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Canonical bundle

In , the canonical bundle of a smooth variety X of dimension n, denoted \omega_X or K_X, is the defined as the nth exterior power of the cotangent sheaf \Omega_X^1, i.e., \omega_X = \bigwedge^n \Omega_X^1. This sheaf of differentials \Omega_X^1 is the of the sheaf T_X, capturing the intrinsic differential structure of X. As a , the canonical bundle encodes forms on X and serves as a fundamental invariant in the study of varieties. The is pivotal in several core results and tools of . The relates it to restrictions on divisors: for a D \subset X, the canonical bundle of D satisfies \omega_D = (\omega_X \otimes \mathcal{O}_X(D))|_D. Serre duality theorem states that for a L on X, the groups satisfy H^n(X, L) \cong H^0(X, L^{-1} \otimes \omega_X)^*, linking global sections and higher to compute invariants like the of curves. These properties extend to singular varieties via dualizing sheaves, though the case remains the primary setting. Beyond definitions, the canonical bundle plays a central role in and the classification of algebraic varieties. Rational maps between smooth projective varieties induce pullbacks on sections of powers of the canonical bundle, with birational maps yielding isomorphisms for sufficiently positive powers. Varieties are classified based on the canonical bundle's ampleness: Fano varieties have ample anticanonical bundle (negative ), Calabi-Yau varieties have trivial canonical bundle, and varieties of general type have ample canonical bundle (positive ). In the , the canonical ring's finite generation (proven in dimension three by 1980s and generally by 2010) yields a , a birational invariant unique up to isomorphism. For curves, the degree of \omega_X is $2g-2, where g is the , directly tying to Riemann-Roch.

Definition and Basics

Definition for smooth varieties

In algebraic geometry, the canonical bundle of a smooth projective variety X of dimension n over an algebraically closed field k is defined as the line bundle K_X = \det(\Omega_{X/k}) = \bigwedge^n \Omega_{X/k}, where \Omega_{X/k} denotes the sheaf of Kähler differentials on X. This construction yields an invertible sheaf on X, reflecting the intrinsic geometry of the variety through its differential structure. The canonical bundle arises naturally as the top exterior power of the cotangent sheaf, providing a sheaf-theoretic generalization of the canonical divisor class familiar from the study of curves. In higher dimensions, it encodes key invariants such as the geometric genus p_g(X) = h^0(X, K_X), which measures the dimension of the space of global sections of holomorphic n-forms on X. For the basic case of a smooth projective C of g, the canonical bundle K_C is a of degree $2g - 2. This degree formula highlights its role in embedding the curve via the complete |K_C|, which is very ample for g \geq 2. The concept of the canonical bundle originated in the development of the Riemann-Roch theorem for curves, with foundational contributions from and Gustav Roch in the 1850s, and further advancements by on canonical systems and their properties.

Relation to differentials and divisors

In algebraic geometry, for a smooth projective variety X over an algebraically closed field, a canonical divisor K is defined as a Cartier divisor such that the associated line bundle \mathcal{O}_X(K) is isomorphic to the canonical sheaf \omega_X, also denoted K_X. The canonical class is then the image of this divisor in the Picard group \operatorname{Pic}(X), represented by the first Chern class c_1(K_X). The global sections of the canonical bundle H^0(X, K_X) correspond to the space of regular holomorphic n-forms on X, where n = \dim X, in the complex analytic setting; algebraically, these are the sections of the sheaf of differentials. For a projective curve C of g, the canonical bundle K_C has global sections forming the space of regular differentials, which is g-dimensional. The degree of the canonical divisor on such a curve C is \deg(K_C) = 2g - 2. By the Riemann-Roch theorem, this degree implies \dim H^0(C, K_C) - \dim H^1(C, K_C) = 2g - 2 + 1 - g = g - 1; combined with \dim H^1(C, K_C) = 1 from Serre duality (since H^1(C, K_C) \cong H^0(C, \mathcal{O}_C)^*), it confirms \dim H^0(C, K_C) = g. In higher dimensions, the canonical bundle K_X is the associated to the canonical class in \operatorname{Pic}(X), generalizing the curve case. Serre duality provides a key relation, stating that for a \mathcal{F} on X, H^i(X, \mathcal{F})^\vee \cong H^{n-i}(X, \mathcal{F}^\vee \otimes K_X), linking of \mathcal{O}_X to that of K_X.

