On the structure of compact K¨ahler manifolds with nef anticanonical bundles

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On the structure of compact K¨ ahler manifolds with nef anticanonical bundles

Jean-Pierre Demailly

Institut Fourier, Universit´e Grenoble Alpes, France

& Acad´emie des Sciences de Paris

Complex Analysis and Geometry – XXIII, June 13, 2017

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Goals / main positivity concepts

Analyze the structure of projective or compact K¨ahler manifolds X with −K_X nef.

As is well known since the beginning of the XX^th century at least, the geometry depends on the sign of the curvature of the canonical line bundle

K_X = ΛⁿT_X^∗, n= dim_CX.

L→X is pseudoeffective(psef) if ∃h=e^−ϕ, ϕ∈L¹_loc, s.t. Θ_L,h=−dd^clogh≥0 on X in the sense of currents

⇔(for X projective) c₁(L)∈Eff.

L→X is semi-positiveif ∃h=e^−ϕ smooth (C^∞) such that ΘL,h=−dd^clogh≥0 on X.

⇐(for X projective) L^⊗m =G⊗H, G semi-ample,H ∈Pic⁰(X). L is nef if ∀ε >0,∃h_ε =e^−ϕ^ε smooth such that

Θ_L,h_ε =−dd^clogh_ε≥ −εω onX

⇔(for X projective) L·C ≥0, ∀C algebraic curve.

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Goals / main positivity concepts

L→X is pseudoeffective(psef) if ∃h=e^−ϕ, ϕ∈L¹_loc, s.t. Θ_L,h=−dd^clogh≥0 on X in the sense of currents

⇔(for X projective) c₁(L)∈Eff.

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Goals / main positivity concepts

L→X is pseudoeffective(psef) if ∃h=e^−ϕ, ϕ∈L¹_loc, s.t.

Θ_L,h=−dd^clogh≥0 on X in the sense of currents

⇔(for X projective) c1(L)∈Eff.

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Goals / main positivity concepts

⇐(for X projective) L^⊗m =G⊗H, G semi-ample,H ∈Pic⁰(X).

L is nef if ∀ε >0,∃h_ε =e^−ϕ^ε smooth such that Θ_L,h_ε =−dd^clogh_ε≥ −εω onX

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Goals / main positivity concepts

⇐(for X projective) L^⊗m =G⊗H, G semi-ample,H ∈Pic⁰(X).

L is nef if ∀ε >0,∃h =e^−ϕ^ε smooth such that

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Complex curves (n = 1) : genus and curvature

K_X = ΛⁿT_X^∗, deg(K_X) = 2g −2

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Comparison of positivity concepts

Recall that for a line bundle

positive ⇔ ample⇒semi-ample⇒semi-positive⇒nef ⇒psef but none of the reverse implications in red holds true.

Example

LetX be the rational surface obtained by blowing up P² in 9 distinct points{p_i} on a smooth (cubic) elliptic curve C ⊂P², µ:X →P² and ˆC the strict transform of C. Then

K_X =µ^∗K_P²⊗ O(P

E_i)⇒ −K_X =µ^∗O_P²(3)⊗ O(−P E_i), thus

−K_X =µ^∗O_P²(C)⊗ O(−P

E_i) =O_X( ˆC). One has

−K_X ·Γ = ˆC ·Γ ≥0 if Γ6= ˆC,

−KX ·Cˆ= (−KX)² = ( ˆC)² =C²−9 = 0 ⇒ −KX nef.

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Comparison of positivity concepts

Example

LetX be the rational surface obtained by blowing up P² in 9 distinct points{p_i} on a smooth (cubic) elliptic curve C ⊂P², µ:X →P² and ˆC the strict transform of C.

Then K_X =µ^∗K_P²⊗ O(P

−K_X =µ^∗O_P²(C)⊗ O(−P

E_i) =O_X( ˆC). One has

−K_X ·Γ = ˆC ·Γ ≥0 if Γ6= ˆC,

−KX ·Cˆ= (−KX)² = ( ˆC)² =C²−9 = 0 ⇒ −KX nef.

