The LYZ equation and the CJY conjecture

Shing-Tung Yau
Yau Mathematical Sciences Center
Tsinghua University

11. Definition.

• At the most fundamental level, mirror symmetry describes aframework for relating complex geometry to symplectic geometry on two Calabi-Yau manifolds.

• It defines a duality between the underlying structures on eachmanifold, and the development of this elegant theory has leadto progress in both physics and mathematics.

• One particular aspect of mirror symmetry we are interested inis the relationship between the derived category of coherent
sheaves on one manifold, and Fukaya’s category of Lagrangiansubmanifolds with local systems on the other.

2• This duality can be approached from a differential geometricperspective.

• In the simple case of a torus, C. Leung, E. Zaslow and I gave
an explicit formulation of an equation on a line bundle that
corresponds to the special Lagrangian equation on the mirror.

• Specifically, let X and X̌ be dual torus fibrations over a base tori
B. In this semi-flat setting, a connection A on a holomorphic
line bundle L over X is dual to a Lagrangian section L of thetorus fibration X̌ → B.

3• If L is a special Lagrangian section, using the Fourier-Mukai
Transform, Leung-Yau-Zaslow show that A must satisfy the equation:
Im(ω − F )n = tan θ̂Re(ω − F )n,(1)

where ω is the Kähler form on X, F is the curvature of theconnection A, and θ̂ is the phase of the special Lagrangian.

• Hence equation (1) is called the LYZ equation instead of thedeformed Hermitian Yang-Mills (dHYM) equation.

• Such an equation is also derived from supersymmetry by M.
Marino, R. Minasian, G. Moore, and A. Strominger.

4• Now let (X, ω) be a compact Kähler manifold of complex dimension n and χ a closed real (1, 1)-form on X.

• The LYZ equation on (X, ω, χ) is
Re(χu +√√n−1ω) = cot θ̂ Im(χu + −1ω)n.

(2)• Hereχu = χ +√−1∂ ∂¯ufor a real smooth function u on M and θ̂ is the argument of thecomplex number
Z

(χ +
X

√

−1ω)n.

5

• Letλ = (λ1, . . . , λn)be the eigenvalues of χu with respect to ω. Let
θi = arccotλi for 1 ≤ i ≤ n.

• Then λi = cot θi and we can compute as follows:√(χu + −1ω)nnY√=(λi + −1) ω n
i=1

√

exp

−1

i=1

=

n
Q

i=1
n
P

i=1
n
Q
i=1



θi

ωn

sin θi

cos
=

n
P

 n
P



θi

sin θi

ωn +

√

sin
−1

i=1
n
Q
i=1



θi

ω n.

sin θi6• So the LYZ equation becomes
X
n

cos

X
n



θi = cot θ̂ sin

i=1



θi ,

i=1

or
X
n

cot



θi = cot θ̂.

i=1• If we defineθω (χu) :=

n
X
i=1

θi =

n
X

arccot λi,i=1then the LYZ equation is simply written asθω (χu) = θ0.

(3)• We assume θ0 = θ̂ ∈ (0, π).

7• In this formulation it is clear that the equation is the complex
version of the special Lagrangian equation for graphs, with θ0
the analog of the Lagrangian angle and λj the analogs of theeigenvalues of the Hessian of the generating function of graph.

• The LYZ equation has been extensively studied by many mathematicians.

• In 2014, Jacob and I initiated to study the LYZ equation.

A. Jacob, S.-T. Yau. A special Lagrangian type equation for holomorphic
line bundles. Math. Ann. 369 (2017), 869-898. arXiv:1411.7457.

8

2. Analytic condition: existence of a subsolution.

• When n = 2, Jacob and I proved that the solution exists if andonly if there exists a function u such thatχu > cot θ0 ω.

• Denote
α = χu − cot θ0 ω.
Then α > 0 and it is a Kähler metric.

• Jacob and I then translated the LYZ equation into the complex
Monge-Ampère equation
12
ωsin2 θ0which was solved by myself in 1976.
(α + i∂ ∂¯u)2 =9• When n ≥ 3, Collins, Jacob and I solved the LYZ equation forθ0 ∈ (0, π) by assuming that the following two conditions hold.
(1) There exists a subsolution u, which means that χu satisfies
the inequality
max

1≤j≤n

X

arccot λi(χu) < θ0(< π).

(4)

i6=j

(2) The function u also satisfies an extra condition:θω (χu) =nX

arccot λi(χu) < π.

