Brief Summary of Calabi-Yau geometry
Shing-Tung Yau
Harvard University and Tsinghua University

June 2018

Outline

1. BCOV Theory and Gromov-Witten Invariants
2. Donaldson-Thomas Invariants
3. Periods and Mirror Map
4. SYZ Conjecture and Special Lagrangian Geometry
5. Non-Kähler Calabi-Yau Manifolds
6. Calabi-Yau Cones
7. Stability in Calabi-Yau Geometry

Introduction

A compact Kähler manifold (X n , ω) is called Calabi-Yau if
c1 (X ) = 0 ∈ H 2 (X , Z). Usually we assume π1 (X ) is finite.
I In 1954, 1957, Calabi conjectured that X carries Ricci-flat metric.
I Calabi reduced his conjecture to a complex Monge-Ampère equation.



∂ 2ϕ
det gi j¯ + i j = e f det gi j¯∂z ∂ z̄with gi j¯ + ∂i ∂j¯ϕ being positive definite.
I Using the tools of geometric analysis, I proved the conjecture in 1976 andapplied the proof to solve some problems in algebraic geometry. This is
the major starting point of the development of geometric analysis.

This provides first known non-locally-homogeneous Einstein manifolds.
I Chern number inequality c2 ∩ [ω]n > 0 and the equality holds if and only
if the universal cover X̃ ∼= Cn .
I The holonomy group is contained in SU(n). This leads to the structure
theorem of Calabi-Yau manifolds.
I There are many other applications in complex and algebraic geometry. For
example, K3 is Kähler by Siu.
I There are abundant examples contributed by both physicists andmathematicians.
I There is no similar theorem in G 2 manifolds. It is more difficult toconstruct examples of G 2 manifolds.
IThe Ricci-flat metric on Calabi-Yau manifolds are solutions of the Einstein field
equation with no matter. The theory of motions of loops inside a Calabi-Yau
manifold provide a model of a conformal field theory. So Calabi-Yau manifolds
became very important in the study of superstring theory. Physics theories
have been inspiring a lot of mathematical studies of Calabi-Yau geometry.
One of the most important topics contributed by both physicists andmathematicians is mirror symmetry. I will discuss the details in next session.

1. BCOV Theory and Gromov-Witten invariantsMirror Symmetry

Mirror symmetry is a duality in string theory originally discovered by Dixon,
Lerche, Vafa, Warner, Witten and many other physicists based on theirstudy of Calabi-Yau manifolds in physics. Based on this, Greene, Plesser
found nontrivial examples of the mirror relationship in terms of Gepnermodels. It is found a very rich theory relating symplectic geometry of a
Calabi-Yau threefold X and complex geometry of its mirror Calabi-Yau
threefold X̌ . Around 1990, the detailed mathematical calculation was carriedout by Candelas, de la Ossa, Green and Parkes to obtain a conjectural
formula of the number of rational curves of arbitrary degree in a quintic
Calabi-Yau threefold by relating it to period integrals of the quintic mirror.
The BCOV theory, by Bershadsky-Cecotti-Ooguri-Vafa, is a vast
generalization of this theory to all genus, all Calabi-Yau threefolds.
Nearly a quarter of century later, it continues to inspire new ideas inmathematical physics. Furthermore, many of its mathematical concepts andconsequences are only beginning to be realized and proved.

