a tropical approach to a generalized hodge conjecture for...
TRANSCRIPT
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A tropical approach to a generalized Hodgeconjecture for positive currents
Farhad Babaee
SNSF/Universite de Fribourg
February 20, 2017 - Toblach
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Are all positive currents with Hodge classes approximable bypositive sums of integration currents? (Demailly 1982)
No! (Joint work with June Huh)
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Are all positive currents with Hodge classes approximable bypositive sums of integration currents? (Demailly 1982)
No! (Joint work with June Huh)
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Currents
X complex smooth manifold of complex dimension n.
• Dk(X ) := Space of smooth differential forms of degree k ,with compact support = test forms
• D′k(X ) = Space of currents of dimension k := Topologicaldual to Dk(X )
• 〈T , ϕ〉 ∈ C (linear continuous action)
• T ∈ D′k(X ) current is closed (= d-closed),〈dT , ϕ〉 := (−1)k+1〈T , dϕ〉 = 0, ∀ϕ ∈ Dk−1(X )
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• Dp,q(X ) : Smooth (p, q)-forms with compact support
• D′p,q(X ) :=(Dp,q(X )
)′• For currents (p, q)-bidimension = (n − p, n − q)-bidegree
• Tj → T in weak limit, if 〈Tj , ϕ〉 → 〈T , ϕ〉 ∈ C
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• Dp,q(X ) : Smooth (p, q)-forms with compact support
• D′p,q(X ) :=(Dp,q(X )
)′• For currents (p, q)-bidimension = (n − p, n − q)-bidegree
• Tj → T in weak limit, if 〈Tj , ϕ〉 → 〈T , ϕ〉 ∈ C
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Integration currents
Example
Let Z ⊂ X a smooth submanifold of dimension p, define theintegration current along Z , denoted by [Z ] ∈ D ′p,p(X )
〈[Z ], ϕ〉 :=
∫Zϕ, ϕ ∈ Dp,p(X ).
This definition extends to analytic subsets Z , by integrating overthe smooth locus.
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Positivity
DefinitionA smooth differential (p, p)-form ϕ is positive if ϕ(x)|S is anonnegative volume form for all p-planes S ⊂ TxX and x ∈ X .
DefinitionA current T ∈ D′p,p(X ) is called positive if
〈T , ϕ〉 ≥ 0
for every positive test form ϕ ∈ Dp,p(X ).
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Examples of positive currents
• An integration current on an analytic subset is a positivecurrent, with support equal to Z
• Convex sum of positive currents
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The generalized Hodge conjecture for positive currents(HC+)
Question/Conjecture: Are all the positive closed currentsapproximable by a convex sum of integration currents alonganalytic cycles?
T + ←−i
∑j
λ+ij [Zij ],
On a smooth projective variety X , and
{T +} ∈ R⊗Z(H2q(X ,Z)/tors ∩ Hq,q(X )
),
where q = n − p.
Demailly, the superhero, 1982: True for p = 0, n − 1, n.
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The generalized Hodge conjecture for positive currents(HC+)
Question/Conjecture: Are all the positive closed currentsapproximable by a convex sum of integration currents alonganalytic cycles?
T + ←−i
∑j
λ+ij [Zij ],
On a smooth projective variety X , and
{T +} ∈ R⊗Z(H2q(X ,Z)/tors ∩ Hq,q(X )
),
where q = n − p.
Demailly, the superhero, 1982: True for p = 0, n − 1, n.
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The generalized Hodge conjecture for positive currents(HC+)
Question/Conjecture: Are all the positive closed currentsapproximable by a convex sum of integration currents alonganalytic cycles?
T + ←−i
∑j
λ+ij [Zij ],
On a smooth projective variety X , and
{T +} ∈ R⊗Z(H2q(X ,Z)/tors ∩ Hq,q(X )
),
where q = n − p.
Demailly, the superhero, 1982: True for p = 0, n − 1, n.
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The Hodge conjecture (HC)
The Hodge conjecture: The group
Q⊗Z(H2q(X ,Z)/tors ∩ Hq,q(X )
),
consists of classes of p-dimensional algebraic cycles withrational coefficients.
Demailly 1982: HC+ =⇒ HC.
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Hodge conjecture for real currents (HC′)
If T is a (p, p)-dimensional real closed current on X withcohomology class
{T } ∈ R⊗Z(H2q(X ,Z)/tors ∩ Hq,q(X )
),
then T is a weak limit of the form
T ←−i
∑j
λij [Zij ],
where λij are real numbers and Zij are p-dimensionalsubvarieties of X .
Demailly 2012: HC′ ⇐⇒ HC
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HC+ not true in general!
Theorem (B - Huh)
There is a 4-dimensional smooth projective toric variety X and a(2, 2)-dimensional positive closed current T + on X with thefollowing properties:
(1) The cohomology class of T + satisfies
{T +} ∈ H4(X ,Z)/tors ∩ H2,2(X ).
(2) The current T + is not a weak limit of the form
T + ←−i
∑j
λ+ij [Zij ],
where λ+ij > 0, Zij are algebraic surfaces in X .
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HC+ not true in general!
Theorem (B - Huh)
There is a 4-dimensional smooth projective toric variety X and a(2, 2)-dimensional positive closed current T + on X with thefollowing properties:
(1) The cohomology class of T + satisfies
{T +} ∈ H4(X ,Z)/tors ∩ H2,2(X ). OK!