(5)

i=1

T. Collins, A. Jacob, S.-T. Yau.

(1, 1) forms with specified Lagrangianphase: a priori estimates and algebraic obstructions. Camb. J. Math. 8(2020), 407-452. arXiv:1508.01934.

10• To be precise, Collins, Jacob and I proved the following• Theorem 1.

(Collins-Jacob-Yau) Let (X, ω) be a compactKähler manifold of dimension n and χ a closed real (1, 1)-formon X with θ0 ∈ (0, π). Suppose there exists a subsolution u of
LYZ equation (2) in the sense of (4) and u also satisfies inequality (5). Then there exists a unique smooth solution of the LYZ
equation.

11• The definition of subsolutions of fully nonlinear equations wasintroduced by B. Guan.

• G. Székelyhidi gave an equivalent version and Collins, Jacob andI used it to the LYZ equation which is equivalent to inequality(4).

B. Guan. Second-order estimates and regularity for fully nonlinear elliptic
equations on Riemannian manifolds. Duke Math. J. 163(2014), 1491-1524.

G. Székelyhidi.

Fully non-linear elliptic equations on compact Hermitian

manifolds. J. Differential Geom. 109(2018), 337-378.

12

• L. Huang-J. Zhang-X. Zhang also considered the solution on acompact almost Hermitian manifold for the case θ0 ∈ (0, π2 ).

• C.-M. Lin generalized Collins-Jacob-Yau’s result to the Hermitian manifold (X, ω) with ∂ ∂¯ω = ∂ ∂¯ω 2 = 0.

L. Huang, J. Zhang, X. Zhang. The deformed Hermitian-Yang-Mills equationon almost Hermitian manifolds.

Sci.

China Math.

65(2021), 127-152.

arXiv:2011.14091.

C.-M. Lin. Deformed Hermitian-Yang-Mills equation on compact Hermitian
manifolds. arXiv:2012.00487.

13

• Now the natural question is whether condition (5) is superfluous.

• When n = 3, Pingali can solve the LYZ equation without condition (5) by translating the LYZ equation into a mixed MongeAmpère type equation.

• When n = 3 and n = 4, C.-M. Lin can solve the LYZ equation
without condition (5).

V. P. Pingali. The deformed Hermitian Yang-Mills equation on three-folds.
arXiv:1910.01870.

C.-M. Lin. The deformed Hermitian-Yang-Mills equation, the positivstellensatz, and the solvability. arXiv:2201.01438v2.

14• Very recently, C.-M. Lin solved the LYZ equation without condition (5).

• Theorem 2. (Lin) Let (X, ω) be a compact Kähler manifold ofdimension n and χ a closed real (1, 1)-form on M with θ0 ∈ (0, π).
If there exists a subsolution u of LYZ equation (2) in the senseof (4), then there exists a unique smooth solution of the LYZ
equation.

C.-M. Lin. On the Solvability of General Inverse σk Equations. arXiv:2310.05339.

15

3. Geometric condition: A Nakai-Moishezon type criterion.

• DenoteΩω (χ, p) = Re(χ +√√p−1ω) − cot θ0Im(χ + −1ω)p• DefineZ
1,1Ωω (χ, p) > 0 for any p-subvarity Y , 0 < p < n}Pω = {[χ] ∈ HR (X) |Y
1,1Sω = {[χ] ∈ HR (X) | the LYZ equation has a smooth solution}.

• Conjecture. (Collins-Jacob-Yau) For any compact Kähler manifold (X, ω),Pω = S ω .

16• G. Chen made an important progress on CJY conjecture.

• A smooth family χt, t ∈ [0, ∞) of real closed (1, 1)-forms is calleda test family if and only if all the following conditions hold.
(A) When t = 0, χ0 = χ.
(B) For all s > t, χs − χt is positive definite.
(C) There exists a large enough number T ≥ 0 such that for allt ≥ T , χt − cot θn0 ω is positive definite.

G. Chen. The J-equation and the supercritical deformed Hermitian-YangMills equation. Invent. Math. 225 (2021), 529-602.

17

• Theorem 3. (Chen) Let (X, ω) be a compact Kähler manifoldof dimension n and χ a closed real (1, 1)-form on X with θ0 ∈(0, π). Then the follwing statements are equivalent.
(1) There exists a smooth solution of LYZ equation (3).
(2) For any smooth test family χt, there exists a constant ε1,1 >0 such that for any t ≥ 0 and p-dimensional subvariety Y ,ZYΩω (χt, p) ≥ (n − p)ε1,1

Z
Y

ω p.