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Comparison of positivity concepts

Example

−K_X =µ^∗O_P²(C)⊗ O(−P

E_i) =O_X( ˆC).

One has

−K_X ·Γ = ˆC ·Γ ≥0 if Γ6= ˆC,

−KX ·Cˆ= (−KX)² = ( ˆC)² =C²−9 = 0 ⇒ −KX nef.

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Comparison of positivity concepts

Example

−K_X =µ^∗O_P²(C)⊗ O(−P

E_i) =O_X( ˆC).

One has

−K_X ·Γ = ˆC ·Γ ≥0 if Γ6= ˆC,

−K_X ·Cˆ= (−K_X)² = ( ˆC)² =C²−9 = 0 ⇒ −K_X nef.

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Rationally connected manifolds

In fact

G := (−K_X)_|_C_ˆ ' O_P²_|C(3)⊗ O_C(−P

p_i)∈Pic⁰(C)

If G is a torsion point in Pic⁰(C), then one can show that

−K_X is semi-ample, but otherwise it is not semi-ample. Brunella has shown that−K_X is C^∞ semi-positive if c₁(G) satisfies a diophantine condition found by T. Ueda, but that otherwise it may not be semi-positive (although nef).

P²# 9 points is an example of rationally connected manifold: Definition

Recall that a compact complex manifold is said to be

rationally connected(or RC for short) if any 2 points can be joined by a chain of rational curves

Remark. X =P² blown-up in≥10 points is RC but −K_X not nef.

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Rationally connected manifolds

In fact

G := (−K_X)_|_C_ˆ ' O_P²_|C(3)⊗ O_C(−P

p_i)∈Pic⁰(C) If G is a torsion point in Pic⁰(C), then one can show that

−K_X is semi-ample, but otherwise it is not semi-ample.

Brunella has shown that−K_X is C^∞ semi-positive if c₁(G) satisfies a diophantine condition found by T. Ueda, but that otherwise it may not be semi-positive (although nef).

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Rationally connected manifolds

In fact

G := (−K_X)_|_C_ˆ ' O_P²_|C(3)⊗ O_C(−P

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Rationally connected manifolds

In fact

G := (−K_X)_|_C_ˆ ' O_P²_|C(3)⊗ O_C(−P

P²# 9 points is an example of rationally connected manifold:

Definition

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Rationally connected manifolds

In fact

G := (−K_X)_|_C_ˆ ' O_P²_|C(3)⊗ O_C(−P

Definition

rationally connected(or RC for short) if any 2 points can be

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Rationally connected manifolds

In fact

G := (−K_X)_|_C_ˆ ' O_P²_|C(3)⊗ O_C(−P

Definition

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Ex. of compact K¨ ahler manifolds with −K

_X

≥ 0

(Recall: By Yau,−K_X ≥0⇔ ∃ω K¨ahler with Ricci(ω)≥0.)

Ricci flat manifolds

–Complex tori T =C^q/Λ

–Holomorphic symplectic manifolds S (also called hyperk¨ahler):

∃σ∈H⁰(S,Ω²_S) symplectic

–Calabi-Yau manifolds Y: π₁(Y) finite and some multiple of K_Y is trivial (may assume π₁(Y) = 1 andK_Y trivial by passing to some finite ´etale cover)

the rather large class of rationally connected manifolds Z with −K_Z ≥0

all products T ×Q

Sj ×Q

Yk ×Q Z`.

Main result. Essentially, this is a complete list !

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Ex. of compact K¨ ahler manifolds with −K

_X

≥ 0

(Recall: By Yau,−K_X ≥0⇔ ∃ω K¨ahler with Ricci(ω)≥0.) Ricci flat manifolds

all products T ×Q

Sj ×Q

Yk ×Q Z`.

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Ex. of compact K¨ ahler manifolds with −K

_X

≥ 0

all products T ×Q

Sj ×Q

Yk ×Q Z`.

