04.09.02 · algebraic-geometry / hodge

Kodaira vanishing theorem

shipped3 tiersLean: partial

Anchor (Master): Kodaira 1953 PNAS; Voisin Vol I; Griffiths-Harris Ch. 1; Lazarsfeld Vol I

Intuition [Beginner]

The Kodaira vanishing theorem says that on a smooth projective complex variety, "positivity" of a line bundle forces all higher cohomology to vanish — in a precise sense. If $L$ is an ample line bundle on a smooth projective variety $X$ of dimension $n$ , then the higher cohomology of the canonical sheaf paired with $L$ is zero in every degree above zero. Equivalently, by Serre duality, the cohomology of the inverse line bundle vanishes in every degree below $n$ .

Why does positivity kill cohomology? Intuitively: an ample line bundle has positive curvature. By a Kodaira-style argument with harmonic forms, sections paired against this positive curvature cannot satisfy the Laplacian equation in higher degree — only zero forms work. This is a quantitative version of the slogan "positivity kills higher cohomology."

Kunihiko Kodaira proved this in 1953 in a short PNAS paper. It is one of the cornerstones of complex algebraic geometry — used in Riemann-Roch computations, the minimal model program, and projective embedding theorems. The Akizuki-Nakano generalisation (1954) extends to forms with higher-degree cotangent powers, and Kawamata-Viehweg (1982) generalises to nef + big line bundles.

Visual [Beginner]

A smooth projective variety with positive curvature, where higher cohomology of the canonical bundle twisted by an ample line bundle vanishes.

Worked example [Beginner]

For $X = P_{C}^{n}$ and $L = O_{P^{n}} (d)$ with $d \geq 1$ (an ample line bundle):

Kodaira vanishing reads: the higher cohomology of $O (d - n - 1)$ on $P^{n}$ vanishes for $d \geq 1$ .

Verification: for $P^{n}$ , the cohomology of $O (m)$ is well-known:

H^{i} (P^{n}; O (m)) = ⎩ ⎨ ⎧ (n n + m) (- m - n - 1 - m - 1) 0 i = 0, m \geq 0 i = n, m \leq - n - 1 otherwise.

For $m = d - n - 1 \geq 1 - n - 1 = - n$ (with $d \geq 1$ ): we are in the range where higher cohomology vanishes, confirming Kodaira.

For $P^{1}$ and $L = O (1)$ (degree 1, ample): the canonical paired with $L$ becomes $O (- 2 + 1) = O (- 1)$ . We need $H^{1} (P^{1}; O (- 1)) = 0$ . Indeed, $O (- 1)$ has zero $H^{0}$ and zero $H^{1}$ on $P^{1}$ .

For an elliptic curve $E$ with $L$ a degree-1 line bundle (ample): the canonical paired with $L$ becomes $L$ itself, since the canonical on $E$ is structure-sheaf-equivalent. We need $H^{1} (E; L) = 0$ . By Riemann-Roch on $E$ : $dim H^{0} (L) - dim H^{1} (L) = de g L + 1 - g = 1 + 1 - 1 = 1$ , with $H^{0} (L)$ nonzero (since $L$ has positive degree), so $H^{1} (L) = 0$ . Verified.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $X$ be a smooth projective variety over $C$ of dimension $n$ , and $L$ a line bundle on $X$ .

Theorem (Kodaira vanishing). If $L$ is ample on $X$ , then

H^{i} (X; ω_{X} \otimes L) = 0 for all i > 0.

Equivalently, by Serre duality:

H^{j} (X; L^{- 1}) = 0 for all j < n = dim X .

Akizuki-Nakano generalisation (1954). For $L$ ample and $0 \leq p \leq n$ :

H^{q} (X; Ω_{X}^{p} \otimes L) = 0 for p + q > n .

The case $p = n$ gives the original Kodaira vanishing (since $Ω_{X}^{n} = ω_{X}$ and $p + q > n$ becomes $q > 0$ ).

Kawamata-Viehweg generalisation (1982). For $L$ a line bundle that is nef and big (asymptotically ample, allowing some boundary fluctuation), the same vanishing holds:

H^{i} (X; ω_{X} \otimes L) = 0 for i > 0.

