Ved V. Datar Research Statement

My research interests center around geometric analysis, and I am especially interested in the interplay between Riemannian, complex and algebraic geometry. My past and ongoing research projects are summarized below. • Existence and degeneration of smooth and singular K¨ahler-Einsteinmetrics, and K¨ahler metrics of constant scalar curvature, and their relation to various notions of stability.

• Gluing problems related to constructing extremal or constant scalar curvature K¨ahler metrics on blow-ups.

• Adiabatic limits of sequences of ASD or Hermitian-Yang-Mills connections with poten- tial applications to mirror symmetry.

• Existence and regularity of solutions to complex Monge-Amp`ereequations. I will now expand on the above points and propose some questions for future investigation.

1. Extremal Kahler¨ metrics. Recall that a Kählermanifold M is a complex manifold with a closed, real, positive (1, 1) form, called the Kählerform or the Kählermetric. The Kählermetric canonically induces a Riemannian metric such that the complex structure is parallel with respect to the Levi-Civita 2 connection. Being a closed form, it determines a cohomology class in H (M, R), also called a Kählerclass. An extremal Kählermetric [9] is a critical point of the Calabi functional Z n 2 ω Ca(ω) = sω M n! as ω varies in a fixed Kählerclass. Here sω denotes the scalar curvature of the Riemannian metric associated to the Kählerform ω. The Euler-Lagrange equation is 1,0 ∂¯∇ sω = 0, 1,0 where ∇ sω is the (1, 0) part of the gradient. An especially important special case is that of constant scalar curvature Kähler(cscK) metrics. The guiding conjecture in the field is Conjecture 1 (Yau-Tian-Donaldson [61, 51, 16]). Let L be an ample line bundle on M. Then M admits a cscK metric in c1(L) if and only if the pair (M,L) is K-stable. One can make a similar conjecture about extremal metrics and relative K-stability [47]. K- stability involves studying degenerations of the manifold to normal Q-Fano varieties. More N k ∗ precisely, consider an embedding M,→ P k given by sections of L for large k, and let λ : C → GL(Nk + 1, C) be a one-parameter subgroup with M0 = limt→0 λ(t) · M being the flat limit. ∗ ∗ λ(t) then induces a C action on M0. One can then associate a number to the C action, called the Futaki invariant Fut(M0, λ). The pair (M,L) is called K-semistable if Fut(M0, λ) ≥ 0 for all ∗ such embeddings and C actions, and K-unstable otherwise. The pair is called K-(poly)stable if equality holds only when M is biholomorphic to M0. Note that in order for the above conjecture to have any hope of being true, the notion of K-stability needs to be refined, but this will not be required in what follows.

1.1. Kähler-Einsteinmetrics on Fano manifolds. A special class of cscK metrics are Kähler-Einstein (KE) metrics, that is Kählermetrics ω whose Ricci form satisfies (1) Ric(ω) = λω 1 2

for some λ = −1, 0, 1. Since Ric(ω) represents the first Chern class c1(M), a necessary condition is that c1(M) either vanishes or has a sign. This trichotomy is reminiscent of the uniformization theorem for Riemann surfaces. It has been known since the 1970s that KE metrics always exist on Kählermanifolds with c1(M) = 0 (by Yau [60]) and c1(M) < 0 (by Aubin [5] and Yau [60]). The remaining case of c1(M) > 0 was settled only recently by the deep work of Chen-Donaldson-Sun [10] and Tian [53]. They proved that a Fano manifold M admits a KE −1 metric if and only if the pair (M,KM ) is K-stable, thus confirming conjecture1 in the case −1 that L = KM . Although this was a major theoretical breakthrough, it did not yield any new examples of KE manifolds. The main reason is that it seems intractable at the moment to check K-stability directly, since in principle one has to study an infinite set of degenerations. To improve upon this, in collaboration with Gábor Székelyhidi we proved an equivariant version of conjecture1 for KE metrics.

