classes

Key Components and Definitions

• The Chern character $ch(F)$ is a ring homomorphism from the Grothendieck group $K(X)$ of coherent sheaves on $X$ to the Chow ring $A(X) \otimes \mathbb{Q}$
  • Encodes important information about the coherent sheaf $F$, such as its rank and Chern classes
• The Todd class $td(TX)$ is a polynomial in the Chern classes of the tangent bundle $TX$
  • Defined using the splitting principle
  • Measures the "twisting" of the variety $X$

Grothendieck-Riemann-Roch Formula and Proof

• The Grothendieck-Riemann-Roch formula states that $\chi(X,F) = deg(ch(F) \cdot td(TX))$
  • $deg$ denotes the degree map from the top-dimensional component of the Chow ring to $\mathbb{Q}$
• The proof involves a reduction to the case of projective space using the Hirzebruch-Riemann-Roch theorem
  • Calculation using the splitting principle and the properties of Chern classes

Riemann-Roch Theorem for Chern Classes

Chern Character and Todd Class

• For a vector bundle $E$, the Chern character is given by $ch(E) = rank(E) + c_1(E) + \frac{1}{2}(c_1(E)^2 - 2c_2(E)) + \cdots$
  • For a coherent sheaf $F$, $ch(F)$ is defined using a locally free resolution of $F$
• The Todd class of the tangent bundle $TX$ is given by $td(TX) = 1 + \frac{1}{2}c_1(TX) + \frac{1}{12}(c_1(TX)^2 + c_2(TX)) + \cdots$

Interpretation and Special Cases

• The Riemann-Roch formula $\chi(X,F) = deg(ch(F) \cdot td(TX))$ can be seen as a "correction" to the naive Euler characteristic
  • Takes into account the Chern classes of $F$ and the geometry of $X$
• For a line bundle $L$, the formula simplifies to $\chi(X,L) = deg(ch(L) \cdot td(TX)) = \frac{1}{2}deg(c_1(L)^{dim(X)} + \text{lower order terms})$
  • Recovers the classical Riemann-Roch theorem

Euler Characteristic Calculations

Definition and Properties

• The Euler characteristic $\chi(X,F)$ is defined as the alternating sum of the dimensions of the cohomology groups $H^i(X,F)$
• Can be computed using the Riemann-Roch theorem in various settings

Examples and Applications

• For a line bundle $L$ on a smooth projective curve $X$, $\chi(X,L) = deg(L) + 1 - g$, where $g$ is the genus of $X$
• For a vector bundle $E$ on a smooth projective surface $X$, $\chi(X,E) = \frac{1}{2}deg(c_1(E)^2 - 2c_2(E)) + \frac{1}{12}deg(c_1(TX)c_1(E)) + \frac{1}{12}deg(c_1(TX)^2 + c_2(TX))$
• In higher dimensions, the Euler characteristic can be computed using the Hirzebruch-Riemann-Roch theorem or the Grothendieck-Riemann-Roch theorem
• The Riemann-Roch theorem is used to study the geometry of moduli spaces of sheaves on varieties
  • Computes expected dimensions and constructs virtual fundamental classes

Riemann-Roch Generalization for Singular Varieties

Higher K-Theory and Chern Character

• The Grothendieck-Riemann-Roch theorem generalizes to singular varieties using intersection theory and the Chern character in higher K-theory
• For a possibly singular variety $X$, the Grothendieck group $K(X)$ is replaced by the higher K-theory group $K_0(X)$
  • $K_0(X)$ is the Grothendieck group of perfect complexes on $X$
• The Chern character $ch: K_0(X) \to A(X) \otimes \mathbb{Q}$ is defined using the Atiyah-Hirzebruch spectral sequence

Virtual Tangent Bundle and Generalizations

• In the presence of singularities, the Todd class is replaced by the Todd class of the virtual tangent bundle
  • Defined using the cotangent complex
• The Riemann-Roch theorem can be generalized to higher K-theory groups $K_i(X)$ for $i > 0$
  • Uses the Chern character in higher K-theory and the Adams operations
• These generalizations have applications in the study of characteristic classes of singular varieties and in the construction of virtual fundamental classes in enumerative geometry