Serre duality

Algebraic theorem Let X be a smooth variety of dimension n over a field k. Define the canonical line bundle K_X to be the bundle of n-forms on X, the top exterior power of the cotangent bundle: :K_X=\Omega^n_X={\bigwedge}^n(T^*X). Suppose in addition that X is proper (for example, projective) over k. Then Serre duality says: for an algebraic vector bundle E on X and an integer i, there is a natural isomorphism: :H^i(X,E)\cong H^{n-i}(X,K_X\otimes E^{\ast})^{\ast} of finite-dimensional k-vector spaces. Here \otimes denotes the tensor product of vector bundles. It follows that the dimensions of the two cohomology groups are equal: :h^i(X,E)=h^{n-i}(X,K_X\otimes E^{\ast}). As in Poincaré duality, the isomorphism in Serre duality comes from the cup product in sheaf cohomology. Namely, the composition of the cup product with a natural trace map on H^n(X,K_X) is a perfect pairing: :H^i(X,E)\times H^{n-i}(X,K_X\otimes E^{\ast})\to H^n(X,K_X)\to k. The trace map is the analog for coherent sheaf cohomology of integration in de Rham cohomology. Differential-geometric theorem Serre also proved the same duality statement for X a compact complex manifold and E a holomorphic vector bundle. Here, the Serre duality theorem is a consequence of Hodge theory. Namely, on a compact complex manifold X equipped with a Riemannian metric, there is a Hodge star operator: :\star: \Omega^p(X) \to \Omega^{2n-p}(X), where \dim_{\mathbb{C}} X = n. Additionally, since X is complex, there is a splitting of the complex differential forms into forms of type (p,q). The Hodge star operator (extended complex-linearly to complex-valued differential forms) interacts with this grading as: :\star: \Omega^{p,q}(X) \to \Omega^{n-q,n-p}(X). Notice that the holomorphic and anti-holomorphic indices have switched places. There is a conjugation on complex differential forms which interchanges forms of type (p,q) and (q,p), and if one defines the conjugate-linear Hodge star operator by \bar{\star}\omega = \star \bar{\omega} then we have: :\bar{\star} : \Omega^{p,q}(X) \to \Omega^{n-p,n-q}(X). Using the conjugate-linear Hodge star, one may define a Hermitian L^2-inner product on complex differential forms, by: :\langle \alpha, \beta \rangle_{L^2} = \int_X \alpha \wedge \bar{\star}\beta, where now \alpha \wedge \bar{\star}\beta is an (n,n)-form, and in particular a complex-valued 2n-form and can therefore be integrated on X with respect to its canonical orientation. Furthermore, suppose (E,h) is a Hermitian holomorphic vector bundle. Then the Hermitian metric h gives a conjugate-linear isomorphism E\cong E^* between E and its dual vector bundle, say \tau: E\to E^*. Defining \bar{\star}_E (\omega \otimes s) = \bar{\star} \omega \otimes \tau(s), one obtains an isomorphism: :\bar{\star}_E : \Omega^{p,q}(X,E) \to \Omega^{n-p,n-q}(X,E^*) where \Omega^{p,q}(X,E)= \Omega^{p,q}(X) \otimes \Gamma(E) consists of smooth E-valued complex differential forms. Using the pairing between E and E^* given by \tau and h, one can therefore define a Hermitian L^2-inner product on such E-valued forms by: :\langle \alpha, \beta \rangle_{L^2} = \int_X \alpha \wedge_h \bar{\star}_E \beta, where here \wedge_h means wedge product of differential forms and using the pairing between E and E^* given by h. The Hodge theorem for Dolbeault cohomology asserts that if we define: :\Delta_{\bar{\partial}_E} = \bar{\partial}_E^* \bar{\partial}_E + \bar{\partial}_E \bar{\partial}_E^* where \bar{\partial}_E is the Dolbeault operator of E and \bar{\partial}_E^* is its formal adjoint with respect to the inner product, then: :H^{p,q}(X,E) \cong \mathcal{H}^{p,q}_{\Delta_{\bar{\partial}_E}} (X). On the left is Dolbeault cohomology, and on the right is the vector space of harmonic E-valued differential forms defined by: :\mathcal{H}^{p,q}_{\Delta_{\bar{\partial}_E}} (X) = \{\alpha \in \Omega^{p,q}(X,E) \mid \Delta_{\bar{\partial}_E} (\alpha) = 