Core Properties and Formulas

Adjunction formula

The provides a fundamental relation between the canonical bundles of a smooth and its ambient smooth . For a smooth Z \subset Y, where Y is a smooth , the formula states that the canonical bundle K_Z of Z is given by K_Z = (K_Y + Z)|_Z, where + denotes the sum in the divisor group on Y, and the restriction is taken to Z. This formula arises from the conormal associated to the of Z in Y. Specifically, since Z is a , it is defined locally by a single equation f = 0, and the conormal sheaf is \mathcal{I}_Z / \mathcal{I}_Z^2 \cong \mathcal{O}_Y(-Z)|_Z. The short of sheaves on Z is $0 \to \mathcal{O}_Z(-Z) \to \Omega_Y|_Z \to \Omega_Z \to 0, where \Omega_Y and \Omega_Z are the cotangent sheaves of Y and Z, respectively. Taking the (top exterior power) of this sequence yields \det(\Omega_Y|_Z) \cong \det(\mathcal{O}_Z(-Z)) \otimes \det(\Omega_Z), since for a short of vector bundles $0 \to A \to B \to C \to 0, the determinants satisfy \det B \cong \det A \otimes \det C. Here, \det(\mathcal{O}_Z(-Z)) = \mathcal{O}_Z(-Z) as it is a , and \det(\Omega_Y|_Z) = K_Y|_Z, \det(\Omega_Z) = K_Z. Rearranging gives K_Z \cong K_Y|_Z \otimes \mathcal{O}_Z(Z) = (K_Y + Z)|_Z. The enables explicit computations of canonical bundles for hypersurfaces in projective spaces. For instance, consider a C \subset \mathbb{P}^2 of d. The canonical bundle of \mathbb{P}^2 is K_{\mathbb{P}^2} = \mathcal{O}_{\mathbb{P}^2}(-3), so by adjunction, K_C = (\mathcal{O}_{\mathbb{P}^2}(-3) + \mathcal{O}_{\mathbb{P}^2}(d))|_C = \mathcal{O}_C(d-3). The of K_C is thus d(d-3), which determines the of C via the degree-genus formula. This extends iteratively to s: for a complete intersection Z of hypersurfaces of degrees d_1, \dots, d_k in \mathbb{P}^n, repeated application yields K_Z = \mathcal{O}_Z\left( \sum d_i - (n+1) \right). For subvarieties of greater than 1, the adjunction formula generalizes by incorporating the normal bundle N_{Z/Y}, relating K_Z to K_Y|_Z \otimes \det(N_{Z/Y}), though the hypersurface case simplifies to N_{Z/Y} \cong \mathcal{O}_Z(Z).

Canonical bundle formula for

The canonical bundle formula for fibrations describes the structure of the canonical bundle on the total space of a relative curve fibration in terms of the base and the fibers, incorporating contributions from singular fibers to capture the geometry of the family. A particularly important case is that of elliptic fibrations (g=1), where the relative canonical bundle is trivial on smooth fibers. For a relatively minimal elliptic fibration f: X \to B, the canonical bundle K_X is expressed as K_X = f^*(K_B + \Delta), where \Delta is a \mathbb{Q}-divisor on B encoding the fundamental line bundle and adjustments for multiple fibers. This formula facilitates the computation of intersection numbers and Kodaira dimensions, relating the arithmetic genus of X to that of B and the fibers. The derivation begins with the short exact sequence of sheaves of relative differentials $0 \to f^* \Omega_B \to \Omega_X \to \Omega_{X/B} \to 0, which, upon taking determinants, yields K_X = f^* K_B \otimes K_{X/B}, where K_{X/B} = \det \Omega_{X/B} is the relative canonical bundle. For an elliptic fibration, the restriction K_{X/B}|_{\text{general fiber}} is trivial, and globally K_{X/B} = f^* \Delta, adjusted for singularities via the dualizing sheaf on singular fibers; the arithmetic genus enters through the preserved topological Euler characteristic across the family. Specifically, for a minimal elliptic surface over a base curve B with no multiple fibers, \Delta = L where \deg_B L = \chi(O_X), the topological Euler characteristic of X; multiple fibers of multiplicity m_i contribute terms (1 - 1/m_i) P_i to \Delta, with P_i points on B. This case exemplifies the role of the formula in classifying singular fiber types and computing invariants like the Noether formula for surfaces. The formula originates in Kodaira's foundational work on compact complex analytic surfaces with elliptic fibrations in the 1960s, providing the complex analytic foundation; algebraic generalizations to fibrations over arbitrary bases were developed by Iitaka in the 1970s–1980s as part of his program on the C_{n,m}, with extensions to certain algebraic fiber spaces appearing in later contributions that resolve cases of Iitaka's for Abelian and K3 fibrations. For fibrations with general fiber g > 1, the relative canonical bundle does not admit such a simple decomposition due to its ampleness on fibers, and more involved logarithmic or relative versions are used.