−1 Theorem 1.1 (D.-Székelyhidi [22]). Let G ⊂ Aut(M) be a compact group. If (M,KM ) is K- polystable with respect to special degenerations that are G-equivariant, then M admits a Kähler- Einstein metric. The advantage of the above theorem is that if the symmetry group is large enough, the number of G-equivariant degenerations can be cut down to only a finite number of possibilities. For instance, if M is toric of dimension n with G = (S1)n, the only equivariant degenerations are the trivial ones. Thus in this case equivariant K-stability reduces to vanishing of the classical Futaki invariant, and Theorem 1.1 recovers a well known result of Wang and Zhu [57]. New examples of KE Fano manifolds were subsequently found in [35] and [26] using Theorem 1.1. In contrast to [10], where a continuity method through conical KE metrics is used, we are forced to use the classical smooth continuity method of Aubin and Yau. We in fact prove a more general theorem relating twisted Kähler-Ricci solitons to the so-called modified G-equivariant K-stability introduced in [27,7]. For a smooth form α ∈ c1(M), and a holomorphic vector field v on M, we say that (M, (1 − t)α, v) admits a Kähler-Ricci soliton if there exists an ω ∈ c1(M) solving

(2) Ric(ω) = tω + (1 − t)α + Lvω, where Lv denotes the Lie derivative. Theorem 1.2 (D.-Székelyhidi [22]). If (M, (1 − s)α, v) is K-semistable for all G-equivariant special degenerations, then (M, (1 − t)α, v) admits a twisted Kähler-Ricci soliton for all t < s. In addition if (M, v) is K-stable, then (M, v) admits a Kähler-Ricci soliton. Even in the case when G is trivial, this result is new. Finally, we can also use Theorem 1.1 to give an algebro-geometric interpretation of the greatest Ricci lower bound invariant of Szëkelyhidi. Recall that for a Fano manifold M we define R(M) := sup {t | (2) has a solution with v = 0}. t∈(0,1) In [46] Székelyhidi showed that R(M) is an invariant independent of the choice of α. By Theorem 1.2, this invariant is equivalent to the supremum taken over all t such that the pair (M, (1−t)α) is K-semistable. In [22] we use this to compute R(M) for toric manifolds, giving a purely algebro-geometric proof of a result of Chi Li [38]. A crucial ingredient in all the existence proofs of Kähler-Einsteinmetrics is an effective version of Kodaira’s embedding theorem, the so-called partial C0 estimate [50, 17]. To prove such an estimate it is necessary to study degenerations of KE metrics, and apply the convergence theory 3 of Cheeger and Colding. A central difficulty in extending this to a sequence of cscK metrics is the possibility of collapse. In the special case of Kählersurfaces with an added non-collapsing hypothesis (say a bound on the Sobolev constant) and some mild topological constrains, one can show that the limit is an orbifold [2, 52]. It is then natural to ask the following.

Question 1. Is there a uniform partial C0 estimate along a sequence of non-collapsed, cscK metrics on Kählersurfaces with uniform bounds on the total volume, scalar curvature and some 2 mild topological constrains (say bounds on c1 and c2)? Note that on Kählersurfaces, a bound on the scalar curvature and Chern numbers immediately implies a bound on L2 norm of the curvature. The main difficulty now is that there is no Ricci lower bound, and one cannot directly apply the Hörmander’s L2 estimates as in [17]. Moving over to the general case, as first step towards understanding possibly collapsed limits of sequences of cscK metrics on Kählersurfaces, I wish to study the following question.

Question 2. Can one extend the ε-regularity theorem of Cheeger-Tian [13] to possibly collapsed cscK metrics on unit volume K¨ahlersurfaces with some mild topological constrains?

1.2. Extremal metrics on blow-ups. The conjecture1 in full generality is still wide open. In light of this, perturbation problems, such as the following question have received considerable attention in recent years.