This is the foundational vanishing theorem in the minimal model program.

Generalisations beyond $C$ . Kodaira vanishing holds:

In characteristic 0: over any algebraically closed field of characteristic 0, by descent from $C$ .
In characteristic $p$ : fails in general (Raynaud 1978). Counter-examples exist for certain Frobenius-related line bundles.
Algebraic proof in char 0: Deligne-Illusie (1987) gave a Frobenius-based proof, working in characteristic $p$ with a Witt-vector lift and using their Hodge degeneration. Esnault-Viehweg simplified this further.

Lefschetz hyperplane theorem. A foundational application: for a smooth ample divisor $Y \subset X$ , the restriction $H^{i} (X; C) \to H^{i} (Y; C)$ is an isomorphism for $i < dim Y = dim X - 1$ , and injective for $i = dim Y$ . Proven via Kodaira vanishing applied to the ideal sheaf of $Y$ .

Kodaira embedding theorem. Another key application: a compact Kähler manifold $X$ admits a holomorphic embedding into projective space iff there exists a positive line bundle on $X$ . The proof uses Kodaira vanishing to ensure global sections of high tensor powers separate points and tangent vectors.

Key theorem with proof [Intermediate+]

Theorem (Kodaira vanishing). Let $X$ be a smooth projective complex variety of dimension $n$ , and $L$ an ample line bundle. Then $H^{i} (X; ω_{X} \otimes L) = 0$ for all $i > 0$ .

Proof outline (Kodaira's harmonic-form approach). Step 1 — positive curvature representative. Since $L$ is ample, by the Kodaira embedding theorem, $L$ admits a Hermitian metric $h$ whose Chern curvature $c_{1} (L, h) = - \frac{i}{2 π} \partial \overset{ˉ}{\partial} lo g h$ is a positive real $(1, 1)$ -form on $X$ . (In Kähler-geometry language: there exists a Kähler metric on $X$ in the cohomology class $c_{1} (L)$ .)

Step 2 — Bochner-Kodaira-Nakano identity. For a Hermitian line bundle $(L, h)$ on a Kähler manifold $X$ , the Bochner-Kodaira-Nakano identity relates the Laplacians on $Ω_{X}^{p} \otimes L$ :

Δ_{\overset{ˉ}{\partial}, L} - Δ_{\partial, L} = [i Θ (L), Λ],

where $Θ (L)$ is the curvature of $L$ and $Λ$ is the contraction with the Kähler form $ω$ of $X$ .

Step 3 — pairing with positive curvature. For an $(n, q)$ -form $α$ with values in $L$ , the Bochner-Kodaira-Nakano identity gives

⟨ Δ_{\overset{ˉ}{\partial}} α, α ⟩ - ⟨ Δ_{\partial} α, α ⟩ = ⟨[i Θ (L), Λ] α, α ⟩ .

For $L$ positive (positive Chern curvature), the right side is positive on $(n, q)$ -forms with $q > 0$ . Specifically, if $i Θ (L) > 0$ (positive Hermitian form), then $[i Θ (L), Λ]$ acts as multiplication by a positive scalar on $(n, q)$ -forms with $q > 0$ .

Step 4 — vanishing for harmonic forms. If $α$ is a $\overset{ˉ}{\partial}$ -harmonic $(n, q)$ -form with values in $L$ (representing a class in $H^{q} (X; ω_{X} \otimes L)$ ), then $Δ_{\overset{ˉ}{\partial}} α = 0$ . The Bochner-Kodaira identity gives

0 - ⟨ Δ_{\partial} α, α ⟩ = ⟨[i Θ (L), Λ] α, α ⟩ .

The left side is non-positive (by self-adjointness of $Δ_{\partial}$ ). The right side is non-negative (by positivity of curvature) and is strictly positive for $α \neq = 0$ in the $(n, q)$ -component with $q > 0$ . The contradiction forces $α = 0$ .