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Ex. of compact K¨ ahler manifolds with −K

_X

≥ 0

–Calabi-Yau manifolds Y: π₁(Y) finite and some multiple of K_Y is trivial (may assume π₁(Y) = 1 andK_Y trivial by passing to some finite ´etale cover

)

all products T ×Q

Sj ×Q

Yk ×Q Z`.

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Ex. of compact K¨ ahler manifolds with −K

_X

≥ 0

the rather large class of rationally connected manifolds Z with −KZ ≥0

all products T ×Q

Sj ×Q

Yk ×Q Z`.

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Ex. of compact K¨ ahler manifolds with −K

_X

≥ 0

the rather large class of rationally connected manifolds Z with −KZ ≥0

all products T ×Q

Sj ×Q

Yk ×Q Z`.

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Structure theorem for manifolds with −K

_X

≥ 0

Theorem [Campana, D., Peternell, 2012]

LetX be a compact K¨ahler manifold with −K_X ≥0. Then:

(a) ∃holomorphic and isometric splitting in irreducible factors Xe = universal cover of X 'C^q×Q

Y_j ×Q

S_k ×Q Z_` whereYj = Calabi-Yau (holonomy SU(nj)),

S_k = holomorphic symplectic (holonomy Sp(n⁰_k/2)), and Z_` = RC with −K_Z_` ≥0 (holonomy U(n⁰⁰_`)).

(b) There exists a finite ´etale Galois cover Xb →X such that the Albanese mapα:Xb →Alb(Xb) is an (isometrically) locally trivial holomorphic fiber bundle whose fibers are productsQ

Y_j ×Q

S_k×Q

Z_`, as described in (a). (c) π₁(Xb)'Z^2qoΓ, Γ finite (“almost abelian” group).

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Structure theorem for manifolds with −K

_X

≥ 0

Y_j ×Q

S_k ×Q

Z_`, as described in (a).

(c) π₁(Xb)'Z^2qoΓ, Γ finite (“almost abelian” group).

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Structure theorem for manifolds with −K

_X

≥ 0

Y_j ×Q

Y ×Q

S ×Q

Z , as described in (a).

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Criterion for rational connectedness

Criterion

LetX be a projective algebraic n-dimensional manifold. The following properties are equivalent.

(a) X is rationally connected.

(b) For every invertible subsheaf F ⊂Ω^p_X :=O(Λ^pT_X^∗), 1≤p ≤n, F isnot psef.

(c) For every invertible subsheaf F ⊂ O((T_X^∗)^⊗p), p≥1, F is not psef.

(d) For some (resp. for any) ample line bundle Aon X, there exists a constant CA >0 such that

H⁰(X,(T_X^∗)^⊗m⊗A^⊗k) = 0 ∀m, k ∈N^∗ with m≥C_Ak.

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Criterion for rational connectedness

Criterion

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Criterion for rational connectedness

Criterion

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Criterion for rational connectedness

Criterion

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Proof of the RC criterion

Proof(essentially from Peternell 2006)

(a)⇒ (d) is easy (RC implies there are many rational curves on whichT_X, soT_X^∗ <0), (d)⇒(c) and (c) ⇒(b) are trivial.

Thus the only thing left to complete the proof is(b) ⇒ (a). Consider theMRC quotient π :X →Y, given by the

“equivalence relation x ∼y if x and y can be joined by a chain of rational curves.

Then (by definition) the fibers are RC, maximal, and a result of Graber-Harris-Starr (2002) implies that Y is not uniruled. By BDPP (2004),Y not uniruled⇒K_Y psef. Then

π^∗K_Y ,→Ω^p_X wherep= dimY, which is a contradiction unless p= 0, and therefore X is RC.

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Proof of the RC criterion

Thus the only thing left to complete the proof is(b) ⇒ (a).

Consider theMRC quotient π :X →Y, given by the

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Proof of the RC criterion

Consider theMRC quotient π:X →Y, given by the

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Proof of the RC criterion

Then (by definition) the fibers are RC, maximal, and a result of Graber-Harris-Starr (2002) implies that Y is not uniruled.

By BDPP (2004),Y not uniruled⇒K_Y psef. Then

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Proof of the RC criterion

Then (by definition) the fibers are RC, maximal, and a result of Graber-Harris-Starr (2002) implies that Y is not uniruled.