Question 3. Suppose M admits an extremal Kählermetric in the class c1(L) for some ample line bundle L. Let π : BlpM → M be the blow-up at p ∈ M with exceptional divisor E. If we ∗ 2 let Lε = π L − ε E, when does (BlpM,Lε) admit an extremal metric? Some general sufficient conditions have been obtained by Arezzo-Pacard [3] and Arezzo- Pacard-Singer [4]. But in applying their results, one often has to blow-up multiple points at the same time. Instead in [48] Székelyhidi proved a version of conjecture1 for blow-ups of extremal manifolds for dimC(M) > 2. It turns out that one only needs to look at degenerations ∗ of BlpM generated by C actions on M. In [23] we have made an attempt to extend this result to dimC(M) = 2. The main difficulty is to construct better approximate solutions than the ones needed by Székelyhidi in [48]. To state our main theorem, we set up some notation.

• Let G ⊂ Aut0(M) be (connected) group of Hamiltonian isometries with Lie(G) = g and moment map µ : M → g∗. • Choosing an inner product on g and identifying g with the corresponding mean-free Hamiltonian functions Ham0(M, ω), we can think of µ : M → Ham0(M, ω). • Let T ⊂ G be any torus, and H it’s centralizer in G with Lie(T ) = t and Lie(H) = h.

The strategy now [4, 48] is as follows. We fix a family of background metrics ω ∈ c1(Lε), and ∞ define a lifting function l : h → C (BlpM) such that for any f ∈ t, ∇ω l(f) is a holomorphic vector field on BlpM. Then √ Ω = ω + −1∂∂u is an extremal metric if and only if there is a f ∈ h solving s(Ω) = l(f) + Re(∇l(f) · ∇u)(3) (4) f ∈ t Solving (3) is relatively easy; see for instance [4]. The difficult part is going from (3) to (4). This is essentially a finite dimensional problem, and here K-stability plays a role. The main observation of Székelyhidi in [48] is that K-stability of (BlpM,Lε) for all small ε > 0 implies 4

a certain finite dimensional GIT stability relative to the polarization L + δKM for small δ. In order to use this, one obtains an expansion of f. For dimC(M) := m > 2 it is shown in [48] that 2 2m−2 2m κ fp = sω + (c1 − c2ε )ε µ(p) + c3ε ∆µ(p) + O(ε ) for some κ > 2m, and constants c1, c2 and c3. The key term is the ∆µ(p) term. In dimension two, from general algebro-geometric considerations it is expected that there will be no ∆µ(p) term of order ε2m = ε4. I was able to confirm this in [23]. In fact, even in obtaining the ε4 term, there are new analytic difficulties to overcome.

Theorem 1.3 (D. [23]). Let (M, ω) be a constant scalar curvature K¨ahlersurface, and let T ⊂ G be a non-trivial torus. With ω and Lε as above, there exists an ε0 depending only M, ω and T such that the following holds. For any ε ∈ (0, ε0) and p ∈ M ﬁxed by T , there exists a ∞ T u ∈ C (BlpM) , and an f ∈ h solving (3) such that f has the expansion s f = s − 2πε2(V −1 + µ(p)) + ω ε4(V −1 + µ(p)) + O(εκ), ω 2 for some κ > 4. Here V denotes the volume of (M, ω), and the constant in O(εκ) depends only on M, ω and T .

The f above is related to the extremal vector ﬁeld of Futaki and Mabuchi [31] on the blow-up. Based on some algebro-geometric computations [48, 25], we have the following conjecture.

Conjecture 2 (D. [25]). With notation as in the theorem above, there exists a solution f with the expansion ε2s ε4π ε6s f = s − ε2 2π − ω + + ε4||µ(p)||2 (V −1 + µ(p)) + ω ∆µ(p) + O(εκ), ω 2 V L2 6 for some κ > 6.

At least when sω 6= 0, resolution of this conjecture should directly lead to a generalization of Sz´ekelyhidi’s theorem to K¨ahlersurfaces.