Step 5 — concluding vanishing. Since the only $\overset{ˉ}{\partial}$ -harmonic $(n, q)$ -form with values in $L$ for $q > 0$ is zero, and harmonic forms represent cohomology, $H^{q} (X; ω_{X} \otimes L) = 0$ for $q > 0$ . $□$

The proof uses positivity of curvature + Bochner-Kodaira-Nakano identity + harmonic-form representation of cohomology to deduce vanishing. The cornerstone is the curvature positivity: the Kähler condition + positivity of $L$ produce the required Hermitian-form positivity.

Bridge. The construction here builds toward later units of the strand, where the same pattern is taken up at higher structure. The defining pattern appears again in those units in a sharpened form, where the local data is glued or quotiented. Putting these together, the foundational insight is that the data of this unit gives the structural signature that the rest of the strand reads off.

Exercises [Intermediate+]

Exercise 3 (medium, proof).

Use Kodaira vanishing to prove that for an ample line bundle $L$ on a smooth projective variety $X$ , the Euler characteristic $χ (X; L^{\otimes m})$ equals $h^{0} (X; L^{\otimes m})$ for $m ≫ 0$ .

Hint

For $m ≫ 0$ , $ω_{X} \otimes L^{\otimes m}$ has vanishing higher cohomology, hence $h^{i} (L^{\otimes m}) = 0$ for $i > 0$ .

Answer

For $m ≫ 0$ , the line bundle $L^{\otimes m} \otimes ω_{X}^{- 1}$ is ample (since $L$ is ample and $ω_{X}^{- 1}$ is fixed). By Kodaira vanishing applied to the ample line bundle $L^{\otimes m} \otimes ω_{X}^{- 1}$ :

H^{i} (X; ω_{X} \otimes (L^{\otimes m} \otimes ω_{X}^{- 1})) = H^{i} (X; L^{\otimes m}) = 0 for i > 0.

So for $m$ large enough: $h^{i} (X; L^{\otimes m}) = 0$ for all $i > 0$ . The Euler characteristic reduces:

χ (X; L^{\otimes m}) = i \sum (- 1)^{i} h^{i} (X; L^{\otimes m}) = h^{0} (X; L^{\otimes m}) .

This is the practical form: for large $m$ , Euler-characteristic computations (which are invariant under deformation by Hirzebruch-Riemann-Roch) give the dimension of global sections — the Hilbert polynomial of $L$ . The result is foundational for the theory of Hilbert series and projective embedding. $□$

Exercise 4 (medium, proof).

Prove the Lefschetz hyperplane theorem: for a smooth ample divisor $Y \subset X$ on a smooth projective complex variety, $H^{i} (X; C) \to H^{i} (Y; C)$ is an iso for $i < dim Y$ .

Hint

Use the long exact sequence in cohomology associated to $0 \to O (- Y) \to O_{X} \to O_{Y} \to 0$ and apply Kodaira to twists.

Answer

Set $dim X = n$ , $dim Y = n - 1$ . The short exact sequence of sheaves:

0 \to O_{X} (- Y) \to O_{X} \to O_{Y} \to 0

gives a long exact sequence:

\dots \to H^{i - 1} (O_{Y}) \to H^{i} (O_{X} (- Y)) \to H^{i} (O_{X}) \to H^{i} (O_{Y}) \to \dots

By Kodaira vanishing applied to the ample $L = O_{X} (Y)$ : $H^{j} (X; O_{X} (- Y)) = H^{j} (X; L^{- 1}) = 0$ for $j < n$ . So the long exact sequence breaks into isomorphisms:

H^{i} (O_{X}) ≅ H^{i} (O_{Y}) for i < n - 1

and an injection $H^{n - 1} (O_{X}) ↪ H^{n - 1} (O_{Y})$ .

By Hodge decomposition, this lifts to the comparable statement on full singular cohomology with $C$ coefficients (using the Hodge-Akizuki-Nakano vanishing for general $Ω^{p}$ , not just $Ω^{0} = O$ ): the restriction $H^{i} (X; C) \to H^{i} (Y; C)$ is an iso for $i < n - 1$ and injective for $i = n - 1$ .

This is the Lefschetz hyperplane theorem. $□$

The theorem implies that low-dimensional cohomology of smooth projective varieties is "inherited" from ample hyperplane sections — the geometry of $X$ is largely controlled by its ample divisors.

