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Adjunction formula

The adjunction formula is a key theorem in algebraic geometry that describes the relationship between the canonical sheaves of a smooth subvariety and its ambient smooth variety, typically expressing the canonical sheaf of the subvariety as the restriction of the ambient canonical sheaf twisted by the line bundle associated to the subvariety.^[1] In its classical form for a smooth effective Cartier divisor D on a smooth projective variety X, the formula states that \omega_D = (\omega_X \otimes \mathcal{O}_X(D))|_D, where \omega_X and \omega_D denote the canonical sheaves of X and D, respectively.^[2] Equivalently, in terms of canonical divisors, K_D = (K_X + D)|_D.^[1] This relation arises from the exact sequence of conormal sheaves for the embedding D \hookrightarrow X, specifically $0 \to \Omega_D \to \Omega_X|_D \to \mathcal{O}_X(-D)|_D \to 0, whose determinant yields the twisting by \mathcal{O}_X(D)|_D upon taking top exterior powers.^[1] The formula holds more generally for smooth subvarieties of any codimension, where \omega_Y = \omega_X|_Y \otimes \det \mathcal{N}_{Y/X} for a smooth subvariety Y \subset X, with \mathcal{N}_{Y/X} the normal bundle.^[1] It extends to singular settings via dualizing sheaves under suitable conditions, such as when D is a Cohen-Macaulay divisor.^[2] The adjunction formula plays a central role in intersection theory and enumerative geometry, enabling computations of invariants like the genus of curves on surfaces—for instance, for a smooth curve C on a smooth surface S, it implies $2g(C) - 2 = (K_S + C) \cdot C, linking topological and algebraic data.^[1] Applications include determining canonical classes of hypersurfaces in projective space, analyzing complete intersections, and studying birational properties of varieties, such as in the classification of algebraic surfaces.^[1] It also underpins tools like Serre duality and Riemann-Roch theorems for subvarieties, facilitating deeper insights into sheaf cohomology and moduli spaces.^[2]

Formulation in Algebraic Geometry

General Formula for Smooth Subvarieties

In algebraic geometry, the adjunction formula relates the canonical sheaf of a smooth variety to that of a smooth subvariety embedded within it. Consider a smooth projective variety X over an algebraically closed field k and a smooth subvariety Y \subset X of codimension c \geq 1, where Y is a local complete intersection (lci) in X. The canonical sheaf \omega_Y of Y is isomorphic to the tensor product of the restriction of the canonical sheaf \omega_X of X to Y with the determinant of the normal bundle N_{Y/X} of Y in X:

\omega_Y \cong \omega_X \vert_Y \otimes \det N_{Y/X}.

This isomorphism holds under the given smoothness and lci assumptions, which ensure that the relevant sheaves are locally free vector bundles. When Y is an effective Cartier divisor (so c=1), the normal bundle simplifies to N_{Y/X} \cong \mathcal{O}_X(Y) \vert_Y, and thus \det N_{Y/X} \cong \mathcal{O}_X(Y) \vert_Y, yielding the restricted form \omega_Y \cong (\omega_X \otimes \mathcal{O}_X(Y)) \vert_Y. In terms of canonical divisors, this corresponds to K_Y = (K_X + Y) \vert_Y. The derivation of the formula proceeds from the exact sequence of cotangent sheaves induced by the lci embedding Y \hookrightarrow X:

$0 \to \mathcal{I}_Y / \mathcal{I}_Y^2 \to \Omega_X \vert_Y \to \Omega_Y \to 0,

where \mathcal{I}_Y \subset \mathcal{O}_X is the ideal sheaf of Y. Since the sheaves involved are locally free under the smoothness assumptions, taking determinants gives

\det \Omega_Y \cong \det(\Omega_X \vert_Y) \otimes (\det(\mathcal{I}_Y / \mathcal{I}_Y^2))^{-1}.