1.3. Further questions. The general Question3 for extremal metrics is still open even in higher dimensions. On would expect that (BlpM,Lε) admits an extremal metric if it is relatively stable with respect to a maximal torus T ⊂ Aut(M), and with respect to degenerations corresponding ∗ to C actions on M. The diﬃculty here is to relate relative K-stability of (BlpM,Lε) to the relative GIT stability of the point p with respect to the polarization L+δKM . With the notation as in the above section, I was able to obtain the following reﬁnement of a theorem in [45].

Theorem 1.4 (D. [25]). Let m > 2, (M, ω) be extremal, and let p ∈ M such that ∇sω(p) = 0. c There exists a δ0 with the following property. If for all δ ∈ (0, δ0), there is no q ∈ G · p such that µ(q) + δ∆µ(q) ∈ gq, then (BlpM,Lε) is relatively unstable for all suﬃciently small ε > 0. Here gq is the stabilizer of q in g.

The hypothesis in the result implies that p is relatively strictly unstable in the GIT sense. In order to address question3, one would need to weaken the hypothesis to relative non- polystability. As suggested in [48], an alternate approach that I am actively exploring is to interpret fp itself as a moment map on the set of T -invariant points p, with respect to a per- turbed K¨ahlerform. 5

Next, one can also explore an analog of question3 for the Calabi flow on blow-ups. Recall that on the level of Kählerpotentials, the Calabi flow is given by ∂ϕ = S − S,¯ ϕ(0) = 0, ∂t t √ where St is the scalar curvature of the metric ω(t) = ω0 + −1∂∂ϕ(t), and S¯ is the average of St, and is a topological constant depending only on [ω0] and c1(M). Apart from short term existence, and a few concrete examples, not much is known about the Calabi flow.

Question 4. Let Aut(M) be discrete, and let ωε ∈ Lε be the family of K¨ahlermetrics on BlpM considered above. Is there a long-time solution to the Calabi ﬂow starting from ωε, for ε << 1? If so, does it converge in the limit to the cscK metrics constructed in [3].

A possible approach is to adapt the arguments in [11] to blow-ups. Another question that I am very interested in investigating in the future is the following.

Question 5 ([49]). Given any K¨ahlermanifold X, can one blow up enough number of points so that the resulting K¨ahlermanifold admits a cscK metric?

An aﬃrmative answer would be analogous to a beautiful result of Taubes’ on anti-self dual metics on connected sums [56].

2. Adiabatic limits of HYM metrics The Strominger-Yau-Zaslow picture of mirror symmetry postulates that mirror Calabi-Yau manifolds are given by compactifying dual torus fibrations over a real base with a singular affine structure. On hyperkählermanifolds, such fibrations often become apparent by taking limits of Ricci flat metrics [32, 54, 33, 55]. Vafa’s extension of the mirror symmetry conjecture [59] to holomorphic bundles, raises a question of how Yang-Mills connections behave under degenerations. A rather general conjectural picture is outlined by Fukaya in [30]. In [24], working with Adam Jacob, we consider a holomorphic SU(n) bundle E over an elliptically fibered K3 1 surface π : X → P . We require E to satisfy the following. ∗ (1) There exists an ε0 such that E is stable with respect to the Kählerclass [ˆωε0 ] := π [ω0]+ 1 ε[ωX ], where ω0 is a metric on P and ωX is a Ricci flat metric on X. (2) The restriction of E to all smooth fibers is semi-stable, and the restriction to a generic fiber admits a flat SU(n) connection. By Friedman-Morgan-Witten [28], the second condition is satisfied by generic holomorphic stable bundles on X, and hence is sufficiently mild. We have the following important consequence of the two conditions.

Lemma 2.1 (D.-Jacob, [24]). E is stable with respect to [ˆωε] for all ε ∈ (0, ε0).