Treatment in Singular Cases

Dualizing sheaf

In algebraic geometry, the notion of the canonical bundle extends to singular varieties through the dualizing sheaf, which provides a coherent replacement that satisfies an appropriate form of Serre duality even in the presence of singularities. For a proper scheme X of dimension n over a field k, the dualizing sheaf \omega_X is a coherent \mathcal{O}_X-module equipped with a trace morphism t: H^n(X, \omega_X) \to k such that, for every coherent sheaf \mathcal{F} on X, the natural map \Hom_{\mathcal{O}_X}(\mathcal{F}, \omega_X) \to \Hom_k(H^n(X, \mathcal{F}), k) is an isomorphism. This formulation generalizes Serre duality to singular settings, where the trace morphism ensures the duality pairing is perfect. When restricted to a smooth open subscheme U \subset X, the dualizing sheaf satisfies \omega_X|_U \cong \Omega^n_{U/k}, the sheaf of differentials of top degree, coinciding with the canonical sheaf on smooth loci. The dualizing sheaf is constructed via Grothendieck duality: if f: X \to \Spec k is the structure morphism, then \omega_X is the zeroth cohomology sheaf of the dualizing complex f^! \mathcal{O}_{\Spec k}. For normal varieties, \omega_X coincides with the reflexive sheaf K_X, defined as the reflexive hull of the sheaf of Kähler differentials or equivalently as the pushforward of the canonical sheaf from a resolution of singularities. In particular, one obtains \omega_X as the direct image g_* \Omega^{\dim X}_{\tilde{X}/k} under a resolution g: \tilde{X} \to X, where \tilde{X} is smooth, provided the higher direct images vanish. A concrete example arises for nodal curves. Let C be a nodal curve with normalization \nu: C^\nu \to C, where each node p_i \in C has preimages \{r_i, s_i\} \in C^\nu. The dualizing sheaf \omega_C is the invertible sheaf on C whose sections over an open U \subset C consist of rational differentials \eta on \nu^{-1}(U) that are regular away from the r_i, s_i and have simple poles there satisfying \Res_{r_i}(\eta) + \Res_{s_i}(\eta) = 0 for each pair. This construction extends the canonical sheaf from the smooth parts while imposing residue conditions at nodes to ensure duality holds globally. For a connected nodal curve of arithmetic genus g, the degree of \omega_C remains $2g-2; for instance, a nodal plane cubic curve, with g=1, has \deg \omega_C = 0.

Properties and reflexivity

On a normal variety X, the dualizing sheaf \omega_X is reflexive, meaning that the natural \omega_X \to \Hom_X(\Hom_X(\omega_X, \mathcal{O}_X), \mathcal{O}_X) is an . This reflexivity holds because schemes are S_2 and R_1, and on such schemes, S_2 coherent sheaves are precisely the reflexive ones. Consequently, the associated divisor K_X is on the smooth locus of X, where \omega_X restricts to the invertible sheaf of differentials. For varieties with simple normal crossing (snc) singularities, the dualizing sheaf \omega_X is invertible, as snc varieties possess only Gorenstein singularities. In this setting, an explicit description arises via log bundles: if X decomposes into irreducible components with snc divisor D at intersections, then on each component X_i, the restriction \omega_X|_{X_i} \cong \mathcal{O}_{X_i}(K_{X_i} + D), reflecting the for log pairs. This structure facilitates computations of residues and duality in log canonical contexts. A key component of the dualizing sheaf \omega_X on a projective Cohen-Macaulay X of dimension n over a k is the map t: H^n(X, \omega_X) \to k, which pairs cohomology groups via Serre duality to yield perfect pairings H^i(X, \mathcal{F}) \times H^{n-i}(X, \mathcal{F}^\vee \otimes \omega_X) \to k for locally free sheaves \mathcal{F}. This enables residue computations, generalizing over varieties by associating residues to cycles on singular loci. Unlike smooth varieties, where \omega_X captures all holomorphic n-forms, singularities on X typically reduce the space of global sections H^0(X, \omega_X), as sections must satisfy compatibility conditions across the singular set. For quotient singularities, such as the A_1 singularity \mathbb{C}^2 / \mathbb{Z}_2 on a surface, the dualizing sheaf consists of \mathbb{Z}_2-invariant differentials from the resolution, yielding fewer global sections than the smooth quotient—for instance, only even-degree forms survive, diminishing the dimension of the form space compared to the smooth case.