Gromov-Witten invariants of Calabi-Yau threefolds
By late 1990s mathematicians had established the foundation ofGromov-Witten (GW) theory as a mathematical theory of A-model
topological strings. In this context, the genus g free energy FgX (t) on a
Calabi-Yau 3-fold X collects the virtual counting NgX,β of stable holomorphic
maps from genus g curve to X , which is a function on a neighborhood aroundthe large radius limit in the complexified Kähler moduli t of X .
 Z
R
X
1
t
3
X

β ,

t
+
N
e
g = 0;

0,β

6 X


β>0






R
 1 Z
X
X
X
tc2 (X ) +N1,βe β t , g = 1;
Fg (t) = −
24 X


β>0






R
X

t

X
X
β ,

N
+
N
e
g > 2.
g ,β
 g ,0β>0where t ∈ H 2 (X ; C) is the complexified Kähler class, andZ
X
1∈Q
Ng ,β :=
[Mg ,0 (X ,β)]vir

J. Li-Tian constructed the virtual fundamental class [Mg ,0 (X , β)]vir in
algebraic geometry. See also Behrend-Fantechi.

GW invariants for quintic Calabi-Yau threefolds
Genus g = 0. (1990) Candelas, de la Ossa, Green and Parkes derived
the generating functions F0X , for X a smooth quintic Calabi-Yau threefolds.
Later, Lian-Liu-Yau and Givental independently proved the genus zero
mirror formula for the quintic Calabi-Yau threefold and further for Calabi-Yau
complete intersections in projective toric manifolds.
Genus g = 1. (1993) Bershadsky, Cecotti, Ooguri, and Vafa derived thegenerating functions F1X , for X a smooth quintic Calabi-Yau threefolds. Thiswas first proved by Zinger, using genus-one reduced GW theory
BCOV Theory. (1993) Bershadsky, Cecotti, Ooguri, and Vafa the socalled BCOV theory, setting the foundation to obtain higher genus generating
functions FgX for X a smooth quintic Calabi-Yau threefolds.
Genus g > 2.Yamaguchi-Yau analyzed BCOV’s holomorphic anomaly
equation and argued the B-model generating function lies in a finitely
generated differential ring. Based on this, Huang, Klemm, andQuackenbush derived an algorithm to determine FgX , for g 6 51,on smooth
quintic threefolds.
Two recent approaches to the BCOV genus-two mirror formula, andYamaguchi-Yau polynomiality conjecture for Fg in all genera:
I (Guo-Janda-Ruan, and Chen-Janda-Ruan): via an extnsion of moduli
of stable maps with p-fields (developed by Chang-Li)) that connects GW
theory to LG theory
I (Chang-Li-Li-Liu, and Chang-Li-Guo): via Mixed-Spin-P fields for amaster moduli space that describes the gauged linear sigma model.

B-model topological strings: BCOV theory

Bershadsky-Cecotti-Ooguri-Vafa (BCOV) theory proposes a description ofhigher genus B-model based on a gauge theory on Calabi-Yau threefold. The
free energy of this gauge theory gives the generating function FgX̌ (τ, τ̄ ) of the
topological B-model living over the moduli of the mirror Calabi-Yau X̌ .
Mirror symmetry predicts that Gromov-Witten generating function FgX (t)coincides with FgX̌ (τ, τ̄ ) of the mirror under the so-called mirror map t ↔ τand limit τ̄ → 0 around the large complex structure
FgX (t) = lim FgX̌ (τ, τ̄ )τ̄ →∞The mirror map and F0 are determined by period integrals of a holomorphic
3-form of the mirror. We will come back to the period integrals later.