Now ﬁx a hermitian metric H0 on E and let Ξ0 be the corresponding Chern connection. By Yau’s resolution of the Calabi conjecture [60], there exists a unique Ricci ﬂat metric ωε ∈ [ˆωε]. Since E is stable with respect to [ˆωε], it follows [15] that there exist connections Ξε complex gauge equivalent to Ξ0, and solving the Hermitian-Yang-Mills equation

FΞε ∧ ωε = 0, where FΞε is the curvature of the connection Ξε. Our main theorem is as follows. 6

1 Theorem 2.1 (D.-Jacob, [24]). For any sequence εk → 0, there exists points b1, ··· bN ∈ P 1 such that for any b ∈ P \{b1, ··· bN }

2 lim ||Ξεk |Eb − Ab||L (E ,H ,g ) = 0. k→∞ 1 b 0 0

Here Eb denotes the fiber over b, g0 is a fixed Riemannian metric on Eb, and Ab is the flat connection on Eb uniquely determined by the holomorphic structure on E. Remark 2.2. (1) We do not need to use gauge transformations in the above theorem. Al- lowing for gauge transformations, we can in fact obtain smooth convergence. −1 (2) In [33] it is proved the restriction of ε ωε to a generic fiber converges to flat metric on the fiber, and so our theorem can be thought of as a vector bundle analog. For SU(2) bundles a similar result was obtained by Nishinou [39]. There are two main components of our proof - a bubbling argument and a new gauge fixing theorem. Based on the work of Dostoglou and Salamon [18, Section 9], it can be expected that after a suitable rescaling, the bubbles can be classified into three types (1) An instanton on S4. (2) An instanton on C × E, where E is an elliptic curve. (3) A holomorphic sphere in the moduli space of flat connections on E. In our case we can show that the bubbles of the first two types can occur only near a finitely many fibers. Working away from these, the curvature of Ξε|Eb goes to zero. Together with our gauge fixing result, we can establish the required convergence. We expect that the type 3 bubbles should also occur at only a finitely many fibers. Proving this would immediately imply that the curvature of the full connection is at most O(ε−1/2), away from finitely many fibers, and consequently we would also obtain a precise rate of convergence of Ξε|Eb to A0. In fact the following stronger result has been conjectured by Fukaya in [30], and we wish to address this in the future.

Question 6. With the same set-up as above, can one choose points b1, ··· bN such that for any −1 1 ∞ compact set K ⊂ π (P \{b1, ··· , bN }), there is a constant such that ||FΞε ||L (K,g0,H0) ≤ C? 2.1. Further questions. A natural question is whether one can improve the fiber-wise conver- 1 gence to a global convergence on compact subsets of P \{b1, ··· bN } for some points b1, ··· bN ∈ 1 P . Taking a slightly different perspective, instead of considering connections complex-gauge equivalent to Ξ0, one can fix the holomorphic structure, and consider Hermitian-Einstein metrics Hε. 0 1 Question 7. With the same setting as above, can one obtain uniform C or L2 bounds on Hε, where the norms are measured with respect to a fixed Hermitian metric H0? Such bounds would imply an absence of bubbling (cf. [29]), and would in particular lead to convergence of the entire family (and not only sequential convergence as in Theorem 2.1). It is particularly interesting to know the role that stability will play in obtaining such an estimate. Finally, one could drop the assumption that all the connections are complex-gauge equivalent, and consider a general family of ASD connections. As a first step towards extending our results to this more general setting, we are working on proving a stronger gauge fixing result that doesn’t rely on complex gauge equivalence of the connections.

3. Conical Kahler-Einstein¨ metrics This section summarizes the work done as a graduate student. 7

3.1. Complex Monge-Ampèreequations. Complex Monge-Ampèreequations play an important role in the study of Kähler-Einsteinmetrics. Let (M, ω) be a Kählermanifold and consider the following equation for ϕ ∈ L∞(X, ω) √ −γϕ √ n e Ω (5) (ω + −1∂∂ϕ) = , ωϕ := ω + −1∂∂ϕ > 0, QN |s |2(1−βj ) j=1 j hj where Ω is a smooth volume form, sj is a section of a line bundle Lj and hj is a Hermitian metric on Lj. Suppose the divisor Dj cut-out by sj is smooth. For technical reasons additionally we P also assume that the divisor D = (1 − βj)Dj is Kawamata log-terminal (that is, βj ∈ (0, 1)) and simple normal crossing. The Ricci curvature of ωϕ then satisfies