The sheaf \mathcal{I}_Y / \mathcal{I}_Y^2 is the conormal sheaf of the embedding, which is the dual of the normal bundle: \mathcal{I}_Y / \mathcal{I}_Y^2 \cong N_{Y/X}^\vee. Therefore, \det(\mathcal{I}_Y / \mathcal{I}_Y^2) \cong \det N_{Y/X}^\vee = (\det N_{Y/X})^\vee, and substituting yields \det \Omega_Y \cong \det(\Omega_X \vert_Y) \otimes \det N_{Y/X}. As the canonical sheaf is the determinant of the cotangent sheaf (\omega_Z = \det \Omega_Z for a smooth variety Z), the adjunction formula follows immediately. The assumptions of projectivity, smoothness of both X and Y, and the lci condition on Y are essential: projectivity ensures properness and compactness for cohomology computations involving canonical sheaves, while smoothness implies that \Omega_X and \Omega_Y are locally free, avoiding torsion or non-free issues. The lci property guarantees the exactness of the conormal sequence, as higher Tor terms vanish, allowing the clean determinant relation; without it, the embedding may not yield a resolution suitable for this calculation. These conditions place the formula in the context of classical algebraic geometry over algebraically closed fields, where Serre duality applies to relate sheaf cohomology. The adjunction formula emerged as part of the foundational development of sheaf cohomology and duality in algebraic geometry during the 1950s and 1960s. Jean-Pierre Serre introduced key concepts of coherent sheaves and their duality in his 1955 paper, laying the groundwork for relating canonical bundles via exact sequences. Alexander Grothendieck extended and formalized these ideas in his Éléments de Géométrie Algébrique (EGA), particularly through the systematic treatment of sheaves on schemes and their derived functors, which solidified the sheaf-theoretic framework for the formula.

Special Case for Smooth Divisors

When considering the special case of a smooth effective Cartier divisor D \subset X in a smooth variety X, the adjunction formula simplifies significantly due to the codimension-one embedding. Here, the canonical sheaf \omega_D on D is isomorphic to the restriction of the twisted canonical sheaf on X: \omega_D \cong (\omega_X \otimes \mathcal{O}_X(D))|_D. Equivalently, in terms of divisors, the canonical divisor satisfies K_D = (K_X + D)|_D. This relation highlights how the geometry of D inherits and adjusts the canonical structure from X via the twisting by the line bundle \mathcal{O}_X(D).^[3]^[4] The codimension-one nature of D is crucial, as it ensures that the line bundle \mathcal{O}_X(D) restricts to the normal bundle N_{D/X} on D, i.e., N_{D/X} \cong \mathcal{O}_X(D)|_D. This identification arises from the short exact sequence of sheaves on X,

$0 \to \mathcal{O}_X(-D) \to \mathcal{O}_X \to \mathcal{O}_D \to 0,

which captures the ideal sheaf of D. Extending this to the sheaf of differentials \Omega_X^\bullet yields a resolution that underlies the adjunction: tensoring the sequence with \Omega_X and taking determinants leads to the isomorphism for the canonical sheaves, reflecting how differentials on D are obtained by "resolving" those on X along the divisor. This construction leverages the local freeness of the structure sheaves in the smooth setting.^[3]^[5] A key property of this formulation is the duality it establishes between global sections of \omega_X(D) and residues of differential forms along D. Sections of \omega_X(D) correspond to meromorphic forms on X with poles bounded by D, and their residues provide holomorphic forms on D. In local coordinates where D = \{z = 0\} and a meromorphic n-form on X takes the shape \omega = (f \, dz / z) \wedge \eta (with \eta a local generator of \Omega_D^{n-1}), the residue map yields \operatorname{Res}_D(\omega) = f|_D \cdot \eta, which is a section of \omega_D. This extends naturally to higher-order poles, such as forms of the type f \, dz / z^{k+1} \wedge \eta for k \geq 1, where residues integrate over cycles in D to produce meromorphic sections on D, preserving the adjunction isomorphism.^[6]^[7] This special case also ties into Serre duality: on a smooth projective variety X of dimension n, Serre duality pairs H^i(X, \mathcal{F}) with H^{n-i}(X, \mathcal{F}^\vee \otimes \omega_X) for coherent \mathcal{F}. The adjunction formula induces a compatible duality on D by restricting twisted sheaves and using the residue map to identify cohomology groups on D with those on X, effectively transferring the duality from the ambient space to the divisor.^[3]