Holomorphic anomaly equations
The BCOV theory, among other things, produced the celebrated holomorphic
anomaly equations which the B-model free energies Fg satisfies:
!
g −1
X
1
¯
Dj Fr Dk Fg −r∂ i Fg = C i¯j¯k̄ e 2K G i j G k k̄ Dj Dk Fg −1 +2
r =1
Yamaguchi-Yau analyzed BCOV’s holomorphic anomaly equation and argued
the B-model generating function lies in a finitely generated differential ring.
Huang-Klemm-Quackenbush analyzed the polynomial structure of
Yamaguchi-Yau and the gap condition at the conifold point to determine thefree energies of B-model on the mirror quintic threefold up to genus 51.
Alim-Scheidegger-Yau-Zhou showed that for the anti-canonical line bundleof a toric semi-Fano surface, the BCOV-Yamaguchi-Yau ring is essentially
identical to the ring of almost-holomorphic modular forms in the sense of
Kaneko-Zagier (a closely related notion of nearly holomorphic modular forms
was systematically studied by Shimura), and the Yamaguchi-Yau functional
equation reduces to an equation for modular forms.
Recent mathematical development of Yamaguchi-Yau functional equation:
I by Lho-Pandharipande for local Calabi-Yau KP 2
I by Ruan et al. for g = 2 GW generating functions for quintic CY
Using the theory of Mixed-Spin-P field (Chang-Li-Li-Liu), Chang-Guo-Li
has observed, via computation, that for all genera (the torus localization of)
MSP reproduces BCOV Feymann diagram directly. If this can be proved, then
MSP should settle Yamaguchi-Yau functional equation for all genera.

Mathematical development of BCOV theory

Must of the work on mathematical foundation of BCOV theory has been led
by Costello-S. Li.
Costello-Li developed a mathematical framework of quantizing BCOV theory
rigorously in terms of effective renormalization method. It also generalized
BCOV theory to Calabi-Yau’s of arbitrary dimension by incorporating withgravitational descendant.
Costello-Li further extended BCOV theory into open-closed string field theory
in topological B-model by coupling with Witten’s holomorphic Chern-Simons
theory and revealed its connection with large N duality.

Homological mirror symmetry (HMS) for CY

The homological mirror symmetry conjecture by Kontsevich for Calabi-Yau
mirror pairs X , X̌ states that Fukaya category of X is equivalent to thederived category of coherent sheaves on X̌ . It is proved by Seidel for the
genus-two curves and quartic surfaces and generalized by Sheridan for all
Calabi-Yau and Fano hypersurfaces in projective spaces. A fully faithful mirror
functor is achieved via family Floer theory by Abouzaid using Lagrangianfibers in the SYZ picture.

2. Donaldson-Thomas Invariants

DT invariants of Calabi-Yau threefolds

Let X be a non-singular projective Calabi-Yau threefold. Donaldson-Thomas(DT) invariants are virtual counts of ideal sheaves of curves in X withholomorphic Euler characteristic n and curve class β ∈ H2 (X ; Z)Z
X
DTn,β :=
1 ∈ Z.
[Hilbn,β (X )]vir

Donaldon-Thomas constructed the virtual fundamental class [Hilbn,β (X )]vir
X
and K. Behrend proved that DTn,βis a weighted Euler characteristic.

GW/DT Correspondence and the Topological Vertex
The GW/DT correspondence, conjectured by
Maulik-Nekrasov-Okounkov-Pandharipande (MNOP), relates the GW
theory and DT theory of a threefold. For nonsingular toric Calabi-Yau
threefolds, this is equivalent to the Aganagic-Klemm-Mariño-Vafa
algorithm of the Topological Vertex derived from the large N duality (by thework of Diaconescu-Florea, Li-Liu-Liu-Zhou, MNOP, andOkounkov-Reshetikhin-Vafa).
MNOP conjecture for dimension zero DT invariants has been proved by
I MNOP: toric threefolds
I Behrend-Fantechi: Calabi-Yau threefolds
I Levine-Pandharipande: projective threefolds
I J. Li: compact complex threefolds
The vertex of GW/DT correspondence for nonsingular toric threefolds
I 1-leg vertex (framed unknot): Liu-Liu-Zhou, Okounkov-Pandhariande,
I 2-leg vertex (framed Hopf link): Liu-Liu-Zhou,
I The full 3-leg case: Maulik-Oblomkov-Okounkov-Pandharipande