Exercise 5 (medium, conceptual).

Why does Kodaira vanishing fail in characteristic $p > 0$ , and what is the Raynaud counterexample?

Hint

Frobenius-pulled-back line bundles can have unusual properties; Raynaud (1978) found a smooth projective surface with an ample line bundle violating Kodaira.

Answer

Kodaira vanishing $H^{i} (X; ω_{X} \otimes L) = 0$ for $i > 0$ and $L$ ample fails in characteristic $p > 0$ :

Raynaud's counterexample (1978). There exists a smooth projective surface $X$ over a field of positive characteristic and an ample line bundle $L$ on $X$ such that $H^{1} (X; ω_{X} \otimes L) \neq = 0$ .

The construction: $X$ is a Tango surface — a specific Frobenius-twisted variety whose cohomology has nonzero torsion behaviour. The Frobenius $F : X \to X^{(p)}$ behaves badly with respect to ample-bundle cohomology in ways that break the harmonic-form argument.

Why characteristic 0 is special.

Hodge theory requires the Kähler condition, which is well-defined over $C$ . In characteristic $p$ , there's no analytic Hermitian-metric notion.
Frobenius pull-back in characteristic $p$ produces unusual phenomena: a line bundle can be ample but have Frobenius-twisted cohomology that contributes to $H^{i}$ in higher degrees.
The Bochner-Kodaira-Nakano identity relies on the Kähler $Δ_{d} = 2 Δ_{\overset{ˉ}{\partial}}$ identity, which has no characteristic- $p$ analogue.

Algebraic proof in characteristic 0 (Deligne-Illusie 1987). Surprisingly, an algebraic proof exists in characteristic 0 — but it works by reducing to characteristic $p$ and using Frobenius decomposition + lifting to $W_{2}$ . The proof requires the variety and line bundle to admit a flat lift to $W_{2} (k)$ where $k$ is the characteristic- $p$ residue field. This lifting condition is automatic in many cases, giving Kodaira vanishing for those.

Modern situation. In char 0: Kodaira-Akizuki-Nakano-Kawamata-Viehweg vanishing all hold. In char $p$ : vanishing holds for Frobenius-split varieties (Mehta-Ramanathan 1985) and for varieties of dimension $\leq p$ (Deligne-Illusie). Beyond these, Raynaud-type counterexamples can occur. $□$

Exercise 6 (hard, proof).

State and prove (or outline) the Akizuki-Nakano vanishing theorem.

Hint

For $L$ ample, $H^{q} (X; Ω^{p} \otimes L) = 0$ for $p + q > n$ .

Answer

Akizuki-Nakano vanishing theorem (1954). Let $X$ be a smooth projective complex variety of dimension $n$ and $L$ an ample line bundle. For all $0 \leq p \leq n$ and $q \geq 0$ :

H^{q} (X; Ω_{X}^{p} \otimes L) = 0 whenever p + q > n .

The case $p = n$ recovers Kodaira vanishing (since $Ω_{X}^{n} = ω_{X}$ and $p + q > n \Leftrightarrow q > 0$ ).

Proof outline. The argument generalises Kodaira's harmonic-form approach. For an $(p, q)$ -form $α$ with values in $L$ , the Bochner-Kodaira-Nakano identity in the more general form reads:

Δ_{\overset{ˉ}{\partial}, L} - Δ_{\partial, L} = [i Θ (L), Λ] .

For $\overset{ˉ}{\partial}$ -harmonic $(p, q)$ -form $α$ with values in $L$ :

0 - ⟨ Δ_{\partial} α, α ⟩ = ⟨[i Θ (L), Λ] α, α ⟩ .

The left side is $- ∥ \partial α ∥^{2} \leq 0$ . The right side, by curvature positivity of $L$ and the Lefschetz operator analysis, is non-negative on $(p, q)$ -forms — and strictly positive when $p + q > n$ .