Illustrative Examples

Hypersurfaces of Degree d

In projective space \mathbb{P}^n over an algebraically closed field, the canonical divisor is given by K_{\mathbb{P}^n} = -(n+1)H, where H denotes the class of a hyperplane.^[8] For a smooth hypersurface X_d \subset \mathbb{P}^n of degree d, the adjunction formula yields the canonical divisor K_{X_d} = (K_{\mathbb{P}^n} + X_d)|_{X_d} = (-n-1 + d)H|_{X_d}.^[8] Equivalently, the canonical bundle is \omega_{X_d} \simeq \mathcal{O}_{X_d}(d - n - 1).^[9] The degree of the canonical divisor on X_d, defined as the intersection number K_{X_d} \cdot H^{n-2}, is then (d - n - 1)d, since the restriction H|_{X_d} has degree d.^[8] This computation highlights the transition in the Kodaira dimension of X_d: for d \leq n+1, X_d has non-positive Kodaira dimension, while for d \geq n+2, it is of general type with \kappa(X_d) = n-1.^[9] In the case of surfaces, where n=3, the degree simplifies to d(d-4); for a quartic surface (d=4), this yields \deg(K_{X_4}) = 0, implying a trivial canonical bundle and thus that X_4 is a Calabi-Yau variety (specifically, a K3 surface).^[10] To verify aspects of this structure, consider the holomorphic Euler characteristic \chi(X_d, \mathcal{O}_{X_d}), computed via the exact sequence $0 \to \mathcal{O}_{\mathbb{P}^n}(-d) \to \mathcal{O}_{\mathbb{P}^n} \to \mathcal{O}_{X_d} \to 0, which gives \chi(X_d, \mathcal{O}_{X_d}) = \chi(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}) - \chi(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(-d)) = 1 - \binom{n-d}{n}.^[11] This aligns with Riemann-Roch applications involving the canonical class derived from adjunction, confirming the bundle's properties without relying on higher cohomology computations via the Bott formula for vector bundles on \mathbb{P}^n.^[11] These results assume X_d is smooth, which holds for a general hypersurface of degree d in characteristic zero by Bertini's theorem on the smoothness of generic sections.^[12] In positive characteristic, smoothness requires additional conditions on general position to avoid singularities.^[13]

Complete Intersections

In algebraic geometry, the adjunction formula extends to smooth complete intersection subvarieties of codimension c in a smooth ambient variety X through successive applications of the hypersurface case. Specifically, let Y \subset X be the common zero locus Y = V(f_1, \dots, f_c) of sections f_i of line bundles L_i on X, where the intersections are transverse, ensuring that the differentials df_1, \dots, df_c are linearly independent along Y. The normal bundle decomposes as the direct sum

N_{Y/X} \cong \bigoplus_{i=1}^c L_i \big|_Y,

with determinant

\det(N_{Y/X}) \cong \bigotimes_{i=1}^c L_i \big|_Y.

The dualizing sheaf of Y is then given by

\omega_Y \cong \omega_X \otimes \det(N_{Y/X}) \big|_Y.

^[1] In terms of canonical divisors, if D_i \subset X denotes the effective divisor defined by the zero locus of f_i, the formula becomes

K_Y = \left( K_X + \sum_{i=1}^c D_i \right) \big|_Y.

This arises iteratively: first apply adjunction to obtain the canonical divisor of the hypersurface Z_1 = V(f_1), yielding K_{Z_1} = (K_X + D_1)|_{Z_1}; then intersect with D_2 to get Z_2 = Z_1 \cap D_2, so K_{Z_2} = (K_{Z_1} + D_2)|_{Z_2} = (K_X + D_1 + D_2)|_{Z_2}; continuing this process up to Y = Z_c produces the summed form. Transversality at each step guarantees that the normal bundle to Z_{i}/Z_{i-1} is L_i|_{Z_i}, preserving the direct sum structure for N_{Y/X}.^[1] Smoothness of Y requires that the defining equations form a regular sequence locally, which is ensured by the transversality condition on the differentials. For generic choices of sections f_i in given linear systems (e.g., homogeneous polynomials of fixed degrees in projective space), Bertini's theorem implies that Y is smooth, as the singular locus of potential intersections has positive codimension.^[1]^[14] A concrete computation arises when X = \mathbb{P}^n and Y is a smooth complete intersection of hypersurfaces of degrees d_1, \dots, d_c, so \dim Y = n - c. Here, K_{\mathbb{P}^n} = \mathcal{O}_{\mathbb{P}^n}(-n-1) and each D_i corresponds to \mathcal{O}_{\mathbb{P}^n}(d_i), yielding

\omega_Y \cong \mathcal{O}_Y\left( \sum_{i=1}^c d_i - n - 1 \right).

The degree of Y (with respect to the hyperplane class) is \prod_{i=1}^c d_i. For instance, if Y is a curve (n - c = 1), the arithmetic genus g satisfies

$2g - 2 = \left( \sum_{i=1}^c d_i - n - 1 \right) \prod_{i=1}^c d_i,

providing a direct link to classical enumerative invariants.^[1]