Ric(ωϕ) = γϕ + [D] + χ, where χ is some smooth form (depending on Ω and hjs), and [D] is the current of integration along D. A local model for ωϕ is a metric with cone singularities. Recall that a flat cone metric on 2 2 2 2 2 C, in polar coordinates,√ is given by ds = dr + β r dθ and the corresponding Kählerform in complex coordinates is −1∂∂¯|z|2β. A Kählercurrent is quasi-isometric to a cone metric along D if it is smooth on X \ D with globally bounded potentials, and locally at any point p ∈ D, it is equivalent to an edge metric of the form √ k √ N X −2(1−βj ) X −1 |zj| dzj ∧ dz¯j + −1 dzj ∧ dz¯j. j=1 j=k+1 When the divisor has only one component, there are very precise regularity results for conical KE metrics [8, 36]. In the case when D has more than one component, essentially the first progress was made by the following theorem of Guenancia-P˘aun[34]. With Jian Song [20], I was able to provide a much shorter proof of this.

Theorem 3.1. Any solution ωϕ to equation (5) is quasi-isometric to a cone metric along D.

Our idea of the proof was to approximate ωϕ by smooth metrics ωη in such a way that we could reduce the theorem to the case when D has only one component. The convergence that we obtain is smooth on compact subsets of X \ D. In [21] I was able to prove a more global convergence result.

Theorem 3.2 (D., [21]). Let ωϕ be a solution of (5), and let dϕ be the induced distance function on the open manifold X \ D. Then there exist uniform constants A, Λ 1, and a sequence ωη ∈ [ω] of smooth K¨ahlermetrics such that

(1) Ric(ωη) > −Aωη ; diam(X, ωη) < Λ. (2)( X, ωη) converges to (X, d) in the Gromov-Hausdorff sense, where as above, (X, d) is the metric completion of (X \ D, dϕ). (3) Xreg = X \ D, where Xreg is the regular set in the sense of Cheeger-Colding consisting n of points whose tangent cones are isometric to C . In particular Xreg is open and dense. Such a result was obtained for smooth anti-canonical divisors in [10]. The main difficulty is that the diameter bound is not as direct, and in fact uses pointed Gromov-Hausdorff convergence along with knowledge that X \ D can be identified as a subspace in the limit. Using Theorem 3.2, since Xreg is open and D is of real co-dimension two, combining with the results in [14] one can immediately conclude that Xreg is convex. 8

Theorem 3.3 (D., [21]). (X \D, dϕ) is geodesically convex (in the sense of Colding-Naber [14]). As a consequence the classical comparison theorems such as Laplacian comparison, Bishop- Gromov and Myers, generalize to the conical setting.

This theorem has proved useful (cf. [19] and [40]) in studying the deformation of conical K¨ahler-Einsteinmetrics along possibly non-smooth simple normal crossing divisors.

3.2. Connecting toric manifolds by Kähler-Einsteinmetrics. Recall that a toric manifold n 1 n X of dimension n is a complex manifold with a Hamiltonian action of the torus, T ≈ (S ) , ∗ n which induces a holomorphic action of (C ) , with a free, open and dense orbit X0 ⊂ X.A n n toric manifold is determined by a Delzant polytope P ⊂ R , and in fact X0 ≈ P × T , where n the identification to the first factor is given by a moment map for the action of T . Conversely, if we fix a Kählerclass α on the toric manifold, then the polytope is also uniquely determined modulo translations. The smooth toric divisors correspond to the co-dimension one faces of the polytope.

2 2 Figure 1. The polytopes for P and Blp(P ) corresponding to the anti-canonical classes 3[H] and 3[H] − [E] respectively, where H is the hyperplane at inﬁnity and E is the exceptional divisor.