By BDPP (2004),Y not uniruled⇒K_Y psef. Then

π^∗K_Y ,→Ω^p_X where p= dimY, which is a contradiction unless p= 0, and therefore X is RC.

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Generalized holonomy principle

Let (E,h)→X be a hermitian holomorphic vector bundle of rankr overX compact/C. Assume that

ΘE,h∧ _(n−1)!^ωⁿ⁻¹ =B ^ω_n!ⁿ, B ∈Herm(E,E), B ≥0 on X.

LetH the restricted holonomy group of (E,h). Then (a) If there exists a psef invertible sheafL ⊂ O((E^∗)^⊗m),

then L is flatand invariant under parallel transport by the connection of (E^∗)^⊗m induced by the Chern connection∇ of (E,h) ; moreover, H acts trivially on L.

(b) If H satisfies H =U(r), then none of the invertible sheavesL ⊂ O((E^∗)^⊗m) can be psef for m ≥1. Proof. L ⊂ O((E^∗)^⊗m) which has trace of curvature≤0 while ΘL ≥0, use Bochner formula.

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Generalized holonomy principle

ΘE,h∧ _(n−1)!^ωⁿ⁻¹ =B ^ω_n!ⁿ, B ∈Herm(E,E), B ≥0 on X. LetH the restricted holonomy group of (E,h). Then

(a) If there exists a psef invertible sheafL ⊂ O((E^∗)^⊗m), thenL is flatand invariant under parallel transport by the connection of (E^∗)^⊗m induced by the Chern connection∇ of (E,h) ; moreover, H acts trivially on L.

(b) If H satisfies H =U(r), then none of the invertible sheavesL ⊂ O((E^∗)^⊗m) can be psef for m ≥1. Proof. L ⊂ O((E^∗)^⊗m) which has trace of curvature≤0 while ΘL ≥0, use Bochner formula.

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Generalized holonomy principle

(b) If H satisfies H =U(r), then none of the invertible sheavesL ⊂ O((E^∗)^⊗m) can be psef for m ≥1.

Proof. L ⊂ O((E^∗)^⊗m) which has trace of curvature≤0 while ΘL ≥0, use Bochner formula.

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Generalized holonomy principle

(b) If H satisfies H =U(r), then none of the invertible sheavesL ⊂ O((E^∗)^⊗m) can be psef for m ≥1.

Proof. L ⊂ O((E^∗)^⊗m) which has trace of curvature≤0 while ΘL ≥0, use Bochner formula.

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Surjectivity of the Albanese morphism

Recall that ifX is a compact K¨ahler manifold, the Albanese map α_X :X →Alb(X) :=C^q/Λ

is the holomorphic map given by z 7→αX(z) =

Z z

z0

uj

1≤j≤q mod period subgroup Λ⊂C^q, where (u₁, . . . ,u_q) is a basis ofH⁰(X,Ω¹_X).

Theorem [Qi Zhang, 2005]

If X is projective and −K_X is nef, thenα_X is surjective. Proof. Based on characteristic p techniques.

Theorem [M. P˘aun, 2012]

If X is compact K¨ahler and −K_X is nef, then α_X is surjective. Proof. Based on variation arguments for twisted

K¨ahler-Einstein metrics.

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Surjectivity of the Albanese morphism

Z z

z0

uj

If X is projective and −K_X is nef, thenα_X is surjective.

Proof. Based on characteristic p techniques. Theorem [M. P˘aun, 2012]

If X is compact K¨ahler and −K_X is nef, then α_X is surjective. Proof. Based on variation arguments for twisted

K¨ahler-Einstein metrics.

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Surjectivity of the Albanese morphism

Z z

z0

uj

Proof. Based on characteristic p techniques.

If X is compact K¨ahler and−K is nef, then α is surjective.

Proof. Based on variation arguments for twisted K¨ahler-Einstein metrics.

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Surjectivity of the Albanese morphism

Z z

z0

uj

Proof. Based on characteristic p techniques.