Curves on Quadric Surfaces

A quadric surface Q \subset \mathbb{P}^3 is defined by a homogeneous quadratic equation and is isomorphic to \mathbb{P}^1 \times \mathbb{P}^1. Its canonical bundle is given by K_Q = -2H|_Q, where H denotes the hyperplane class pulled back from \mathbb{P}^3. For a smooth curve C \subset Q of bidegree (a, b), the line bundle associated to C on Q is \mathcal{O}_Q(C) = aH_1 + bH_2, where H_1 and H_2 are the classes of the two rulings on Q, satisfying H_1 \cdot H_2 = 1, H_1^2 = H_2^2 = 0, and H = H_1 + H_2. The degree of C in \mathbb{P}^3 is then a + b.^[15] Applying the adjunction formula yields the canonical bundle of C as K_C = (K_Q + C)|_C = (-2H + aH_1 + bH_2)|_C = ((a-2)H_1 + (b-2)H_2)|_C. The degree of K_C is thus (a-2) \cdot b + (b-2) \cdot a = 2ab - 2(a + b), which equals $2g - 2 where g = (a-1)(b-1) is the genus of C. A curve of bidegree (1,1) is a conic with \deg(K_C) = -2, confirming it is rational (genus 0). A smooth curve of bidegree (3,3) has genus 4 and \deg(K_C) = 6; its embedding in \mathbb{P}^3 via the complete linear system |\mathcal{O}_Q(3H_1 + 3H_2)| is the canonical embedding, as non-hyperelliptic genus-4 curves lie on a unique quadric surface of this type. The rulings intersect C in 3 points each, giving two g^1_3 linear series on C.^[16] Geometrically, since Q \cong \mathbb{P}^1 \times \mathbb{P}^1 with canonical bundle K_Q = -2F_1 - 2F_2 (where F_1, F_2 are the fiber classes, identified with the rulings), the adjunction formula on C reflects the product structure, as the computation aligns with the adjunction for divisors on product varieties.^[15]

Residue and Inversion Aspects

Poincaré Residue Map

The Poincaré residue map serves as the explicit mechanism that realizes the adjunction formula for smooth divisors in algebraic geometry, by mapping meromorphic differential forms on the ambient variety to holomorphic forms on the divisor. For a smooth divisor D \subset X in a smooth variety X of dimension n, the map is defined as \operatorname{Res}: \Omega_X^{p+1}(\log D) \to \Omega_D^p for $0 \leq p \leq n-1, where \Omega_X^k(\log D) denotes the sheaf of logarithmic k-forms along D. Locally, in coordinates where D is given by x_1 = 0, a section f \cdot \frac{dx_1}{x_1} \wedge dx_2 \wedge \cdots \wedge dx_{p+1} maps to \operatorname{Res}(f \cdot \frac{dx_1}{x_1} \wedge dx_2 \wedge \cdots \wedge dx_{p+1}) = f|_D \cdot dx_2 \wedge \cdots \wedge dx_{p+1}. This construction arises from the short exact sequence $0 \to \Omega_X^{p+1} \to \Omega_X^{p+1}(\log D) \to \Omega_D^p \to 0, ensuring the residue map is surjective and induces the desired sheaf isomorphism.^[17] In the context of canonical sheaves, the Poincaré residue map induces a surjective map H^0(X, \Omega_X^n(\log D)) \twoheadrightarrow H^0(D, \Omega_D^{n-1}) on global sections, which is an isomorphism when H^0(X, \Omega_X^n) = 0, aligning with the adjunction relation where the dualizing sheaf \omega_X(D)|_D \cong \omega_D. This follows from the identification \omega_X(D) \cong \Omega_X^n(\log D) for smooth D, providing a concrete link between the logarithmic differentials on X and the cotangent sheaf on D. The map is holomorphic in the complex analytic setting, where it extracts residues of meromorphic n-forms with simple poles along D, and algebraic over fields of characteristic zero, preserving the structure of differentials in the scheme-theoretic framework.^[6]^[18] Key properties include compatibility with transverse intersections: if D_1 and D_2 intersect properly, the residue along D_1 + D_2 composes iteratively with residues along each component. In the algebraic setting, the map extends to singular divisors via normalization, where for a normalization \tilde{D} \to D, the pullback of residues yields an isomorphism \omega_X(\tilde{D})|_{\tilde{D}} \cong \omega_{\tilde{D}}, ensuring consistency in birational geometry. Historically, the concept originated in Henri Poincaré's work on residues for multiple integrals in the 1880s, particularly in his studies of Fuchsian functions and automorphic forms, and was formalized in modern algebraic geometry by Alexander Grothendieck in the Éléments de géométrie algébrique (EGA), where logarithmic sheaves and residue maps are developed systematically.^[17]^[6]

Inversion of Adjunction

The inversion of adjunction serves as a converse to the adjunction formula, linking the singularities of a subvariety to those of the ambient space in the framework of log pairs. For a normal variety X and a subvariety Y lying in the smooth locus of X, the multiplier ideal sheaf \mathcal{J}(Y) \subset \mathcal{O}_X captures the failure of log canonicity of the pair (X, Y) via log-canonical thresholds, which quantify the largest \lambda > 0 such that (X, \lambda Y) remains sub log canonical along Y. Specifically, the pair (X, K_X + Y) is log canonical along Y if and only if K_Y is canonical on Y.^[19] In the logarithmic setting, consider a log pair (X, \Delta) where X is normal, \Delta is an effective \mathbb{R}-divisor such that K_X + \Delta is \mathbb{R}-Cartier, and Y is a prime divisor on X with no common components with the non-klt part of \Delta. Under semi log canonical (slc) assumptions on (X, \Delta), the inversion of adjunction asserts that if \nu: Y^\nu \to Y denotes the normalization of Y, then