DT/SW correspondence

The DT/SW correspondence was first conjectured using Gauge theory
reduction approach by Gukov-Liu-Sheshmani-Yau (GLSY). It relates theDT theory of sheaves with 2 dimensional support in a non-compact local
surface threefold X : L → S, L ∈ Pic(S) over a smooth projective surface S,
to the Seiberg-Witten (SW) invariants of the surface S and invariants ofnested Hilbert scheme on S.
I Gholampour-Sheshmani-Yau (GSY) proved GLSY conjecture and showed
modular property of invariants of Nested Hilbert scheme of points.
I GSY proved that Nested Hilbert scheme invariants can specialize toPoincaré invariants of Dürr-Kabanov-Okonek (DKO) and the stable pairinvariants of Kool-Thomas (KT).
I GSY related the Vafa-Witten (VW) invariants to SW invariants of surface+ correction terms governed by invariants of nested Hilbert schemes.

3. Periods and Mirror Map

Geometric set-up

Consider a family π : Y → B of d-dimensional compact complex manifolds,
with Yb := π −1 (b). The H d (Yb ) form a vector bundle E → B with a flat
connection ∇. Fix a global section ω ∈ Γ(B, E ), it generates a period sheaf
Π(ω) given by integrals over geometric cyclesZω.
γ

The study of period map has a long history by Euler, Gauss, Riemann, andthere were works of Picard, Fuchs, Leray, Griffiths, Dwork and also
Gelfand-Kapranov-Zelevinsky. Period integrals is a very important componentof computations in mirror symmetry as pioneered by Candelas, de la Ossa,
Green and Parkes.

Large Complex Structure Limits
Hosono-Lian-Yau ’96 showed for hypersurfaces Y in a toric variety X , a
Large Complex Structure Limit (LCSL) Y∞ is given by the toric divisor
D = ∪Di of X . They also computed the periods of Y near Y∞ by firstfinding complete solutions to a GKZ system.
I By [Lian-Todorov-Yau ’00], a LCSL Y∞ is characterized by the property
that, there is a global meromorphic top form, such that its restriction to
Y∞ has the form (z1 ...zn )−1 dz1 ∧ ... ∧ dzn in a neighborhood.
I Mirror Symmetry Conjecture:I

periods near Y∞ =⇒ counting curves on mirror Y ∗.
General Problem: Construct an explicit complete linear PDE system τ forthe sheaf Π(ω). That is a system τ such thatsol(τ ) = Π(ω).
Main goals & applications:R
To compute explicitly periods γ ω as power series and determine local,
and even global monodromy of periods
I To count curves in algebraic varieties by mirror symmetryITautological systems

Lian-Yau in 2010 wrote global residue formula for period integrals. Period
integrals for the family Y are annihilated by the Fourier transform of thedefining ideal of the tautological embedding of X , and first order symmetry
operators induced by the ambient space symmetry. The resulting system ofdifferential equations τ is called a tautological system. τ specializes to a GKZ
system if X is a toric variety and G = T is the dense torus.
The tautological system τ is complete for X = Pn by
Bloch-Huang-Lian-Srinivas-Yau 2013. Moreover, for any homogeneous
G -variety, τ is complete iff PH n (X ) = 0 (Huang-Lian-Zhu 2014) . The
construction also describes solutions to the differential system τ in topologicalterms. This result is applied to the large complex structure limit forhomogeneous G -variety of a semisimple group G for the purpose of mirror
symmetry.

A recent application: Hasse-Witt matrices andperiodsI

Let X be a Fano toric variety or G /P of dimension n defined over Z, andπ : Y → B the universal family of Calabi-Yau (or general type)hypersurfaces in X . Given any prime p, after reduction mod p, we have
Frobenius endomorphism F : OY → OY by raising functions to p-th
power. This gives a p-semilinear morphism
R n−1 π∗ (OY ) → R n−1 π∗ (OY ).