If X is compact K¨ahler and−K_X is nef, then α_X is surjective.

Proof. Based on variation arguments for twisted K¨ahler-Einstein metrics.

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Approach via generically nef vector bundles (J.Cao)

Definition

LetX compact K¨ahler manifold,E →X torsion free sheaf.

(a) E isgenerically nef with respect to a K¨ahler classω if µ_ω(S) =ω-slope ofS :=

R

X c₁(S)∧ωⁿ⁻¹ rankS ≥0 for all torsion free quotients E → S →0.

If E isω-generically nef for all ω, we simply say thatE is generically nef.

(b) Let

0 = E₀ ⊂ E₁ ⊂. . .⊂ E_s =E

be a filtration ofE by torsion free coherent subsheaves such that the quotientsEi+1/Ei areω-stable subsheaves of E/E_i of maximal rank. We call such a sequence a refined Harder-Narasimhan (HN) filtration w.r.t. ω.

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Approach via generically nef vector bundles (J.Cao)

Definition

LetX compact K¨ahler manifold,E →X torsion free sheaf.

(a) E isgenerically nef with respect to a K¨ahler classω if µ_ω(S) =ω-slope ofS :=

R

X c₁(S)∧ωⁿ⁻¹ rankS ≥0 for all torsion free quotients E → S →0.

If E isω-generically nef for all ω, we simply say thatE is generically nef.

(b) Let

0 = E0 ⊂ E1 ⊂. . .⊂ Es =E

be a filtration ofE by torsion free coherent subsheaves such that the quotientsEi+1/Ei areω-stable subsheaves of E/E_i of maximal rank. We call such a sequence a refined Harder-Narasimhan (HN) filtration w.r.t. ω.

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Characterization of generically nef vector bundles

It is a standard fact that refined HN-filtrations always exist, moreover

µ_ω(E_i/Ei−1)≥ν_ω(E_i+1/E_i) for all i.

Proposition

Let (X, ω) be a compact K¨ahler manifold and E a torsion free sehaf onX. Then E isω-generically nef if and only if

µω(Ei+1/Ei)≥0 for some refined HN-filtration.

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A result of J. Cao about manifolds with −K

_X

nef

Theorem

(Junyan Cao, 2013) Let X be a compact K¨ahler manifold with −K_X nef. Then the tangent bundle T_X is ω-generically nef for all K¨ahler classesω.

Proof. use the fact that ∀ε >0,∃ K¨ahler metric with Ricci(ω_ε)≥ −ε ω_ε (Yau, DPS 1995).

From this, one can deduce Theorem

LetX be a compact K¨ahler manifold with nef anticanonical bundle. Then the bundlesT_X^⊗m areω-generically nef for all K¨ahler classesω and all positive integers m. In particular, the bundlesS^kT_X and Vp

T_X areω-generically nef.

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A result of J. Cao about manifolds with −K

_X

nef

Theorem

(Junyan Cao, 2013) Let X be a compact K¨ahler manifold with −K_X nef. Then the tangent bundle T_X is ω-generically nef for all K¨ahler classesω.

Proof. use the fact that ∀ε >0,∃ K¨ahler metric with Ricci(ω_ε)≥ −ε ω_ε (Yau, DPS 1995).

From this, one can deduce Theorem

LetX be a compact K¨ahler manifold with nef anticanonical bundle. Then the bundlesT_X^⊗m areω-generically nef for all

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A lemma on sections of contravariant tensors

Lemma

Let (X, ω) be a compact K¨ahler manifold with −K_X nef and η∈H⁰(X,(Ω¹_X)^⊗m⊗ L)

whereL is a numerically trivial line bundle on X.

Then the filtered parts ofη w.r.t. the refined HN filtration are parallel w.r.t. the Bando-Siu metric in the 0 slope parts, and the<0 slope parts vanish.