Canonical Maps and Embeddings

General canonical map

The canonical map of a smooth projective variety X of dimension n over an is the rational map \phi_{|K_X|}: X \dashrightarrow \mathbb{P}(H^0(X, K_X)^*), where |K_X| denotes the complete associated to the canonical bundle K_X = \bigwedge^n \Omega_X, and the projective space has dimension h^0(X, K_X) - 1. This map is defined by sending a point x \in X to the point in the projective space corresponding to the in H^0(X, K_X) consisting of sections vanishing at x, or dually via the evaluation of a basis of global sections of K_X at points of X. If the |K_X| is basepoint-free, the map is a ; otherwise, it is rational and may require blowing up base points to resolve into a . For varieties of general type, the canonical map exhibits birational properties: specifically, the associated pluricanonical maps \phi_{|mK_X|} are birational onto their images for sufficiently large m, reflecting the growth of plurigenera h^0(X, mK_X) \sim c m^n with c > 0. This birationality follows from the finite generation of the canonical ring R(X, K_X) = \bigoplus_{m \geq 0} H^0(X, mK_X), which allows the construction of the canonical model as \mathrm{Proj} R(X, K_X), onto which X maps birationally via the morphism induced by the graded ring structure. Clifford's theorem provides dimension bounds for the restriction of the canonical system to subvarieties: for an irreducible subvariety Y \subset X of dimension k, the restricted system |K_X|_Y| satisfies h^0(Y, K_X|_Y) \leq \frac{\deg(K_X|_Y)}{2} + 1 under suitable specialness conditions, generalizing the classical bound for curves and constraining the geometry of the image. The of the image of the is at most \min(n, h^0(X, K_X) - 1), as the factors through a linear projection from the ambient , and the 's limits the span of the image unless the is exceptionally large. The is non-degenerate—meaning its image does not lie in a proper of the target —when K_X is ample, since the global sections of K_X then generate the bundle without common zeros outside a proper , ensuring the surjects onto the fiber coordinates. In higher dimensions, such as for surfaces (n=2), the often fails to be birational or , but relates closely to pluricanonical maps \phi_{|mK_X|} for m \geq 2, which resolve indeterminacies and embed the surface into higher-dimensional ; within the , this culminates in the \mathrm{Proj} R(X, K_X), a singular with terminal singularities to which the minimal maps birationally, capturing the of general type surfaces.

Canonical models for curves

For smooth projective curves of genus g = 0, which are isomorphic to \mathbb{P}^1, the canonical bundle is \mathcal{O}_{\mathbb{P}^1}(-2), which has no global sections, so the canonical linear system is empty and there is no . For genus g = 1, elliptic curves have trivial canonical bundle \omega_C \cong \mathcal{O}_C, with h^0(C, \omega_C) = 1, so the canonical map sends the curve to a point in \mathbb{P}^0. Curves of genus g = 2 are all hyperelliptic, and their canonical map is the degree-2 hyperelliptic morphism to \mathbb{P}^1, whose image is the rational normal of degree 1 (itself \mathbb{P}^1) in \mathbb{P}^{1}. For hyperelliptic curves of g \geq 2, the \phi_{|K_C|}: C \to \mathbb{P}^{g-1} is not an but a degree-2 onto its image, which is a rational curve of degree g-1. This image arises as the composition of the degree-2 hyperelliptic map C \to \mathbb{P}^1 with the Veronese of \mathbb{P}^1 into \mathbb{P}^{g-1} as the rational curve of degree g-1. If the image is projected from a point on the rational curve, it yields a rational scroll surface containing the projected curve. For non-hyperelliptic curves of g \geq 3, the embeds C as a curve of degree $2g-2 in \mathbb{P}^{g-1}, known as the . By Petri's , for general such curves of g \geq 4, the homogeneous of this embedded curve in \mathbb{P}^{g-1} is generated by quadrics. For example, in genus 3, it is a quartic ( of no additional quadrics beyond the ); in genus 4, a space curve of degree 6 on a surface in \mathbb{P}^3. The canonical model of a non-hyperelliptic of g \geq 3 is projectively unique up to of \mathbb{P}^{g-1}, as it is determined by the complete |K_C| with no base points. Trigonal exceptions occur when the curve admits a degree-3 to \mathbb{P}^1, in which case the canonical image lies on a rational normal scroll of degree g-2 in \mathbb{P}^{g-1}, generated by the lines joining points in the g^1_3.

