Vanishing theorems for projective
morphisms between complex analyticspaces
Osamu Fujino

Kyoto University
May 3, 2022

Osamu Fujino

Vanishing theorems for projective morphisms

Background and Motivation
I would like to discuss the minimal model program for
projective morphisms between complex analytic spaces.
I will use it for the study of complex analytic singularities,
degenerations of projective varieties, and so on.
The Kawamata–Viehweg vanishing theorem can beformulated and proved for projective morphisms of complex
analytic spaces.
We can use L2 -method for kawamata log terminal pairs.
In order to treat complex analytic log canonical pairs,
quasi-log structures on complex analytic spaces, and so on,
we have to establish some vanishing theorems.

Osamu Fujino

Vanishing theorems for projective morphisms

Kollár’s theorem
Theorem 1 (Kollár’s torsion-freeness and vanishing theorem)
Let f : X → Y be a surjective morphism of complex projective
varieties such that X is smooth.
(i) (Torsion-freeness). Rq f∗ ωX is torsion-free for every q.
(ii) (Vanishing theorem). Let A be an ample line bundle on Y .
Then H p (Y, A ⊗ Rq f∗ ωX ) = 0 holds for every p > 0 and q.
Kollár’s vanishing theorem is a generalization of the Kodairavanishing theorem for projective varieties.
Theorem 1 is known to be equivalent to Kollár’s injectivity
theorem.
Theorem 1 holds even when X is a compact Kähler manifold.
We can prove Theorem 1 by the E1 -degeneration ofHodge-to-de Rham spectral sequence.
Osamu Fujino

Vanishing theorems for projective morphisms

Takegoshi’s theorem
Theorem 2 (Takegoshi’s theorem)
Let f : X → Y be a proper surjective morphism from a connectedKähler manifold X to a complex analytic space Y .
(i) (Torsion-freeness). Then Rq f∗ ωX is torsion-free for every q.
(ii) (Vanishing theorem). Let π : Y → Z be a projective morphismof complex analytic spaces and let A be a π-ample line
bundle on Y . Then R p π∗ (A ⊗ Rq f∗ ωX ) = 0 holds for everyp > 0 and every q.
This is a complex analytic generalization of Kollár’s
torsion-freeness and vanishing theorem.
In Theorem 2, X and Y are not necessarily compact.
Takegoshi’s result is much more general than Theorem 2.
Theorem 2 follows from the theory of harmonic forms.
Osamu Fujino

Vanishing theorems for projective morphisms

Simple normal crossing pairs
Definition 3 (Globally embedded simple normal crossing pairs)
Let X be a simple normal crossing divisor on a complex manifold
M and let B be an R-divisor on M such that Supp(B + X) is asimple normal crossing divisor on M and that B and X have nocommon components. Then we put D := B|X and call (X, D) an
analytic globally embedded simple normal crossing pair.
Definition 4 (Simple normal crossing pairs)
If the pair (X, D) is locally isomorphic to a globally embedded
simple normal crossing pair at any point and the irreducible
components of X and D are all smooth, then (X, D) is called ananalytic simple normal crossing pair.

Osamu Fujino

Vanishing theorems for projective morphisms

Decomposition of (X, D)
Definition 5
Let (X, D) be an analytic simple normal crossing pair such that D isreduced. For any positive integer k, we put

X [k] := {x ∈ X| mult x X ≥ k}∼ ,where Z ∼ denotes the normalization of Z . Then X [k] is the disjoint
union of the intersections of k irreducible components of X , and issmooth. We have a reduced simple normal crossing divisor
D[k] ⊂ X [k] defined by the pull-back of D by the natural morphism
X [k] → X . For any l ∈ Z≥0 , we put{}∼D[k,l] := x ∈ X [k] mult x D[k] ≥ l .

We note D[k,0] = X [k] and dim D[k,l] = dim X + 1 − k − l.
Osamu Fujino

Vanishing theorems for projective morphisms

Spectral sequence
Theorem 6 (Fujino–Fujisawa–Saito, 2014)
Let (X, D) be an analytic simple normal crossing pair such that D isreduced and let f : X → Y be a proper morphism to a complex
manifold Y . Assume that f is Kähler on each irreducible
component of X . Then there is the weight spectral sequence−q,i+q

F E1=⊕Ri f∗ ωD[k,l] /Y ⇒ Ri f∗ ωX/Y (D),k+l=dim X+q+1

degenerating at E2 , and its E1 -differential d1 splits so that the−q,i+q−q,i+q
are direct factors of F E1.
F E2
Theorem 6 follows from Saito’s theory of mixed Hodgemodules.
The proof of Theorem 6 becomes simpler when f isprojective.
Osamu Fujino