(3) There exist a test family χt and a constant ε1,1 > 0 suchthat for any t ≥ 0 and p-dimensional subvariety Y ,ZYΩω (χt, p) ≥ (n − p)ε1,1

J.-P. Demailly, M. Paun.

Z
Y

ω p.

Numerical characterization of the Kähler cone.

Ann. of Math. (2) 159 (2004), 1247-1274.

J. Song. Nakai-Moishezon criterions for complex Hessian equations. arxiv:
2012.07956.
18• Motivated by Chen and Song, Chu-Lee-Takahashi established• Theorem 4.

(Chu-Lee-Takahashi) The LYZ equation on acompact Kähler manifold (X, ω) with complex dimension n issolvable for θ0 ∈ (0, π) if and only if there exists a Kähler metric
γ on X such that for any 1 ≤ p ≤ n,ZXΩω (χ, p) ∧ γ n−p ≥ 0

and for any proper m-dimensional subvariety Y of X and 1 ≤p ≤ m,ZYΩω (χ, p) ∧ γ m−p > 0.

• Chu-Lee-Takahashi then confirmed the CJY conjecture for projective manifolds.

J. Chu, M.-C. Lee, R. Takahashi.

A Nakai-Moishezon type criterion forsupercritical deformed Hermitian-Yang-Mills equation. arxiv:2105.10725.
19• Junsheng Zhang proved that Sω is a both open and closed subset of Pω . He then disproved the CJY conjecture in the nonprojective case. The counter-example is the blow up of C3/Λat a point. Here C3/Λ is a three dimension torus without any
positive dimensional proper analytic subvariety.

• The situation is very much like the work of Demailly-Paun,
where they used my solution to the Calabi conjecture to give a
differential geometric proof of the Nakai-Moishezon numericalcriterion for ample line bundles in algebraic geometry.

J. Zhang. A note on the supercritical deformed Hermitian-Yang-Mills equation. arXiv:2302.06592.

20

4. Algebraic condition: Stability.

• Collins and I developed the mirror of the infinite dimensional
GIT picture for the LYZ equation.

• We described a infinite dimensional symplectic manifold, admitting an action by a group of symplectomorphisms, together witha space H ⊂ C ∞(X, R) and a Riemannian structure on H, which
can be thought of as analogous to G/K in the finite dimensional
GIT.

• We computed the geodesic equation, and introduce a notion of
ε-geodesics, which solve an approximate version of the geodesic
equation.

T. Collins and S.-T. Yau. Moment maps, nonlinear PDE, and stability inmirror symmetry. arXiv:1811.04824.
21

• We also introduced the complexified Calabi-Yau functional, andextract from this functional analogues of the C, and J functionals, as well as a C-valued functional Z. The J functional is
the Kempf-Ness functional for the GIT problem; it has criticalpoints at solutions of the LYZ equation, and is convex alongsmooth geodesics.

• A fundamental issue in the analogy with finite dimensional GIT
is that smooth geodesics need not exist.

• Thus, our main analytic contribution is to prove, in the hypercritical phase case, the existence of weak geodesics connecting
points in H, with C 1,α regularity.

With this much regularity,

we can show that the functionals J , C, Z are well-defined and
we prove that they are convex/concave along these generalized
geodesics.
22• With these results in hand, we are in the setting of an infinite
dimensional GIT problem with geodesics playing the role of oneparameter subgroups.

• Using algebraic geometry we constructed model infinite rays,
analogous to one-parameter subgroups in the space H, and evaluate the limit slope of the Calabi-Yau functional along thesemodel curves in terms of algebraic data.

• Using the existence of regular geodesics these model curves giverise to algebro-geometric obstructions to the existence of solutions to LYZ.

23

• For example, we established the following theorem in the hypercritical phase case, i.e. θ0 ∈ (0, π2 ).

• Theorem 5. (Collins-Yau) Let J1 ⊂ J2 ⊂ · · · ⊂ Jr−1 ⊂ Jr = OX
be a sequence of ideal sheaves, and define
I = J0 + t · J1 + · · · + tr−1 · Jr−1 + (tr ) ⊂ OX ⊗ C[t].
Let µ : X → X × 4 be a log resolution of I, so that µ−1I =OX (−E) for a simple normal crossing divisor E. If [χ] admits asolution of the LYZ equation then
"
#
√
∗
∗
n
(µ [ω] + −1(µ [χ] − δE))
√
E.Im
≥0
n
(ω + −1χ) .[X]24

5. Parabolic version.

• For the parabolic flow method, there are also several results.