Under certain basis by adjunction formula, this gives rise to the
Hasse-Witt matrices HWp .
I [Huang-Lian-Yau-Yu’18] An appropriate limit of a rescalling of HWp as
p → ∞ recovers the unique holomorphic period of the family over thecomplex numbers, at the rank 1 point (LCSL candidate) s∞ . Conversely,
HWp is equal to an appropriate truncation of Taylor series expansion of
the complex period at s∞ mod p.
I In the case of X being a toric variety, this explains part of the work by
Candelas, de la Ossa, Rodriguez-Villegas in 2000, regarding Calabi-Yau
varieties over finite fields.

4. SYZ Conjecture and Special Lagrangian Geometry

T-duality and SYZI

T -duality relates- type IIA string theory on the circle SA of radius R
T -dual l- type IIB string theory on the circle SB of radius 1/R.

Inspired by the role of D-branes under T-duality on Calabi-Yau

Conjecture (Stominger-Y.-Zaslow, 1996)
Let X and X̌ be a mirror pair of CY manifolds near the large structure limits.
1. X and X̌ admit dual special Lagrangian torus fibrations µ : X → B andµ̌ : X̌ → B over the same base B.
2. There exists a fiberwise Fourier-Mukai transform which maps Lagrangiansubmanifolds of X to coherent sheaves on X̌ .

e.g .

D3: SLag torus fibers ←→ D0: skyscraper sheaves.

FM transform of special Lagrangian equation

I

Leung-Yau-Zaslow worked out fiberwise Fourier-Mukai transform ofsupersymmetric cycles when quantum correction is absent (semi-flat).

Theorem (Leung-Y.-Zaslow)
Let X and X̌ be a mirror pair of CY manifolds (in the sense of SYZ). Thenthe fiberwise Fourier-Mukai transform of a special Lagrangian section in X
produces a holomorphic line bundle on X̌ which satisfies the deformed
Hermitian Yang-Mills equation (dHYM in short).
I

More specifically,0,2(i) ω|C = 0
FM (i) FA = 0−→(ii) Im Ω|C = tan θ Re Ω|C(ii) Im (ω + FA )k = tan θ Re (ω + FA )kConstruction of SYZ fibrations

W.D. Ruan asserted a Lagrangian torus fibration on the quintic CY
constructed by gradient flow and toric degenerations. Goldstein and
Gross constructed Lagrangian fibrations for toric Calabi-Yau n-folds,
generalizing the fibrations of Harvey-Lawson on C3 . They showed thatthese fibrations are special with respect to certain holomorphic volume
forms.
I Castano-Bernard and Matessi constructed Lagrangian fibrations on
local models around singular fibers in dimension 2 and 3, and applied
symplectic methods to glue them to give a global Lagrangian fibration.
I Near a singular fiber, the genuine CY metric should be approximated bygluing local CY models around singular fibers with the semi-flat metric.
I The semi-flat metric with singularities were constructed by
Greene-Shapere-Vafa-Yau. The local CY model around each singularfiber is constructed by Ooguri-Vafa. For K3 surface, Gross-Wilson
attempts to glue together the models based on the original proof of
Calabi conjecture, but failed to contain enough information of instantoncorrections.

I

SYZ construction with quantum corrections

I

Auroux proposed that the counting of holomorphic disks (that bounds
smooth torus fibers) drastically changes when one goes across a certain
wall in the SYZ base, displaying a wall-crossing phenomenon.

Chan-Lau-Leung used this technique to construct the SYZ mirrors of
toric CY manifolds. A typical example is KPn .
Abouzaid-Auroux-Katzarkov constructed toric CY manifolds as SYZ
mirrors of blow-ups of toric varieties.
I Recently Lau-Zheng constructed the SYZ mirrors of hypertoric varieties.
A typical example is T ∗ Pn . They serve as local models for holomorphic
symplectic manifolds.

IQuantum corrections and scattering

I

Quantum correction come from holomorphic disks emanating fromsingular fibers, forming the wall structure on the SYZ base.