Vanishing theorems for projective morphisms

Strata of (X, D)
Definition 7 (Strata)
Let (X, D) be an analytic simple normal crossing pair. Let
ν : X ν → X be the normalization. We put

KX ν + Θ = ν∗ (KX + D),
that is, Θ is the union of ν∗−1 D and the inverse image of the singularlocus of X . If W is an irreducible component of X or the ν-image ofsome log canonical center of (X ν , Θ), then W is called a stratum of(X, D).
When D is reduced, W is a stratum of (X, D) if and only if W is theimage of an irreducible component of D[k,l] for some k > 0 andl ≥ 0.

Osamu Fujino

Vanishing theorems for projective morphisms

Standard setting
Theorem 8 (Fujino, 2022)
Let (X, D) be an analytic simple normal crossing pair such that D isreduced and let f : X → Y be a proper morphism of complex
analytic spaces. Assume that f is Kähler on each irreducible
component of X . Then(i) (Strict support condition). Every associated subvariety ofRq f∗ ωX (D) is the f -image of some stratum of (X, D) for everyq.
(ii) (Vanishing theorem). Let π : Y → Z be a projective morphismof complex analytic spaces and let A be a π-ample line
bundle on Y . Then()R p π∗ A ⊗ Rq f∗ ωX (D) = 0holds for every p > 0 and every q.
Osamu Fujino

Vanishing theorems for projective morphisms

Theorem 8 (i) is almost obvious by Theorem 6.
Theorem 8 (i) is a consequence of the strict support conditionof polarizable pure Hodge modules.
Theorem 8 (ii) follows from Takegoshi’s vanishing theorem
with the aid of Theorem 6.

Osamu Fujino

Vanishing theorems for projective morphisms

Injectivity theorem
Theorem 9 (Injectivity theorem, Fujino, 2022)
Let (X, D) and f : X → Y be as in Theorem 8. Let L be anf -semiample line bundle on X . Let s be a nonzero element ofH 0 (X, L⊗k ) for some k ∈ Z≥0 such that the zero locus of s does notcontain any strata of (X, D). Then, for every q, the map()()×s : Rq f∗ ωX (D) ⊗ L⊗l → Rq f∗ ωX (D) ⊗ L⊗k+l
induced by ⊗s is injective for every l ∈ Z>0 .
Theorem 9 is a generalization of Kollár’s injectivity theorem.
Theorem 9 can be proved by Theorem 8 without difficulties.
For various geometric applications, Theorem 8 seems to be
more useful than Theorem 9.
Osamu Fujino

Vanishing theorems for projective morphisms

Main theorem
Theorem 10 (Main theorem, Fujino 2022)
Let (X, ∆) be an analytic simple normal crossing pair such that ∆ isa boundary R-divisor on X . Let f : X → Y be a projective
morphism to a complex analytic space Y and let L be a line bundle
on X . Let q be an arbitrary nonnegative integer. Then(i) (Strict support condition). If L − (ωX + ∆) is f -semiample, thenevery associated subvariety of Rq f∗ L is the f -image of somestratum of (X, ∆).
(ii) (Vanishing theorem). If L − (ωX + ∆) ∼R f ∗ H holds for some
π-ample R-line bundle H on Y , where π : Y → Z is a projective
morphism to a complex analytic space Z , then we haveR p π∗ Rq f∗ L = 0for every p > 0.
Osamu Fujino

Vanishing theorems for projective morphisms

When f : X → Y and π : Y → Z are algebraic, Theorem 10 iswell known and has already played a crucial role for the studyof log canonical pairs and quasi-log schemes.
Theorem 10 will play a crucial role for the study of analytic log
canonical pairs and quasi-log structures for complex analyticspaces.

Osamu Fujino

Vanishing theorems for projective morphisms

Reid–Fukuda type
Theorem 11 (Vanishing theorem of Reid–Fukuda type)
Let (X, ∆) be an analytic simple normal crossing pair such that ∆ isa boundary R-divisor on X . Let f : X → Y and π : Y → Z be
projective morphisms of complex analytic spaces and let L be aline bundle on X . If L − (ωX + ∆) ∼R f ∗ H holds such that H is anR-line bundle, which is nef and log big over Z with respect tof : (X, ∆) → Y , on Y , thenR p π∗ Rq f∗ L = 0
holds for every p > 0 and every q.

Osamu Fujino

Vanishing theorems for projective morphisms

Thank you very much!

Osamu Fujino

Vanishing theorems for projective morphisms