Proof. By Cao’s theorem there exists a refined HN-filtration 0 =E₀ ⊂ E₁ ⊂. . .⊂ E_s =T_X^⊗m

with ω-stable quotients E_i+1/E_i such that µ_ω(E_i+1/E_i)≥0 for alli. Then we use the fact that any section in a

(semi-)negative slope reflexive sheaf E_i+1/E_i ⊗ L is parallel w.r.t. its Bando-Siu metric (resp. vanishes).

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A lemma on sections of contravariant tensors

Lemma

with ω-stable quotients E_i+1/E_i such that µ_ω(E_i+1/E_i)≥0 for alli. Then we use the fact that any section in a

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A lemma on sections of contravariant tensors

Lemma

with ω-stable quotients E_i+1/E_i such that µ_ω(E_i₊₁/E_i)≥0 for alli. Then we use the fact that any section in a

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Smoothness of the Albanese morphism (after Cao)

Theorem (Junyan Cao 2013)

Non-zero holomorphicp-forms on a compact K¨ahler manifold X with −K_X nef vanish only on the singular locus of the refined HN filtration ofT^∗X.

This already implies the following result. Corollary

LetX be a compact K¨ahler manifold with nef anticanonical bundle. Then the Albanese mapα_X :X →Alb(X) is

a submersion on the complement of the HN filtration singular locus in X [⇒α_X surjects ontoAlb(X) ].

Proof. The differentialdα_X is given by (du₁, . . . ,du_q) where (u₁, . . . ,u_q) is a basis of 1-forms, q = dimH⁰(X,Ω¹_X). Cao’s thm ⇒rank of (du₁, . . . ,du_q) is =q generically.

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Smoothness of the Albanese morphism (after Cao)

This already implies the following result.

Corollary

Proof. The differentialdα_X is given by (du₁, . . . ,du_q) where (u₁, . . . ,u_q) is a basis of 1-forms, q = dimH⁰(X,Ω¹_X). Cao’s thm ⇒rank of (du₁, . . . ,du_q) is =q generically.

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Smoothness of the Albanese morphism (after Cao)

Corollary

Proof. The differentialdα_X is given by (du₁, . . . ,du_q) where

Cao’s thm ⇒rank of (du₁, . . . ,du_q) is =q generically.

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Smoothness of the Albanese morphism (after Cao)

Corollary

Proof. The differentialdα_X is given by (du₁, . . . ,du_q) where (u₁, . . . ,u_q) is a basis of 1-forms, q= dimH⁰(X,Ω¹_X).

Cao’s thm⇒ rank of (du₁, . . . ,du_q) is =q generically.

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Isotriviality of the Albanese map

Theorem [J. Cao, arXiv:1612.05921]

LetX be a projective manifold with nef anti-canonical bundle.

Then the Albanese map α_X :X →Y =Alb(X) is locally isotrivial, i.e., for any small open set U ⊂Y, α⁻¹_X (U) is biholomorphic to the productU ×F, where F is the generic fiber ofαX.

Moreover−K_F is again nef.

Proof. LetA be a (large) ample line bundle on X and E = (α_X)∗Aits direct image. Then E = (α_X)∗(mK_X_/Y +L) with L=A−mK_X_/Y =A−mK_X nef. By results of

Berndtsson-P˘aun on direct images, one can show thatdetE is pseudoeffective.

Using arguments of [DPS95], one can infer that

E⁰ =E ⊗(detE)^−1/r, r = rank(E), is numerically flat, hence a locally constant coefficient system(Simpson, Deng Ya). However, ifA0,E provides equations of the fibers.

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Isotriviality of the Albanese map

Then the Albanese map α_X :X →Y =Alb(X) is locally isotrivial, i.e., for any small open set U ⊂Y, α⁻¹_X (U) is biholomorphic to the productU ×F, where F is the generic fiber ofαX. Moreover−K_F is again nef.