The condition $p + q > n$ is essential: $[i Θ (L), Λ]$ acts on $(p, q)$ -forms by a quantity proportional to $p + q - n$ (from the Lefschetz $sl_{2}$ -algebra action). When $p + q > n$ , the action is positive; when $p + q = n$ , it vanishes; when $p + q < n$ , it can be negative.

For $p + q > n$ and $\overset{ˉ}{\partial}$ -harmonic $α$ in the form $Ω^{p} \otimes L$ : positivity forces $α = 0$ . Hence $H^{q} (X; Ω^{p} \otimes L) = 0$ for $p + q > n$ . $□$

The Akizuki-Nakano vanishing is more flexible than Kodaira: it gives vanishing in many bidegrees, not just $(n, q)$ . This extra flexibility is essential for the Lefschetz hyperplane theorem (above) and for Hodge-theoretic computations.

Exercise 7 (hard, conceptual).

State the Kawamata-Viehweg generalisation of Kodaira vanishing and explain its role in the minimal model program.

Hint

Kawamata-Viehweg: $L$ nef and big suffices.

Answer

Kawamata-Viehweg vanishing (Kawamata 1982; Viehweg 1982). Let $X$ be a smooth projective complex variety of dimension $n$ and $L$ a line bundle on $X$ that is nef ( $L \cdot C \geq 0$ for every curve $C \subset X$ ) and big (the asymptotic growth $h^{0} (X; L^{\otimes m}) \sim m^{n}$ ). Then $H^{i} (X; ω_{X} \otimes L) = 0$ for $i > 0$ .

Why it's important. Kodaira vanishing requires ample, a strict positivity condition. Kawamata-Viehweg replaces this with nef + big, much weaker:

Nef ( $L \cdot C \geq 0$ ): allows $L$ to "miss" some curves (numerical-zero intersection on some classes).
Big: ensures asymptotic growth of sections to $n$ -dimensional ( $dim X$ ).

Together, nef + big = pseudoeffective and ample on a Zariski open. The vanishing holds despite the failure of strict ampleness on a finite collection of curves.

Role in the minimal model program (Mori's program). The MMP seeks to find a minimal model in each birational class — a variety $X_{m i n}$ such that $ω_{X_{m i n}}$ is nef (intersected positively with every curve). Mori showed (1982) that one can run the MMP: contract extremal $K_{X}$ -negative curves repeatedly, eventually arriving at a minimal model.

Kawamata-Viehweg vanishing is essential at every step:

Existence of minimal models. Vanishing controls the cohomology of pulled-back line bundles after contractions.
Termination. Vanishing-based techniques (multiplier ideals, Kawamata-Viehweg) ensure the MMP terminates in the relevant cases.
Birkar-Cascini-Hacon-McKernan (2010). The proof of existence of minimal models for varieties of general type uses Kawamata-Viehweg as a key technical input — without it, the techniques fail.

Generalisations. Multiplier-ideal vanishing (Nadel 1990; Esnault-Viehweg 1992): for $L$ with possibly singular metric of positive curvature, the cohomology of $ω_{X} \otimes L \otimes J (ϕ)$ vanishes, where $J (ϕ)$ is the multiplier ideal sheaf of the metric's potential. This is a core technical tool of higher-dimensional birational geometry. $□$

Lean formalization [Intermediate+]

lean_status: partial — Mathlib has ample line bundles and basic positivity; the named Kodaira vanishing theorem is partial.

[object Promise]

Advanced results [Master]

Akizuki-Nakano-Kawamata-Viehweg. The full vanishing programme: Akizuki-Nakano refines the bidegree, Kawamata-Viehweg weakens the ampleness hypothesis. Together they cover essentially all positivity-based vanishing in characteristic 0.

Multiplier-ideal vanishing. Nadel vanishing (Nadel 1990): for a singular Hermitian metric $h$ on $L$ with semi-positive curvature, $H^{i} (X; ω_{X} \otimes L \otimes J (h)) = 0$ for $i > 0$ , where $J (h)$ is the multiplier ideal sheaf. Foundational for $L^{2}$ -methods in birational geometry.

