5.2 Serre duality and Riemann-Roch theorem

Serre duality and the Riemann-Roch theorem are powerful tools in algebraic geometry. They connect cohomology groups of sheaves, allowing us to compute dimensions and relate different geometric objects on varieties.

These results have far-reaching applications in studying curves, surfaces, and higher-dimensional varieties. They're essential for understanding linear systems, divisors, and the interplay between geometry and topology in algebraic geometry.

Serre Duality for Varieties

Statement and Proof

• State Serre duality for smooth projective varieties relates the cohomology groups of a coherent sheaf to the cohomology groups of its dual sheaf twisted by the canonical bundle
• Specifically, for a smooth projective variety $X$ of dimension $n$ and a coherent sheaf $F$ on $X$, Serre duality states that there is a natural isomorphism $H^i(X, F) \cong H^{n-i}(X, F^{\vee} \otimes \omega_X)^{\vee}$
  • $\omega_X$ is the canonical bundle of $X$
  • $(-)^{\vee}$ denotes the dual
• The proof relies on constructing a trace map $tr: H^n(X, \omega_X) \to k$, where $k$ is the base field
  • Uses the derived functor formalism to establish the duality isomorphism
• Serre duality generalizes to the relative setting, where $X$ is a smooth projective morphism over a base scheme $S$
  • Relates the higher direct image sheaves of $F$ and its dual twisted by the relative canonical bundle

Generalizations and Relative Setting

• Serre duality extends to the relative setting for a smooth projective morphism $X$ over a base scheme $S$
  • Relates the higher direct image sheaves of a coherent sheaf $F$ and its dual twisted by the relative canonical bundle
• The relative version of Serre duality is crucial in the study of families of varieties and moduli spaces
  • Allows for the comparison of cohomology groups across fibers of the family
• Generalizations of Serre duality, such as Grothendieck duality, apply to more general morphisms (proper morphisms) and settings (derived categories)
  • Grothendieck duality relates the derived functors of the pushforward and pullback functors

Applications of Serre Duality

Computing Cohomology Groups

• Serre duality reduces the computation of cohomology groups of a coherent sheaf $F$ to the cohomology groups of its dual sheaf $F^{\vee}$ twisted by the canonical bundle
  • Simplifies computations and provides vanishing results
• On a smooth projective curve $C$ of genus $g$, Serre duality implies $H^0(C, F) \cong H^1(C, F^{\vee} \otimes \omega_C)^{\vee}$ and $H^1(C, F) \cong H^0(C, F^{\vee} \otimes \omega_C)^{\vee}$
  • $\omega_C$ is the canonical bundle of $C$
• In higher dimensions, Serre duality relates the cohomology groups of a sheaf to its dual twisted by the canonical bundle
  • Example: on a smooth projective surface $S$, $H^i(S, F) \cong H^{2-i}(S, F^{\vee} \otimes \omega_S)^{\vee}$ for $i = 0, 1, 2$

Studying Natural Sheaves

• Serre duality applies to the study of cohomology of natural sheaves on smooth projective varieties
  • Tangent bundle, cotangent bundle, differential forms
• For the tangent bundle $T_X$ on a smooth projective variety $X$, Serre duality gives $H^i(X, T_X) \cong H^{n-i}(X, \Omega_X^1)^{\vee}$
  • $\Omega_X^1$ is the sheaf of differential 1-forms (cotangent bundle)
• Serre duality relates the cohomology of the sheaf of differential $p$-forms $\Omega_X^p$ to the cohomology of $\Omega_X^{n-p}$
  • Useful in studying the Hodge theory and deformation theory of varieties

Riemann-Roch Theorem for Curves

Statement and Proof

• The Riemann-Roch theorem relates the dimension of the space of global sections of a line bundle on a smooth projective curve to its degree and the genus of the curve
• For a smooth projective curve $C$ of genus $g$ and a line bundle $L$ on $C$, the Riemann-Roch theorem states $h^0(C, L) - h^1(C, L) = \deg(L) - g + 1$
  • $h^i(C, L)$ denotes the dimension of the $i$-th cohomology group of $L$
• The proof involves using Serre duality to relate the cohomology groups of $L$ to its dual twisted by the canonical bundle
  • Uses properties of the canonical bundle to compute the relevant dimensions

Applications

• The Riemann-Roch theorem has important applications in the study of linear systems, divisors, and the geometry of curves
• Determines the dimension of the complete linear system associated to a divisor
  • For a divisor $D$ on a curve $C$, $\dim |D| = \deg(D) - g + 1$ if $\deg(D) \geq 2g - 1$
• Helps in the existence of special divisors (canonical, theta characteristics) and the study of Weierstrass points
• Plays a crucial role in the classification of curves (gonality, Clifford index) and the theory of algebraic curves
  • Example: a curve of genus $g \geq 2$ has a $g^1_2$ (a linear system of degree 2 and dimension 1) if and only if it is hyperelliptic

Riemann-Roch Theorem for Higher Dimensions

Hirzebruch-Riemann-Roch Theorem

• The Riemann-Roch theorem generalizes to higher-dimensional smooth projective varieties using Chern classes and the Todd class
• For a smooth projective variety $X$ of dimension $n$ and a coherent sheaf $F$ on $X$, the Hirzebruch-Riemann-Roch theorem states that the Euler characteristic of $F$ is the integral of the product of the Chern character of $F$ and the Todd class of the tangent bundle of $X$
  • $\chi(X, F) = \int_X \operatorname{ch}(F) \cdot \operatorname{td}(T_X)$
• Reduces to the classical Riemann-Roch theorem for curves when $X$ is a curve and $F$ is a line bundle
• The proof involves the Grothendieck-Riemann-Roch theorem, relating the Chern character of the pushforward of a coherent sheaf to the Chern character of the sheaf itself and the relative Todd class

Applications and Generalizations

• The generalized Riemann-Roch theorem has applications in the arithmetic and geometry of higher-dimensional varieties
  • Computing intersection numbers, studying the Picard group, and the theory of characteristic classes
• Allows for the computation of the Euler characteristic of coherent sheaves on smooth projective varieties
  • Useful in enumerative geometry and the study of moduli spaces
• Further generalizations, such as the Grothendieck-Riemann-Roch theorem, apply to more general morphisms and settings
  • Relates the Chern character of the pushforward of a coherent sheaf to the Chern character of the sheaf itself and the relative Todd class of the morphism
• The Atiyah-Singer index theorem is an analytic analogue of the Hirzebruch-Riemann-Roch theorem, relating the index of an elliptic differential operator to topological invariants of the manifold and the operator
  • Connects the Riemann-Roch theorem to differential geometry and topology