0\}. Using this description, the Serre duality theorem can be stated as follows: The isomorphism \bar{\star}_E induces a complex linear isomorphism: :H^{p,q}(X,E) \cong H^{n-p,n-q}(X,E^*)^*. This can be easily proved using the Hodge theory above. Namely, if [\alpha] is a cohomology class in H^{p,q}(X,E) with unique harmonic representative \alpha \in \mathcal{H}^{p,q}_{\Delta_{\bar{\partial}_E}} (X), then: :(\alpha, \bar{\star}_E \alpha) = \langle \alpha, \alpha \rangle_{L^2} \ge 0 with equality if and only if \alpha = 0. In particular, the complex linear pairing: :(\alpha, \beta) = \int_X \alpha \wedge_h \beta between \mathcal{H}^{p,q}_{\Delta_{\bar{\partial}_E}} (X) and \mathcal{H}^{n-p,n-q}_{\Delta_{\bar{\partial}_{E^*}}} (X) is non-degenerate, and induces the isomorphism in the Serre duality theorem. The statement of Serre duality in the algebraic setting may be recovered by taking p=0, and applying Dolbeault's theorem, which states that: :H^{p,q}(X,E) \cong H^q(X, \boldsymbol{\Omega}^p \otimes E) where on the left is Dolbeault cohomology and on the right sheaf cohomology, where \boldsymbol{\Omega}^p denotes the sheaf of holomorphic (p,0)-forms. In particular, we obtain: :H^q(X,E) \cong H^{0,q}(X,E) \cong H^{n,n-q}(X,E^*)^* \cong H^{n-q}(X, K_X \otimes E^*)^* where we have used that the sheaf of holomorphic (n,0)-forms is just the canonical bundle of X. ==Algebraic curves==

A fundamental application of Serre duality is to algebraic curves. (Over the complex numbers, it is equivalent to consider compact Riemann surfaces.) For a line bundle L on a smooth projective curve X over a field k, the only possibly nonzero cohomology groups are H^0(X,L) and H^1(X,L). Serre duality describes the H^1 group in terms of an H^0 group (for a different line bundle). That is more concrete, since H^0 of a line bundle is simply its space of sections. Serre duality is especially relevant to the Riemann–Roch theorem for curves. For a line bundle L of degree d on a curve X of genus g, the Riemann–Roch theorem says that: :h^0(X,L)-h^1(X,L)=d-g+1. Using Serre duality, this can be restated in more elementary terms: :h^0(X,L)-h^0(X,K_X\otimes L^*)=d-g+1. The latter statement (expressed in terms of divisors) is in fact the original version of the theorem from the 19th century. This is the main tool used to analyze how a given curve can be embedded into projective space and hence to classify algebraic curves. Example: Every global section of a line bundle of negative degree is zero. Moreover, the degree of the canonical bundle is 2g-2. Therefore, Riemann–Roch implies that for a line bundle L of degree d>2g-2, h^0(X,L) is equal to d-g+1. When the genus g is at least 2, it follows by Serre duality that h^1(X,TX)=h^0(X,K_X^{\otimes 2})=3g-3. Here H^1(X,TX) is the first-order deformation space of X. This is the basic calculation needed to show that the moduli space of curves of genus g has dimension 3g-3. ==Serre duality for coherent sheaves==

Another formulation of Serre duality holds for all coherent sheaves, not just vector bundles. As a first step in generalizing Serre duality, Grothendieck showed that this version works for schemes with mild singularities, Cohen–Macaulay schemes, not just smooth schemes. Namely, for a Cohen–Macaulay scheme X of pure dimension n over a field k, Grothendieck defined a coherent sheaf \omega_X on X called the dualizing sheaf. (Some authors call this sheaf K_X.) Suppose in addition that X is proper over k. For a coherent sheaf E on X and an integer i, Serre duality says that there is a natural isomorphism: :\operatorname{Ext}^i_X(E,\omega_X)\cong H^{n-i}(X,E)^* of finite-dimensional k-vector spaces. Here the Ext group is taken in the abelian category of O_X-modules. This includes the previous statement, since \operatorname{Ext}^i_X(E,\omega_X) is isomorphic to H^i(X,E^*\otimes \omega_X) when E is a vector bundle. In order to use this result, one has to