Kollár's vanishing and injectivity. Kollár (1986, Annals of Math. 123): for a proper surjective $f : X \to Y$ between smooth varieties with connected fibres, $R^{i} f_{*} ω_{X} = 0$ for $i >$ relative dimension. The Kollár injectivity theorem: $H^{i} (X; ω_{X})$ injects into $H^{i}$ of a divisor pull-back. Foundational for the Kollár-Shokurov framework of birational geometry.

Esnault-Viehweg algebraic proof. Lectures on Vanishing Theorems (1992): Esnault-Viehweg gave a purely algebraic proof of Kawamata-Viehweg using Hodge-theoretic methods — Deligne-Illusie + cyclic covers + Frobenius. Works in characteristic 0 without Hodge analysis.

Hodge degeneration and vanishing. Kodaira vanishing follows from Akizuki-Nakano + Hodge symmetry via:

H^{q} (X; Ω^{p} \otimes L) = H_{\overset{ˉ}{\partial}, L}^{p, q} (X) = 0 for p + q > n .

The Hodge framework systematises vanishing as a consequence of Hodge type on cohomology.

Cohomology and positivity dictionary. A line bundle's positivity is detected by its cohomology:

$L$ very ample $\Rightarrow$ embedding into $P^{N}$ with $H^{0}$ separating points.
$L$ ample $\Rightarrow$ Kodaira-vanishing of higher cohomology of $ω \otimes L$ .
$L$ nef + big $\Rightarrow$ Kawamata-Viehweg vanishing.
$L$ nef $\Rightarrow$ partial vanishing (Kollár).
$L$ pseudoeffective $\Rightarrow$ multiplier-ideal vanishing.

This dictionary is the foundation of modern positivity theory (Lazarsfeld 2004, Positivity in Algebraic Geometry).

Synthesis. This construction generalises the pattern fixed in 04.03.01 (sheaf cohomology), with the symmetric data replaced by its skew or twisted analogue. Read in the opposite direction, the construction is dual to the metric story: complements and orthogonality are taken with respect to the bilinear datum of this unit, not a metric, and the resulting category of subobjects is the one the rest of the strand classifies. The central insight is that this datum identifies algebra with geometry: functions become vector fields, subspaces become quotients, and invariants become cohomology classes — and that identification is the engine driving every theorem downstream.

Full proof set [Master]

Kodaira vanishing is sketched in the formal-definition section. The Bochner-Kodaira-Nakano identity is proved in Voisin Vol I §6.3 or Demailly's Complex Analytic and Differential Geometry §VI. Akizuki-Nakano proof is parallel; Kawamata-Viehweg uses cyclic covers (Esnault-Viehweg LNM 1992); Deligne-Illusie's algebraic proof of Kodaira is in Invent. Math. 89 (1987).

Connections [Master]

Sheaf cohomology 04.03.01 — Kodaira vanishing is a fundamental theorem about sheaf cohomology vanishing.
Ample line bundle 04.05.05 — ampleness is the hypothesis driving the vanishing.
Hodge decomposition 04.09.01 — the proof uses harmonic-form representatives and the Hodge-theoretic framework.
Canonical sheaf 04.08.02 — the canonical sheaf $ω_{X}$ appears in the conclusion.
Serre duality 04.08.03 — gives the dual form $H^{j} (X; L^{- 1}) = 0$ for $j < n$ .
Riemann-Roch theorem for curves 04.04.01 — Kodaira vanishing on curves recovers $H^{1} (O (D)) = 0$ for $de g D > 2 g - 2$ .
Moduli of curves 04.10.01 — vanishing theorems on moduli spaces are essential for tautological-class computations.

Historical & philosophical context [Master]

Kunihiko Kodaira's 1953 On a differential-geometric method in the theory of analytic stacks (Proc. Nat. Acad. Sci. USA 39, 1268–1273) was a short paper introducing the vanishing theorem. The setting was complex Hermitian manifolds with positive Chern curvature; Kodaira used harmonic-form methods, building on Hodge's 1941 monograph and his own work with D. C. Spencer on deformation theory.

The methodology was strikingly transcendental: differential geometry, harmonic forms, the Bochner-Kodaira-Nakano identity. There was no obvious algebraic statement — yet the result had immediate algebraic consequences: dimension formulas, projective embedding theorems, classification of compact complex surfaces. Kodaira's embedding theorem (1954) — a compact Kähler manifold admits a holomorphic embedding into projective space iff there is a positive line bundle — followed shortly, using vanishing to construct global sections.