K_{Y^\nu} + \Delta_{Y^\nu} = \nu^*(K_X + \Delta + Y),

where \Delta_{Y^\nu} is the (possibly fractional) trace of \Delta on Y^\nu, often realized as Shokurov's different. This formula inverts the direct adjunction by propagating log canonical properties from the ambient pair to the restricted pair on Y.^[20] The inversion of adjunction plays a crucial role in the minimal model program (MMP), where it facilitates the analysis of birational transformations such as flipping contractions by relating log discrepancies across exceptional loci. For example, when Y is a curve in a surface X, the inversion detects rational singularities on X by verifying whether the restricted pair (Y, \Delta_Y) inherits canonical singularities from the log canonical ambient pair (X, \Delta + Y). This tool has been essential in establishing termination of flips and abundance results in higher dimensions.^[21] János Kollár's work in the 1990s on log adjunction extended the classical smooth case to singular pairs, conjecturing the inversion in the context of discrepancies and minimal model theory, which spurred subsequent proofs and generalizations. Recent work, such as the proof of inversion of adjunction for higher rational singularities (Xu, 2025), continues to extend these results to broader classes of singularities.^[22]^[23]

Specific Cases and Applications to Curves

Canonical Divisor of Plane Curves

For a smooth plane curve C \subset \mathbb{P}^2 of degree d, the canonical divisor K_C is obtained via the adjunction formula applied to the hypersurface embedding in the projective plane, where the canonical divisor of \mathbb{P}^2 is K_{\mathbb{P}^2} = -3H with H the hyperplane class.^[24] Thus, K_C = (K_{\mathbb{P}^2} + C)|_C = (-3H + dH)|_C = (d-3)H|_C.^[24] The degree of K_C follows from the degree of the hyperplane restriction H|_C, which intersects C in d points by Bézout's theorem, yielding \deg(K_C) = (d-3) \cdot d = d(d-3).^[25] This degree computation aligns with the Riemann-Roch theorem on curves, confirming \deg(K_C) = 2g - 2 where g = \frac{(d-1)(d-2)}{2} is the genus of C.^[25] The complete linear system |K_C| provides the canonical embedding of C into \mathbb{P}^{g-1}, where the sections of \mathcal{O}_C(d-3) are restrictions of homogeneous polynomials of degree d-3 on \mathbb{P}^2.^[24] For d \geq 4, smooth plane curves are non-hyperelliptic, so |K_C| is very ample and embeds C as a curve of degree $2g-2 = d(d-3) in \mathbb{P}^{g-1}.^[25] In the special case d=3, C is an elliptic curve with g=1, and K_C is trivial as a divisor class (degree 0), consistent with the canonical map degenerating to a point in \mathbb{P}^0.^[24] For d=4, the plane embedding of the quartic is precisely the canonical model in \mathbb{P}^2.^[25] The dual curve C^\vee \subset (\mathbb{P}^2)^\vee parametrizes the tangent lines to C, and bitangent lines—those tangent at two distinct points—correspond to effective divisors D of degree 2 such that $2D \sim K_C.^[26] These bitangents realize the odd theta characteristics of C, square roots of the canonical bundle.^[26] For a smooth plane quartic (d=4, g=3), there are exactly 28 bitangents, each defining an odd theta characteristic on the canonical model.^[26]