• Jacob-Yau and Collins-Jacob-Yau proposed the line bundle mean
curvature flow (LBMCF)
 ut = θ0 − θω (χu)(6)

 u(0) = u

where u is a subsolution of the LYZ equation.

• They proved the existence and convergence of the long-time
solution to the LBMCF for the case θ0 ∈ (0, π2 ) and also θω (χu) ∈(0, π2 ).

25

• Takahashi proposed the tangent Lagrangian phase flow (TLPF)
 ut = tan(θ0 − θω (χu))(7)

 u(0) = u

where u is a subsolution of the LYZ equation.

• He proved the existence and convergence of the long-time solution to the TLPF for the case θ0 ∈ (0, π2 ) and also θω (χu) − θ0 ∈(− π2 , π2 ).

R. Takahashi. Tan-concavity property for Lagrangian phase operators andapplications to the tangent Lagrangian phase flow. Internat. J. Math. 31(2020), 26 pp.

26

• Motivated by the concavity of cot θω (χu) by G. Chen, Fu, Zhang
and I considered a new flow: ut = cot θω (χu) − cot θ0,(8)

 u(x, 0) = u (x)0withθω (χu0 ) < π.

G. Chen. The J-equation and the supercritical deformed Hermitian-YangMills equation. Invent. Math. 225 (2021), 529-602.

J. Fu, S.-T. Yau and Dekai Zhang. A deformed Hermitian Yang-Mills flow.
Accepted for publication in JDG. arXiv:2105.13576v3.

27

• We first proved the existence theorem of the long-time solutionof flow (8).

• Theorem 6. (Fu-Yau-Zhang) Let (X, ω) be a compact Kähler
manifold and χ a closed real (1, 1)-form with θ0 ∈ (0, π). If u0
satisfies the inequalityθω (χu0 ) < π,then flow (8) has a unique smooth long-time solution u.

28• Then we considered the convergence of the long-time solutionof flow (8).

• Theorem 7. (Fu-Yau-Zhang) Let (X, ω) be a compact Kähler
manifold and χ a closed real (1, 1) form with θ0 ∈ (0, π). Suppose that exists a subsolution u of LYZ equation (3) in the senseof (4) which also satisfies (5). Then there exists a long-time
solution u(x, t) of flow (8) with u0 = u and it converges to asmooth solution u∞ to the LYZ equation:θω (χu∞ ) = θ0.

29

• Hence we reproved the Collins-Jacob-Yau’s existence theorem.
Our parabolic version seems rather natural and the proof issimpler.

• The advantage of the new flow is that the imaginary part of theCalabi-Yau functional is constant along the flow, which is the
key to do the C 0 estimate.

• We in fact identified which assumption is needed for the longtime existence part and which one is needed for the convergence
part.

30

• The second motivation of our paper is to use flow (8) to study
the equation under the existence of a semi-subsolution:
max

1≤j≤n

X

arccot λi(χu) ≤ θ0(< π).

i6=j• We can solve the two dimensional case. In this case, a smoothfunction u is called a semi-subsolution of the LYZ equation ifχu ≥ cot θ0 ω.

(9)• We hope our flow is also useful to solve the higher dimensionalcase. This is analog to the semi-stable case of the HYM equation. The HYM flow proposed by Donaldson is very useful instudy of the HYM equation in the semi-stable case.

31

6. Further Discussions.

• If for some reason, we can solve the LYZ equation, then we canderive important consequences for the algebraic manifolds. Letme give examples.

• Five years ago, T. Collins and I observed that if the projective
manifold is covered by a homogenous manifold, we can solve
the LYZ equation by using homogeneity to reduce to pointwise
calculations. (Good examples are abelian varieties or Shimuravarieties.)32• Once it is done, the invariant Kähler forms and the invariant(1,1)-forms together give rise to a range of Chern number inequalities over algebraic subvarieties. These are not trivial inequalities.

• It would be nice to find out the cases when those inequalities
become equalities. If a closed (p, p)-form can be represented byan effective algebraic subvariety§those inequalities should givenon trivial information about the Hodge conjecture.

33

• On the other hand, for the complexified Kähler cone of X, weconsider its open subset
C = {ω + iχ | ω ∈ K and [χ] ∈ Sω }.

• We propose the following equations:– Is the open set C is pseudo-convex (Stein)?
– If not, what is the automorphism group of the pseudo-convex
hull of C? In general, is it discrete?

34

Thank you!

35