The Akizuki-Nakano generalisation to forms with $Ω^{p}$ (1954, Proc. Jap. Acad. 30) appeared almost immediately. The Kodaira-Spencer deformation framework (1958, three foundational papers) used vanishing to control infinitesimal deformations of complex structures.

The modern development split into two streams:

Algebraic. Hartshorne's Ample Subvarieties of Algebraic Varieties (1970), Lazarsfeld's Positivity (2004), and Esnault-Viehweg Lectures on Vanishing Theorems (1992) developed algebraic-geometric proofs and generalisations. Kawamata-Viehweg vanishing (1982) extended Kodaira to nef + big line bundles, becoming the workhorse of the minimal model program (Mori 1982; Reid; Kollár; Birkar-Cascini-Hacon-McKernan 2010).

Algebraic-arithmetic. Deligne-Illusie's 1987 paper (Invent. Math. 89, 247–270) gave an algebraic proof of Kodaira vanishing in characteristic 0 by reduction mod $p$ , leveraging their proof of Hodge degeneration. This opened p-adic Hodge theory and made vanishing accessible without analysis.

Counter-examples in positive characteristic. Raynaud (1978) constructed a smooth projective surface in characteristic $p > 0$ with an ample line bundle violating Kodaira vanishing. This showed the analytic / Kähler input was essential — characteristic- $p$ Kodaira fails. Subsequent work (Mehta-Ramanathan 1985 on Frobenius-split varieties, Hara 1998 on $F$ -singularity classes) characterised when vanishing holds in characteristic $p$ .

In current research, Kodaira-type vanishing is foundational for:

Minimal model program (Mori 1982; BCHM 2010): existence of minimal models requires vanishing at every step.
Birational geometry of higher-dimensional varieties.
L-functions and arithmetic geometry: vanishing of $H^{i}$ in mixed-characteristic $p$ -adic Hodge theory.
Complex Monge-Ampère equations and the Calabi conjecture (Yau 1978): positivity of curvature ⇔ existence of Kähler-Einstein metrics, with vanishing controlling integrability.
Mirror symmetry: vanishing of multiplier-ideal sheaves controls Gromov-Witten invariants.

Kodaira's 1953 paper was a short PNAS announcement. It has become one of the most-cited theorems in algebraic geometry, alongside Riemann-Roch and Serre duality. Kodaira was awarded the Fields Medal in 1954 — partly for this work, partly for his classification of complex surfaces.

Bibliography [Master]

Kodaira, K., On a differential-geometric method in the theory of analytic stacks, Proc. Nat. Acad. Sci. USA 39 (1953), 1268–1273.
Akizuki, Y. & Nakano, S., Note on Kodaira-Spencer's proof of Lefschetz theorems, Proc. Japan Acad. 30 (1954), 266–272.
Kawamata, Y., A generalization of Kodaira-Ramanujam's vanishing theorem, Math. Ann. 261 (1982), 43–46.
Viehweg, E., Vanishing theorems, J. reine angew. Math. 335 (1982), 1–8.
Raynaud, M., Contre-exemple au "vanishing theorem" en caractéristique $p > 0$ , in C. P. Ramanujam: A Tribute, Tata 1978, 273–278.
Voisin, C., Hodge Theory and Complex Algebraic Geometry I, Cambridge 2002.
Lazarsfeld, R., Positivity in Algebraic Geometry I, II, Springer Ergebnisse 48–49, 2004.
Esnault, H. & Viehweg, E., Lectures on Vanishing Theorems, DMV Seminar 20, Birkhäuser 1992.
Demailly, J.-P., Complex Analytic and Differential Geometry, online manuscript 1997-present.
Deligne, P. & Illusie, L., Relèvements modulo $p^{2}$ et décomposition du complexe de de Rham, Invent. Math. 89 (1987), 247–270.
Birkar, C., Cascini, P., Hacon, C., McKernan, J., Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), 405–468.