Genus Calculations for Curves

For a smooth curve C embedded in a smooth projective surface X, the adjunction formula relates the canonical divisor K_C of C to that of X by K_C = (K_X + C)|_C, where the degree of K_C equals $2g - 2 with g the genus of C.^[27] By intersection theory on X, this degree is \deg(K_C) = (K_X + C) \cdot C = K_X \cdot C + C^2, yielding the fundamental relation $2g - 2 = K_X \cdot C + C^2.^[27] In the specific case of a smooth plane curve C of degree d in \mathbb{P}^2, the canonical divisor is K_{\mathbb{P}^2} = -3H with H the hyperplane class, and C = dH, so C^2 = d^2 and K_{\mathbb{P}^2} \cdot C = -3d.^[2] Substituting into the adjunction relation gives $2g - 2 = d^2 - 3d, or equivalently g = \frac{(d-1)(d-2)}{2}.^[2] This formula, derived directly from adjunction, provides the genus for any smooth plane curve and serves as a cornerstone for enumerative geometry. For curves embedded in more general surfaces, the adjunction-derived genus formula takes the arithmetic form g = 1 + \frac{1}{2}(C^2 + K_X \cdot C), applicable whenever intersection numbers are known.^[27] A representative example is a smooth plane section C of a cubic surface X \subset \mathbb{P}^3, where X has hyperplane class H|_X and canonical divisor K_X = -H|_X. Here, C = H|_X satisfies C^2 = 3 and K_X \cdot C = -3, so $2g - 2 = 3 - 3 = 0, confirming g = 1 and that C is elliptic. The adjunction formula extends to singular curves via the arithmetic genus p_a(C) = 1 + \frac{1}{2}(C^2 + K_X \cdot C), defined as \dim H^1(X, \mathcal{O}_C), which coincides with the smooth case formula but counts multiplicities at singularities. Adjunction-derived genera underpin bounds on curve complexity, such as the Castelnuovo inequality, which limits g for curves of given degree on rational surfaces using C^2 + K_X \cdot C, though the core computations remain rooted in the basic intersection formula.^[27] Similarly, Brill-Noether theory employs these genera to classify linear systems on curves, with adjunction providing the initial degree-genus link for embedded examples.^[27]

Extensions to Topology

Adjunction in Low-Dimensional Manifolds

In low-dimensional topology, particularly in the smooth category, the adjunction formula has analogs for embedded surfaces in 4-manifolds. For a smooth closed oriented surface \Sigma of genus g embedded in a smooth oriented 4-manifold X with b^+(X) > 1 and nonvanishing Seiberg-Witten invariant for a spin^c structure \mathfrak{s}, the Seiberg-Witten adjunction inequality states that

$2g - 2 \geq \Sigma \cdot \Sigma + |c_1(\mathfrak{s}) \cdot \Sigma|,

where c_1(\mathfrak{s}) is the first Chern class of the determinant line bundle of \mathfrak{s}. This inequality provides a lower bound on the genus of embedded surfaces and mirrors the algebraic geometry adjunction formula by relating the topology of the surface to intersection data in the ambient manifold.^[28] The self-intersection number \Sigma \cdot \Sigma equals the Euler class of the normal bundle N_\Sigma evaluated on the fundamental class of \Sigma, i.e., e(N_\Sigma)[\Sigma] = \Sigma \cdot \Sigma. Since N_\Sigma is an oriented rank-2 real vector bundle over \Sigma, the inequality constrains possible embeddings using characteristic classes and gauge-theoretic invariants. In simply connected 4-manifolds, additional constraints arise from the intersection form and basic classes from Seiberg-Witten theory. Equality holds for certain canonical representatives, such as when c_1(\mathfrak{s}) \cdot \Sigma = 0 and \Sigma minimizes the genus in its homology class. This leverages properties of the normal bundle and Thom isomorphism in the analysis of embeddings.^[28] In simply connected smooth 4-manifolds with b^+ > 1, equality in the adjunction inequality is achieved for certain surfaces. For example, an embedded sphere (g = 0) satisfies \Sigma^2 = -2, corresponding to exceptional spheres in blow-ups or rational surfaces. Similarly, an embedded torus (g = 1) achieves \Sigma^2 = 0, as seen in elliptic fibrations or torus bundles over surfaces. These cases illustrate the inequality's role in constraining smooth embedding possibilities, distinguishing smooth from topological categories where Freedman's results allow more flexible genus realizations without gauge-theoretic obstructions.^[28] Historically, adjunction inequalities played a key role in Michael Freedman's classification of simply connected topological 4-manifolds in the 1980s, where genus bounds informed topological surgery techniques to establish existence of embeddings and detect differences between topological and smooth structures. In the smooth category, gauge theory provides sharper obstructions.

Applications in 4-Manifold Topology

In definite 4-manifolds, Donaldson's diagonalizability theorem, when combined with the adjunction inequality derived from Seiberg-Witten invariants, prohibits the existence of embedded surfaces of genus g \geq 2 with self-intersection \Sigma^2 < 2g - 2. This follows because the Seiberg-Witten adjunction inequality states that for an embedded closed oriented surface \Sigma of genus g in a 4-manifold X with b^+ > 1 and nonvanishing Seiberg-Witten invariant for spin^c structure \mathfrak{s}, one has