Proof. LetA be a (large) ample line bundle on X and E = (α_X)∗Aits direct image. Then E = (α_X)∗(mK_X_/Y +L) with L=A−mK_X_/Y =A−mK_X nef. By results of

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Isotriviality of the Albanese map

Proof. LetA be a (large) ample line bundle on X and E = (α_X)∗A its direct image. Then E = (α_X)∗(mK_X_/Y +L) with L=A−mK_X_/Y =A−mK_X nef. By results of

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Isotriviality of the Albanese map

Proof. LetA be a (large) ample line bundle on X and E = (α_X)∗A its direct image. Then E = (α_X)∗(mK_X_/Y +L) with L=A−mK_X_/Y =A−mK_X nef. By results of

E⁰ =E ⊗(detE)^−1/r, r = rank(E), is numerically flat, hence a locally constant coefficient system(Simpson, Deng Ya).

However, ifA0,E provides equations of the fibers.

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The simply connected case

The above results reduce the study of projective manifolds with −K_X nef to the case when π₁(X) = 0.

Theorem [Junyan Cao, Andreas H¨oring, 2 days ago!] LetX be a projective manifold such that −K_X is nef and π₁(X) = 0. Then X =W ×Z with K_W ∼0 and Z is a rationally connected manifold.

Corollary [Junyan Cao, Andreas H¨oring]

LetX be a projective manifold such that −K_X is nef. Then after replacingX with a finite ´etale cover, the Albanese map α_X is isotrivial and its fibers are of the form

QS_j ×Q

Y_k ×Q

Z_` with S_j holomorphic symplectic, Y_k Calabi-Yau and Z_` rationally connected.

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The simply connected case

Theorem [Junyan Cao, Andreas H¨oring, 2 days ago!]

LetX be a projective manifold such that −K_X is nef and π₁(X) = 0. Then X =W ×Z with K_W ∼0 andZ is a rationally connected manifold.

QS_j ×Q

Y_k ×Q

Z_` with S_j holomorphic symplectic, Y_k Calabi-Yau and Z_` rationally connected.

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The simply connected case

Theorem [Junyan Cao, Andreas H¨oring, 2 days ago!]

LetX be a projective manifold such that −K_X is nef and π₁(X) = 0. Then X =W ×Z with K_W ∼0 andZ is a rationally connected manifold.

QS ×Q

Y ×Q

Z with S holomorphic symplectic,

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Further problems (I)

Partly solved questions

Develop further the theory of singular Calabi-Yau and singular holomorphic symplectic manifolds (work of Greb-Kebekus-Peternell).

Show that the “slope±ε” part corresponds to blown-up tori, singular Calabi-Yau and singular holomorphic symplectic manifolds (as fibers and targets).

The rest of T_X (slope<0) should yield a general type orbifold quotient (according to conjectures of Campana). Expected more general definition

A compact K¨ahler manifold X is a singular Calabi-Yau ifX has a non singular modelX⁰ satisfyingπ₁(X⁰) = 0 and K_X⁰ =E for an effective divisorE of numerical dimension 0, and H⁰(X⁰,Ω^p_X0) = 0 for 0<p <dimX.

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Further problems (I)

The rest of T_X (slope<0) should yield a general type orbifold quotient (according to conjectures of Campana). Expected more general definition

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Further problems (I)

The rest of T_X (slope<0) should yield a general type orbifold quotient (according to conjectures of Campana).

Expected more general definition

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Further problems (I)

The rest of T_X (slope<0) should yield a general type orbifold quotient (according to conjectures of Campana).

Expected more general definition

A compact K¨ahler manifold X is a singular Calabi-Yau ifX has a non singular modelX⁰ satisfyingπ (X⁰) = 0 and

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Further problems (II)

Definition

A compact K¨ahler manifold X =X^2p is a singular hyperk¨ahler manifold ifX has a non singular model X⁰ satisfying

π₁(X⁰) = 0 and possessing a sectionσ ∈H⁰(X⁰,Ω²_X0) such that the zero divisor E = div(σ^p) has numerical dimension 0 (so that KX⁰ =E again).

Conjecture (known by BDPP for X projective!)

LetX be compact K¨ahler, and letX →Y be the MRC fibration (after taking suitable blow-ups to make it a genuine morphism). Then K_Y is psef.

Proof ? Take the part of slope>0 in the HN filtration of TX, w.r.t. to classes in the dual of the psef cone, show that this corresponds to the MRC fibration, and apply duality.