determine the dualizing sheaf explicitly, at least in special cases. When X is smooth over k, \omega_X is the canonical line bundle K_X defined above. More generally, if X is a Cohen–Macaulay subscheme of codimension r in a smooth scheme Y over k, then the dualizing sheaf can be described as an Ext sheaf: :\omega_X\cong\mathcal{Ext}^r_{O_Y}(O_X,K_Y). When X is a local complete intersection of codimension r in a smooth scheme Y, there is a more elementary description: the normal bundle of X in Y is a vector bundle of rank r, and the dualizing sheaf of X is given by: :\omega_X\cong K_Y|_X\otimes {\bigwedge}^r(N_{X/Y}). In this case, X is a Cohen–Macaulay scheme with \omega_X a line bundle, which says that X is Gorenstein. Example: Let X be a complete intersection in projective space {\mathbf P}^n over a field k, defined by homogeneous polynomials f_1,\ldots,f_r of degrees d_1,\ldots,d_r. (To say that this is a complete intersection means that X has dimension n-r.) There are line bundles O(d) on {\mathbf P}^n for integers d, with the property that homogeneous polynomials of degree d can be viewed as sections of O(d). Then the dualizing sheaf of X is the line bundle: :\omega_X=O(d_1+\cdots+d_r-n-1)|_X, by the adjunction formula. For example, the dualizing sheaf of a plane curve X of degree d is O(d-3)|_X. Complex moduli of Calabi–Yau threefolds In particular, we can compute the number of complex deformations, equal to \dim(H^1(X,TX)) for a quintic threefold in \mathbb{P}^4, a Calabi–Yau variety, using Serre duality. Since the Calabi–Yau property ensures K_X \cong \mathcal{O}_X Serre duality shows us that H^1(X,TX) \cong H^2(X, \mathcal{O}_X\otimes \Omega_X) \cong H^2(X, \Omega_X) showing the number of complex moduli is equal to h^{2,1} in the Hodge diamond. Of course, the last statement depends on the Bogomolev–Tian–Todorov theorem which states every deformation on a Calabi–Yau is unobstructed. ==Grothendieck duality==

Grothendieck's theory of coherent duality is a broad generalization of Serre duality, using the language of derived categories. For any scheme X of finite type over a field k, there is an object \omega_X^{\bullet} of the bounded derived category of coherent sheaves on X, D^b_{\operatorname{coh}}(X), called the dualizing complex of X over k. Formally, \omega_X^{\bullet} is the exceptional inverse image f^!O_Y, where f is the given morphism X\to Y=\operatorname{Spec}(k). When X is Cohen–Macaulay of pure dimension n, \omega_X^{\bullet} is \omega_X[n]; that is, it is the dualizing sheaf discussed above, viewed as a complex in (cohomological) degree −n. In particular, when X is smooth over k, \omega_X^{\bullet} is the canonical line bundle placed in degree −n. Using the dualizing complex, Serre duality generalizes to any proper scheme X over k. Namely, there is a natural isomorphism of finite-dimensional k-vector spaces: :\operatorname{Hom}_X(E,\omega_X^{\bullet})\cong \operatorname{Hom}_X(O_X,E)^* for any object E in D^b_{\operatorname{coh}}(X). More generally, for a proper scheme X over k, an object E in D^b_{\operatorname{coh}}(X), and F a perfect complex in D_{\operatorname{perf}}(X), one has the elegant statement: :\operatorname{Hom}_X(E,F\otimes \omega_X^{\bullet})\cong\operatorname{Hom}_X(F,E)^*. Here the tensor product means the derived tensor product, as is natural in derived categories. (To compare with previous formulations, note that \operatorname{Ext}^i_X(E,\omega_X) can be viewed as \operatorname{Hom}_X(E,\omega_X[i]).) When X is also smooth over k, every object in D^b_{\operatorname{coh}}(X) is a perfect complex, and so this duality applies to all E and F in D^b_{\operatorname{coh}}(X). The statement above is then summarized by saying that F\mapsto F\otimes \omega_X^{\bullet} is a Serre functor on D^b_{\operatorname{coh}}(X) for X smooth and proper over k. Serre duality holds more generally for proper algebraic spaces over a field. ==Notes==