$2g - 2 \geq \Sigma \cdot \Sigma + |c_1(\mathfrak{s}) \cdot \Sigma|,

where c_1(\mathfrak{s}) is the first Chern class of the determinant line bundle associated to \mathfrak{s}. In definite manifolds, the intersection form's definiteness and the structure of basic classes restrict |c_1(\mathfrak{s}) \cdot \Sigma| such that violations of the bound imply vanishing invariants, contradicting the theorem's constraints on smooth realizations. For instance, in simply-connected positive definite 4-manifolds, only spheres can represent classes of negative self-intersection, excluding higher-genus surfaces that would otherwise embed smoothly. The adjunction inequality exhibits non-additivity under connected sums, providing a tool to detect exotic smooth structures on 4-manifolds. Wall's non-additivity formula for the signature in gluings along embedded surfaces of genus g and self-intersection k adjusts the naive sum by a term involving $2\chi(\Sigma) + k, reflecting how the surface's topology alters the overall intersection form. In connected sums, this manifests in bounds on embedded surfaces spanning both summands, where the minimal genus in X \# Y can undercut the sum of minimal genera in X and Y, as the connecting sphere allows "shortcut" embeddings. This non-additivity detects exotic smoothness in examples like elliptic surfaces with multiple fibers, such as the Dolgachev surface E(1)_{2,3}, which is homeomorphic but not diffeomorphic to the standard rational elliptic surface E(1) = \mathbb{CP}^2 \# 9\overline{\mathbb{CP}^2}; here, the adjunction inequality, applied via Seiberg-Witten invariants, reveals discrepancies in representable surface classes that violate expected additivity for the standard structure. Heegaard Floer homology refines the adjunction inequality, particularly in bounding slice genera for knots in contact 3-manifolds bounding 4-manifolds.^[29] Ozsváth and Szabó's contact invariant c(\xi) \in \widehat{HF}(-Y), which vanishes for overtwisted contact structures and is nonzero for tight ones, induces a relative adjunction inequality: for a knot K in a contact 3-manifold (Y, \xi) bounding a surface \Sigma in a 4-manifold W with \partial W = -Y, the slice genus g_4(K) satisfies $2g_4(K) - 1 \geq |c_1(\mathfrak{s}) \cdot [\Sigma]| when c(\xi) pairs nontrivially with the relevant Floer class.^[29] This refines the classical bound by incorporating contact geometry, obstructing slice disks for Legendrian knots in tight contacts and providing sharper constraints on concordance than Seiberg-Witten alone.^[29] Contemporary extensions via Floer theory address limitations in classical adjunction inequalities for classifying fake 4-balls—contractible smooth 4-manifolds with boundary S^3 but exotic relative to the standard ball. Heegaard Floer homology detects such exotics through correction terms d(W, \mathfrak{s}), where nonvanishing relative invariants or violations of refined adjunction bounds (e.g., for properly embedded disks) obstruct the standard smooth structure; for instance, cork twists altering embedded surfaces while preserving topology yield fake 4-balls distinguished by Floer non-additivity. These inequalities, combined with mapping class group actions on Floer groups, classify infinite families of fake 4-balls up to diffeomorphism, revealing the incompleteness of gauge-theoretic adjunction alone in low-dimensional fillings.