Let D be a simple normal crossing toric divisor on X. We then say that (X,D) is a log-Fano P pair if D = ajDj with aj ∈ [0, 1) and | − (KX + D)| is an ample class. Here −KX is the anti- canonical class of X. If (X,D) is a toric log-Fano pair, then a Kählercurrent ω ∈ | − (KX + D)| is said to be a toric conical Kähler-Ricci soliton if there exists a toric holomorphic vector field ξ ∈ H0(X,T 1,0X) solving

(6) Ric(ω) = ω + Lξω + [D] where [D] is the current of integration along D. The current ω is called a toric conical Kähler- Einstein metric if one can choose ξ = 0. In collaboration with Bin Guo, Jian Song and Xiaowei Wang [19], we were able to show that all toric log-Fano pairs admit a conical Kähler-Riccisoliton, and we classified the ones admitting a toric conical KE metric. This is a complete generalization of a fundamental theorem of Wang-Zhu [57] (cf. also [6, 37, 42] for related results).

Theorem 3.4 (D.-Guo-Song-Wang, [19]). Let (X,D) be a toric log Fano pair, where D = PN j=1(1 − βj)Dj. (1) Then (X,D) admits a toric conical K¨ahlerRicci soliton. Moreover, if we set n P = {x ∈ R | lj(x) = vj · x + βj}, Pn i ∂ then the vector ﬁeld ξ is given by ξ = i=1 ciz ∂zi where ~c = (c1, ··· , cn) is the unique solution to Z xec·x dx = 0 P 1 n ∗ n and (z , ··· , z ) are the standard coordinates on (C ) ≈ X0 ⊂ X. 9

2 2 Figure 2. The degeneration of BlpP to P at the level of the polytopes, or equivalently at the level of the K¨ahlerclasses

(2) Consequently, (X,D) admits a conical Kähler-Einstein metric (i.e. ~c = 0) if and only if the barycenter of P is the origin. (3) In particular, any toric manifold admits a conical Kähler-Einsteinmetric for a suitable choice of divisor D. Our main result in [19] demonstrates that any two toric manifolds of the same dimension can be connected by a family of conical KE toric manifolds. Moreover, this family is continuous in the Gromov-Hausdorff topology.

Theorem 3.5 (D.-Guo-Song-Wang, [19]). Let X0 and X1 be two n-dimensional toric manifolds. Then, there exist a family {(Xt, ωt)}t∈[0,1] of n-dimensional toric manifolds Xt with toric conical KE metrics ωt for t ∈ [0, 1], such that (Xt, ωt) is a continuous path in Gromov-Hausdorff topology for t ∈ [0, 1]. 3.3. Further questions. The above theorem can be considered as a differential geometric counterpart to the weak factorization theorem of Abramovich et al.[1] that toric manifolds can be connected by a sequence of blow-ups or blow-downs. A natural next step is to extend Theorem 3.5 to del Pezzo surfaces, or to more general families of birational manifolds. As a first step, we would like to address the following question. Question 8. Suppose X is a Kählermanifold admitting a conical KE metric ω ∈ α with singularities along a simple normal crossing divisor. Let π : X˜ → X be a blow up along some sub-variety. (1) Does X˜ admit a conical KE metric in α − [E] for small > 0? (2) Denoting the conical KE metrics by ω, does (X,˜ ω) → (X, ω) in the Gromov-Hausdorff distance?

4. Conclusion My research has focussed on existence and degeneration of smooth and singular Káhler- Einstein metrics, extremal Kählermetrics and Hermitian-Yang-Mills connections. I have suc- ceeded in solving some open problems, while my investigations have also led to several other interesting questions that I wish to explore in the future. Along the way, I have learnt some tech- niques from complex Monge-Ampèreequations, Ricci flow, Yang-Mills theory, toric geometry and Riemannian geometry, including structure theory of Gromov-Hausdorff limits. I would like to continue some of these threads, and also branch out into other subjects such as Kähler-Ricci flow and the analytic minimal model program of Tian-Song [41], and the applications of complex geometric methods such as L2-estimates to the study of stable minimal surfaces in manifolds with positive isotropic curvatures, and to problems in algebraic geometry in the spirit of [44].

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