Kontsevich-Soibelman provided a universal wall-crossing formula andasserted that the scattering of holomorphic disks obey the formula.
I Gross-Siebert developed a reconstruction program of mirrors which uses
the wall crossing formula to capture quantum corrections combinatorially.
I Chan-Cho-Lau-Tseng computed all the quantum corrections for toricCY via the mirror map. Chan-Lau-Leung-Tseng found a relation with
Seidel representations for semi-Fano toric manifolds.

I5. Non-Kähler Calabi-Yau Manifolds

Non-Kähler Calabi-Yau Manifolds
It is interesting to also consider manifolds that are not Kähler but whose
canonical bundle is still trivial. These non-Kähler, almost Hermitian manifolds
with c1 = 0 are referred to as non-Kähler Calabi-Yau manifolds. Interest inthem within the context of complex threefolds dates back to the mid-1980s.
In 1987, Reid made a proposal (commonly called Reid’s fantasy) that all
Calabi-Yau threefolds, that can be deformed to a Moishezon manifold, fit intoa single universal moduli space where Calabi-Yaus of different homotopy types
are connected to one another through conifold transitions. These conifold
transition made use of the work of Clemens and Friedman: first contract
disjoint rational curves which results in a Calabi-Yau threefold withdouble-point singularities and then deform the singular space into a smoothcomplex manifold.
The complex manifold that results from a Clemens-Friedman conifold howeverneed not be Kähler. For instance, we may contract enough rational curves
such that the second Betti number becomes zero. The smoothed-out complex
manifold then can only be non-Kähler. In fact, it would be diffeomorphic to ak-connected sum of S 3 × S 3 with k > 2.

Non-Kähler Calabi-Yau manifolds and the Strominger system

For non-Kähler Calabi-Yaus, we would like to impose some additional
geometric conditions which replaces the Kähler condition. One such set of
conditions is that from a system of supersymmetry equations from heterotic
strings known as the Strominger system. This system requires theconsideration of also a stable bundle over the complex manifold. The
hermitian metric is required to satisfy the balanced condition plus an anomaly
equation that relates the hermitian metric with the Hermitian-Yang-Mills
metric on the stable bundle.
Many solutions of the Strominger systems on complex threefolds are now
known. Fu-Yau (2006) gave the first solution on a compact, non-Kähler
threefold – a torus bundle over a K 3 surface. Recently, Fei-Huang-Picard(2017) found Strominger system solutions on compact, non-Kähler threefolds
of infinitely many topological types and Hodge numbers.

Non-Kähler symplectic Calabi-Yaus and mirror symmetry

For symplectic manifolds, we can use a compatible almost complex structure
on it to define a first Chern class. A symplectic manifold is called symplectic
Calabi-Yau if its first Chern class is trivial. Symplectic Calabi-Yau manifolds
are generally non-Kähler manifolds. Consideration of such non-Kähler
symplectic Calabi-Yau spaces is also natural as mirror duals of non-Kähler
complex Calabi-Yau spaces.
Indeed, a symplectic mirror of the Clemens-Friedman’s conifold transition forcomplex threefolds was proposed by Smith-Thomas-Yau in 2002. In thesymplectic version, disjoint Lagrangian three-spheres of a Kähler Calabi-Yau
would be collapsed and replaced by symplectic two-spheres. The resulting real
six-manifold would be still symplectic but its third Betti number may be zero.
Hence, such a manifold would be in general be non-Kähler, but nevertheless, a
sympletic Calabi-Yau.

Smith-Thomas-Yau used such symplectic conifold transitions to construct
many real six-dimensional symplectic Calabi-Yaus. They showed that if such a
conifold transition collapsed all disjoint three-spheres, then the resulting space
is a manifold that is diffeomorphic to connected sums of CP3 . This mirrors
the complex case, where after the collapsed of all disjoint rational curves gives
a connected sums of S 3 × S 3 .
Symplectic Calabi-Yaus can also be considered from the perspective of SYZ
mirror symmetry of non-Kähler Calabi-Yaus. In the semi-flat limit,
Lau-Tseng-Yau (2015) related symplectic, non-Kähler, SU(3)supersymmetry conditions of type IIA strings to those of complex, non-Kähler,
SU(3) supersymmetry conditions of type IIB strings via SYZ and
Fourier-Mukai transform. In real dimensions eight and higher, they alsopresented a mirror pair system of equations - a symplectic system and acomplex system for non-Kähler Calabi-Yaus - that are dual to each other
under SYZ mirror symmetry.