Canonical Rings

Definition and construction

In , the canonical ring of a smooth X over a k is defined as the graded k- R(X, K_X) = \bigoplus_{m \geq 0} H^0(X, \mathcal{O}_X(m K_X)), where K_X is the canonical divisor class of X. The applied to this ring yields the X_{\mathrm{can}} = \mathrm{Proj}_k R(X, K_X), which is a capturing the birational invariants of X related to its canonical sheaf. The canonical ring is constructed via the pluri-canonical systems |m K_X|, which provide the graded pieces, and the associated morphism \phi_{|m K_X|}: X \dashrightarrow \mathbb{P}^N for sufficiently large m embeds X into projective space as a Veronese subvariety, with the image stabilizing to X_{\mathrm{can}} as m increases. Finite generation of R(X, K_X) as a k-algebra holds for varieties of log general type, as established by the minimal model program; specifically, for a Kawamata log terminal pair (X, \Delta) of log general type, the ring is finitely generated, ensuring X_{\mathrm{can}} is projective. If X is normal, then R(X, K_X) is integrally closed in its fraction field, reflecting the normality of X_{\mathrm{can}}. The R(X, K_X) has \dim X + 1, and thus the X_{\mathrm{can}} has dimension \dim X. The \phi: X \dashrightarrow X_{\mathrm{can}}, defined by the |m K_X| for large m, factors through the Proj of the canonical ring, providing a birational that resolves any non-normalities.

Structure and applications for curves

For a smooth projective C of g \geq 2 over an k of characteristic zero, the ring R(C, K_C) is generated as a k- by the sections of the canonical bundle K_C, which form a basis \{x_1, \dots, x_g\} for H^0(C, K_C) of dimension g. By , this generation occurs in degree 1 for non-hyperelliptic curves, making R(C, K_C) \cong k[x_1, \dots, x_g]/I where I is a homogeneous ideal. Petri's theorem further specifies that, for non-hyperelliptic curves with g \geq 4, I is generated solely by quadratic forms arising from the kernel of the Petri map \wedge^2 H^0(K_C) \to H^0(2K_C), which encodes the syzygies of the canonical \phi_{K_C}: C \hookrightarrow \mathbb{P}^{g-1}. In the hyperelliptic case, the canonical ring requires additional generators of degree 2 beyond the degree-1 section space, with relations extending up to degree 4; this structure imposes extra constraints, rendering R(C, K_C) isomorphic to the Veronese subring of the in g-1 variables associated to the rational image under the . These relations reflect the 2:1 nature of the onto a rational scroll, distinguishing the hyperelliptic locus from the general case. The algebraic structure of canonical rings for curves provides a concrete parametrization of the moduli space \mathcal{M}_g. Specifically, the open subset \mathcal{M}_g^{nh} of non-hyperelliptic curves embeds into the GIT quotient classifying g-dimensional subspaces of quadrics in \mathbb{P}^{g-1} modulo the action of \mathrm{SL}(g), where the quadratic ideals I determine isomorphism classes via the canonical embeddings; this realizes \mathcal{M}_g^{nh} as a determinantal variety capturing the Petri map's injectivity. The hyperelliptic locus \mathcal{H}_g \subset \mathcal{M}_g similarly arises from quotients incorporating the Veronese subring relations, completing the description of \mathcal{M}_g. Explicit computations, including via Gröbner bases, yield concrete generators for low-genus cases. For instance, in genus 4 (non-hyperelliptic), R(C, K_C) is minimally generated by four linear forms with three independent quadrics forming a complete intersection ideal. In genus 5, the ring involves five generators and a quadratic ideal of codimension 5, with relations verifiable through syzygy computations that confirm Koszul properties. These examples illustrate how the ring's presentation resolves the local geometry of \mathcal{M}_g near special points, filling gaps in classical descriptions by enabling algorithmic verification of moduli points.