References

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If X is a Serre duality space (i.e. a space where Serre duality holds), and D is an effective Cartier divisor, then ωD ...
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[PDF] arXiv:alg-geom/9511015v1 24 Nov 1995
The adjunction formula relates canonical divisors of varieties. Let X be a smooth variety and S a smooth divisor. Then we have (KX + S)|S ∼ KS. But if X ...
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[PDF] adjunction for the grauert–riemenschneider canonical sheaf and ...
May 20, 2016 · The adjunction formula for a smooth divisor. Let M be a complex mani ... short exact sequence (1.2). By use of the long exact ...
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[PDF] Residues and Hodge theory
Suppose X is a complex manifold, D → X is a smooth divisor. ... g(ρ2η,−ρ1η)=(ρ2η)|U12 −(−ρ1η)|U12 = η. From the above short exact sequence we obtain the long ...
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[PDF] adjunction formula, poincaré residue and holomorphic differentials ...
R. Hartshorne, Algebraic geometry, Springer-Verlag, New York—Heidelberg—Berlin, 1977. 8. J. Jost, Compact Riemann surfaces.
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[PDF] Uniform Boundedness of Rational Points. - Berkeley Math
Let X be a hypersurface in Pn of degree d. By the Adjunction formula,. KX ∼= KPn ⊗ OPn (X)|X. = OPn (−n − 1 + d)|X ...
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[PDF] notes for 483-3: kodaira dimension of algebraic varieties
Let X ⊂ Pn be a smooth hypersurface of degree d. If. OX(1) is the ... Let D be a divisor on a smooth projective surface X. Then χ(OX(D)) = D · (D − KX).
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[PDF] General introduction to K3 surfaces - Yale Math
Hartshorne defines Chow ring and Chern classes of an algebraic variety and states the. Hirzebruch–Riemann–Roch formula. Chern classes of vector bundles are ...
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[PDF] FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37
Feb 27, 2008 · But the pullback of a hyperplane in PN to Pn is a degree d hypersurface. ... Caution: The Euler characteristic of the structure sheaf doesn't ...
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[0910.4105] Bertini Type Theorems - arXiv
Oct 21, 2009 · Bertini's Theorem states that a general hyperplane H intersects X with an irreducible smooth subvariety of X.Missing: smoothness | Show results with:smoothness
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[PDF] bertini theorems over finite fields - MIT Mathematics
Then there exists a smooth projective geometrically integral hypersurface H ⊂ Pn such that H ∩ X is smooth of dimension m − 1 and contains F. Remarks. (1) If m ...Missing: smoothness source
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[PDF] Lecture 22 Bertini's Theorem, Coherent Sheves on Curves
Let's consider some ways to construct smooth varieties. Theorem 1.1 (Bertini's Theorem). Let X ⊆ PV be a smooth subvariety. Then for a generic hyperplane H,. Y ...
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[PDF] CURVES ON VARIETIES These are (rough) notes from a 4-part ...
We get many examples by taking curves lying on small degree surfaces. If Q = P1 × P1 is a quadrics surface, then a curve of bidegree (a, b) has degree d = a + b ...
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[PDF] The arithmetic and geometry of genus four curves
4 corresponds to a curve whose canonical embedding lines on a smooth quadric surface. It then follows that this curve is of type (3, 3) in P1 × P1. A (3, 3) ...Missing: bidegree | Show results with:bidegree<|control11|><|separator|>
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[PDF] Contents - Math (Princeton)
Jun 1, 2010 · By the adjunction formula, deg ωE = (E·E)+(E·KX) is ... János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cam-.
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[PDF] arXiv:1107.2863v3 [math.AG] 14 Nov 2012
Nov 14, 2012 · Introduction. Let X be a smooth variety and S ⊂ X a smooth hypersurface. The Poincaré residue map is an isomorphism. R : ωX(S)|S ∼= ωS.
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### Summary of Statement Relating Multiplier Ideal J(Y) to Inversion of Adjunction and Log-Canonical Thresholds
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[PDF] Log canonical inversion of adjunction
Abstract: This is a short note on the log canonical inversion of adjunction. Key words: inversion of adjunction; adjunction; log canonical singularities; ...<|control11|><|separator|>
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On inversion of adjunction - Project Euclid
Key words: Adjunction; inversion of adjunction; minimal model program. 1. Introduction. In [9], we established the following adjunction and inversion of ...
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[PDF] Flips and abundance for algebraic threefolds - Numdam
problem of inversion of adjunction. The following theorem shows that in the notation of (17.3) totaldiscrep(S,Diff(B)) > -1 {:::> discrep(CenternS =/= 0,X,S ...
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[PDF] Geometry of Algebraic Curves
the adjunction formula, has genus one. 5.10 Example. Recall that a quadric in P3 is isomorphic to P1 × P1, so it has two rulings of lines on it. If C ⊂ P3 ...<|control11|><|separator|>
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[PDF] Geometry of Algebraic Curves - staff.math.su.se
The canonical divisor on a smooth plane curve. 30. 6.2 ... as the difference of the expected genus of a smooth degree d plane curve and the actual genus.
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[PDF] On Some Geometric Constructions Related to Theta Characteristics
If C is not hyperelliptic, the canonical series embeds C as a smooth quartic in P2. The 28 odd theta characteristics q correspond to the 28 bitangent lines to ...
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[PDF] Divisors on a surface - Purdue Math
1 (Adjunction formula). Let C ⇢ X be a smooth curve of genus g, then. 2g. 2 ... If Y is a surface with an curve E ⇠= P1 with. E2 = 1, then it must be the blow up ...
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[PDF] Riemann Surfaces and Algebraic Curves
Dec 11, 2001 · Apply the Riemann Hurwitz formula (Theorem 1.9) to the composition of this projection with the normalization map. For more details see [1] ...
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The symplectic Thom conjecture - Annals of Mathematics
In this paper, we demonstrate a relation among Seiberg-Witten invariants which arises from embedded surfaces in four-manifolds whose self-intersection number is ...Missing: 4- | Show results with:4-
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[PDF] Six Lectures on Four 4-Manifolds
These lectures review the existence and uniqueness of smooth and symplectic structures on 4-manifolds, focusing on surgical techniques and Seiberg-Witten ...Missing: submanifold | Show results with:submanifold
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Heegaard Floer homologies and contact structures - math - arXiv
Oct 8, 2002 · Given a contact structure on a closed, oriented three-manifold Y, we describe an invariant which takes values in the three-manifold's Floer homology \HFa.Missing: adjunction inequality slice genus