(2) The current T + is not a weak limit of the form
T + ←−i
∑j
λ+ij [Zij ],
where λ+ij > 0, Zij are algebraic surfaces in X .
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Extremality in the cone of closed positive currents
DefinitionA (p, p)-closed positive current T is called extremal if for anydecomposition T = T1 + T2 , there exist λ1, λ2 ≥ 0 such thatT = λ1T1 and T = λ2T2. (Ti closed, positive and samebidimension).
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Extremality reduces the problem to sequences
LemmaX an algebraic variety, T + be a (p, p)-dimensional current on Xof the form
T + ←−i
∑j
λ+ij [Zij ],
where λ+ij > 0, Zij are p-dimensional irreducible analytic subsets of
X . If T is extremal then
T + ←−iλ+i [Zi ].
for some λ+i > 0 and Zi irreducible analytic sets.
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Obstruction by the Hodge index theorem in dimension 4
Proposition
Let {T } be a (2, 2) cohomology class on the 4 dimensional smoothprojective toric variety X . If there are nonnegative real numbers λiand 2-dimensional irreducible subvarieties Zi ⊂ X such that
{T } = limi→∞{λi [Zi ]},
then the matrix[Lij ]{T } = −{T }.Dρi .Dρj ,
has at most one negative eigenvalue.
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Our goal
A (2, 2)-current on a 4-dimensional smooth projective toric varietywhich is
• Closed
• Positive
• Extremal, and
• Its intersection form has more than one negative eigenvalues
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Tropical currents
Log : (C∗)n → Rn
(z1, . . . , zn) 7→ (− log |z1|, . . . ,− log |zn|)
• Log−1({pt}) ' (S1)n,
• dimR Log−1(rationalp-plane) = n + p
• Log−1(rational p-plane) has a natural fiberation over (S1)n−p
with fibers of complex dimension p
• Similarly for any p-cell σ, Log−1(σ) has a natural fiberationover (S1)n−p
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Tropical currents
Log : (C∗)n → Rn
(z1, . . . , zn) 7→ (− log |z1|, . . . ,− log |zn|)
• Log−1({pt}) ' (S1)n,
• dimR Log−1(rationalp-plane) = n + p
• Log−1(rational p-plane) has a natural fiberation over (S1)n−p
with fibers of complex dimension p
• Similarly for any p-cell σ, Log−1(σ) has a natural fiberationover (S1)n−p
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n = 2, p = 1
w=21
3
12
3 S 1
R (C*)2 2
Q
Support TC = Log−1(C ), TC =∑σ wσ
∫Sn−p [fibers of Log−1(σ)] dµ
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Dimension n
1 2 1 2
C ⊂ Rn, dim(C ) = p TC ∈ D′p,p((C∗)n), Support TC = Log−1(C )
1 21+2
{T C } = rec(C ) ∈ Hn−p,n−p(XΣ) T C ∈ D′p,p(XΣ)
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1 2 1 2
C ⊂ Rn, dim(C ) = p TC ∈ D′p,p((C∗)n), Support TC = Log−1(C )
1 21+2
{T C } = rec(C ) ∈ Hn−p,n−p(XΣ) T C ∈ D′p,p(XΣ)
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A (2, 2)-current on a 4-dimensional smooth projective toric varietywhich is
• Closed
Balanced complex
• Positive
Positive weights
• Extremal
?
• Its intersection form has more than one negative eigenvalues
?
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Extremality of tropical currents in anydimension/codimension
Weights unique up to a multiple + Not contained in any proper affinesubspace
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Examples of extremal currents
Lelong 1973: Integration currents along irreducible analytic subsetsare extremal. Is that all?Demailly 1982: i
π∂∂ log max{|z0|, |z1|, |z2|} is extremal on P2, andits support has real dimension 3, thus cannot be an integrationcurrent along any analytic set.
Dynamical systems (usually with fractal supports, thusnon-analytic):Codimension 1: Bedford and Smillie 1992, Fornaess and Sibony1992, Sibony 1999, Cantat 2001, Diller and Favre 2001, Guedj2002...Higher Codimension: Dinh and Sibony 2005, Guedj 2005, Dinhand Sibony 2013
Complicated structures, easily seen to be approximable!
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Extremal if: weights unique up to a multiple + Not contained in anyproper affine subspace
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Manipulation of signatures for 2-cells in dimension 4
The operation F 7−→ F−ij produces one new positive and one newnegative eigenvalue for its intersection matrix
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A (2, 2)-current on a 4-dimensional smooth projective toric varietywhich is
• Closed
Balanced complex
• Positive
Positive weights
• Extremal
Non-degenerate + weights unique up to a multiple
• Its intersection form has more than one negative eigenvalues
The operation on two cells provides one new negativeand one new positive eigenvalue
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A concrete example
Consider G ⊆ R4 \ {0}
e1 e2 e3 e4
f1 f2 f3 f4,
where e1, e2, e3, e4 are the standard basis vectors of R4 andf1, f2, f3, f4 the rows of
M :=
0 1 1 11 0 −1 11 1 0 −11 −1 1 0
.
The weights of solid (resp. dashed) edges are +1 (resp. −1).
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Thank you for your attention, indeed!