6. Calabi-Yau ConesCalabi-Yau Cones

A Kähler manifold (X 2n , J, g ) is called a Kähler cone if
I

There is a function r : X → R>0 so thatg = dr 2 + r 2 ḡfor some metric ḡ on the Link of the cone, {r = 1}.
I The vector fields r ∂r , and ξ := J(r ∂r ) are real holomorphic.
√
I In particular, r ∂r −−1ξ generates a (C∗ )k action on X for some k > 1.
(X 2n , J, g ) is a Kähler cone if and only if the link (S, g ) is a Sasakian
manifold with Reeb vector field ξ.

Calabi-Yau Cones

TheoremIf (X 2n , J, g ) is a Kähler cone manifold, then (X 2n , J) can be embedded as anaffine variety in CN for some large N, so that ξ is induced by a holomorphic
vector field in u(N).
Motivated by the AdS/CFT correspondence, it is natural to ask:

Question
When does (X 2n , J, g ) admit a Ricci-flat Kähler cone metric?
As in the compact case, there are restrictions on which X we can consider. X
is admissible if :⊗`I X is Q-Gorenstein (ie. K
X = OX for some ` > 0)
I X has at worst log-terminal singularities.

Calabi-Yau Cones

In fact, there are many examples of such affine varieties, defined by weighted
homogeneous polynomials.

ExampleFor each p, q the affine variety Zp,q = {xy + z p + w q = 0} ⊂ C4 is
Q-Gorenstein, log terminal, with a Reeb vector field ξp,q induced by the action
λ.(x, y , z, w ) = (λpq x, λpq x, λ2q z, λ2p w )Example
If Y is a projective variety with −KY ample, then X = KY \{0} is admissible,
with C∗ action by scaling the fibers. X admits a Ricci-flat cone metric withrespect to the scaling Reeb field if and only if Y admits a K”ahler-Einstein
metric with positive Ricci curvature.
Kähler-Einstein metrics with positive Ricci curvature are obstructed in general.

Conjecture (Yau-Tian-Donaldson)
A Kähler manifold Y admits a Kähler-Einstein metric with positive Ricci
curvature if and only if (Y , −KY ) is K-stable.
The notion of K -stability is purely algebraic.

Theorem (Chen-Donaldson-Sun)
The Yau-Tian-Donaldson conjecture is true.

For conical Ricci-flat metrics:
I Martelli-Sparks-Yau found obstructions to existence of Ricci-flat cone
metrics, and explained how these obstructions could be interpreted interms of the AdS/CFT dual field theory.
I Futaki-Ono-Wang proved that Ricci flat Kähler cone metrics exist on alladmissible toric varieties, for appropriately chosen Reeb field.
I Collins-Székelyhidi found a notion of K-stability for Ricci-flat cone
metrics on an admissible affine variety X .
I If X = {f1 = · · · = fn = 0} ⊂ CN , K -stability is defined in terms of subtle
algebraic properties of the ringC[x1 , . . . , xN ](f1 , . . . , fn )graded by the Reeb vector field.

Theorem (Collins-Székelyhidi)
The affine variety X , with Reeb vector field ξ, admits a Ricci-flat Kähler cone
metric if and only if (X , ξ) is K -stable.
Collins-Székelyhidi show that the affine variety
Zp,q = {xy + z p + w q = 0} ⊂ C4 with Reeb vector field a multiple of ξp,q is
K -stable if and only if 2p > q and 2q > p.
Using the AdS/CFT correspondence, Collins-Xie-Yau proposed aninterpretation of K -stability in field theory terms. The interpretation is interms of the chiral ring and a generalized central charge maximization.

7. Stability in Calabi-Yau Geometry

Stability conditions

Motivated by Π-stability of D-branes studied by Douglas, in 2007,
Bridgeland introduced the notion of stability conditions on a triangulated
category D.
I Given a Calabi–Yau manifold X , we can associate two triangulated
categories to it:
I (A): The derived Fukaya category Fuk(X ), which only depends on thesymplectic structure on X .
I (B): The derived category of coherent sheaves Db Coh(X ), which only
depends on the complex structure on X .
I Bridgeland stability conditions are supposed to recover the “other half” of
the geometric structure from the category: Stability conditions on Fuk(X )recover the complex structure, and stability conditions on Db Coh(X )recover the symplectic structure.
I

Deformed Hermitian–Yang–Mills equation
To study Bridgeland stability conditions on Db Coh(X ), we consider the
mirror of special Lagrangian equation, which is the deformed
Hermitian–Yang–Mills equation.
I Consider a holomorphic line bundle L over X (supposedly dual to a
Lagrangian section in the mirror).

I

IQ) Does there exist a metric h on L such thatIme −i θ̂ (ω − F )n = 0,
where F = −∂ ∂¯ log h, and e i θ̂ ∈ S 1 is a topological constant determinedby ω, c1 (L)?

I

Note e i θ̂ ∈ S 1 is determined byZ√(ω + −1c1 (L))n ∈ R>0 e i θ̂X

I

Jacob-Yau and Collins-Jacob-Yau studied the existence and theuniqueness of the deformed Hermitian-Yang-Mills equation.

Lagrangian phase θ of line bundles

I

Consider the relative endomorphism K of T 1,0 (X ) given by
K :=

I

√

−1g j k̄ Fk̄l∂
⊗ dz l .
∂z j

In terms of normal coordinates with gk̄j = δkj and Fk̄j =eigenvalue of K ), definePΘω (h) = j arctan(λj ).

√−1λj δkj (λj =The dHYM equation is θ(h) = Θ for a constant Θ ∈ (−n π2 , n π2 ).
I If there exists a solution of dHYM, then the constant Θ is unique, andlifts θ̂ to R: e i θ̂ = e iΘ .
I Jacob–Y.: For given a line bundle L on X , suppose that (L, h) has
oscX Θω (h) < π, then there exists a unique lift Θ ∈ R of θ̂ ∈ [0, 2π).

IAnalysis on the phase of dHYM equation

Theorem (Collins–Jacob–Y.)
Suppose that the lifted angle Θ satisfies the critical phase conditionπΘ > (n − 2) .
2
Then there exists a solution to the deformed Hermitian-Yang-Mills equation if
and only if there exists a metric h on L → X so thatπI Θω (h) > (n − 2) , and2
I For every 1 6 k 6 nXπarctan(λj (h)) > Θ −
2
j6=k

√
where λj (h) are the eigenvalues of −1g j k̄ F (h)k̄l .
Furthermore, the solution is unique up to multiplication by a positive constant.

Obstructions to the existence of solutions of dHYM

From the conditions in the theorem, we find obstructions to the existence of asolution:

Theorem (Collins-Jacob-Y.)
If there exists a solution to the dHYM equation, then for every proper,
irreducible analytic subvariety V ⊂ X with dim V = p we have
ZV (L)Im>0ZX (L)√√
R
R
where ZV = V e − −1ω ch(L), and ZX = X e − −1ω ch(L).
Note: (1) This obstruction appears to be related to Bridgeland stability.
(2) In dimension 2, these inequalities are equivalent to existence of a solution
to the dHYM equation (Collins-Jacob-Y.).

Thanks!