Gluing problems related to constructing extremal or constant scalar curvature Kähler metrics on blow-ups.

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 Kähler-Einstein metrics, and Kähler metrics of constant scalar curvature, and their relation to various notions of stability. Gluing problems related to constructing extremal or constant scalar curvature Kähler metrics on blow-ups. Adiabatic limits of sequences of ASD or Hermitian-Yang-Mills connections with potential applications to mirror symmetry. Existence and regularity of solutions to complex Monge-Ampère equations. I will now expand on the above points and propose some questions for future investigation. 1. Extremal Kähler metrics. Recall that a Kähler manifold M is a complex manifold with a closed, real, positive (1, 1) form, called the Kähler form or the Kähler metric. The Kähler metric canonically induces a Riemannian metric such that the complex structure is parallel with respect to the Levi-Civita connection. Being a closed form, it determines a cohomology class in H 2 (M, R), also called a Kähler class. An extremal Kähler metric [9] is a critical point of the Calabi functional Ca(ω) = s 2 ω n ω n! as ω varies in a fixed Kähler class. Here s ω denotes the scalar curvature of the Riemannian metric associated to the Kähler form ω. The Euler-Lagrange equation is 1,0 s ω = 0, where 1,0 s ω 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 c 1 (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 precisely, consider an embedding M P N k given by sections of L k for large k, and let λ : C GL(N k + 1, C) be a one-parameter subgroup with M 0 = lim t 0 λ(t) M being the flat limit. λ(t) then induces a C action on M 0. One can then associate a number to the C action, called the Futaki invariant Fut(M 0, λ). The pair (M, L) is called K-semistable if Fut(M 0, 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 M 0. 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-Einstein metrics on Fano manifolds. A special class of csck metrics are Kähler-Einstein (KE) metrics, that is Kähler metrics ω whose Ricci form satisfies (1) Ric(ω) = λω 1 M

2 for some λ = 1, 0, 1. Since Ric(ω) represents the first Chern class c 1 (M), a necessary condition is that c 1 (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 Kähler manifolds with c 1 (M) = 0 (by Yau [60]) and c 1 (M) < 0 (by Aubin [5] and Yau [60]). The remaining case of c 1 (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 metric if and only if the pair (M, K 1 M ) is K-stable, thus confirming conjecture 1 in the case that L = K 1 M. 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 conjecture 1 for KE metrics. Theorem 1.1 (D.-Székelyhidi [22]). Let G Aut(M) be a compact group. If (M, K 1 M ) 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 = (S 1 ) 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 c 1 (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 c 1 (M) solving (2) Ric(ω) = tω + (1 t)α + L v ω, where L v 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-Einstein metrics is an effective version of Kodaira s embedding theorem, the so-called partial C 0 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 Kähler surfaces 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 C 0 estimate along a sequence of non-collapsed, csck metrics on Kähler surfaces with uniform bounds on the total volume, scalar curvature and some mild topological constrains (say bounds on c 2 1 and c 2)? Note that on Kähler surfaces, a bound on the scalar curvature and Chern numbers immediately implies a bound on L 2 norm of the curvature. The main difficulty now is that there is no Ricci lower bound, and one cannot directly apply the Hörmander s L 2 estimates as in [17]. Moving over to the general case, as first step towards understanding possibly collapsed limits of sequences of csck metrics on Kähler surfaces, 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 Kähler surfaces with some mild topological constrains? 1.2. Extremal metrics on blow-ups. The conjecture 1 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 Kähler metric in the class c 1 (L) for some ample line bundle L. Let π : Bl p M M be the blow-up at p M with exceptional divisor E. If we let L ε = π L ε 2 E, when does (Bl p M, L ε ) 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 conjecture 1 for blow-ups of extremal manifolds for dim C (M) > 2. It turns out that one only needs to look at degenerations of Bl p M generated by C actions on M. In [23] we have made an attempt to extend this result to dim C (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 Aut 0 (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 Ham 0 (M, ω), we can think of µ : M Ham 0 (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 ɛ c 1 (L ε ), and define a lifting function l : h C (Bl p M) such that for any f t, ωɛ l(f) is a holomorphic vector field on Bl p M. Then Ω = ω ɛ + 1 u is an extremal metric if and only if there is a f h solving (3) (4) s(ω) = l(f) + Re( l(f) u) 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 (Bl p M, L ε ) for all small ε > 0 implies

4 a certain finite dimensional GIT stability relative to the polarization L + δk M for small δ. In order to use this, one obtains an expansion of f. For dim C (M) := m > 2 it is shown in [48] that f p = s ω + (c 1 c 2 ε 2 )ε 2m 2 µ(p) + c 3 ε 2m µ(p) + O(ε κ ) for some κ > 2m, and constants c 1, c 2 and c 3. 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 ε 2m = ε 4. I was able to confirm this in [23]. In fact, even in obtaining the ε 4 term, there are new analytic difficulties to overcome. Theorem 1.3 (D. [23]). Let (M, ω) be a constant scalar curvature Kähler surface, and let T G be a non-trivial torus. With ω ɛ and L ε as above, there exists an ε 0 depending only M, ω and T such that the following holds. For any ε (0, ε 0 ) and p M fixed by T, there exists a u C (Bl p M) T, and an f h solving (3) such that f has the expansion f = s ω 2πε 2 (V 1 + µ(p)) + s ω 2 ε4 (V 1 + µ(p)) + O(ε κ ), for some κ > 4. Here V denotes the volume of (M, ω), and the constant in O(ε κ ) 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 f = s ω ε 2( 2π ε2 s ω + ε4 π ) 2 V + ε4 µ(p) 2 L (V 1 + µ(p)) + ε6 s ω 2 6 µ(p) + O(εκ ), for some κ > 6. At least when s ω 0, resolution of this conjecture should directly lead to a generalization of Székelyhidi s theorem to Kähler surfaces. 1.3. Further questions. The general Question 3 for extremal metrics is still open even in higher dimensions. On would expect that (Bl p M, L ε ) 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 (Bl p M, L ε ) to the relative GIT stability of the point p with respect to the polarization L+δK M. 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 s ω (p) = 0. There exists a δ 0 with the following property. If for all δ (0, δ 0 ), there is no q G c p such that µ(q) + δ µ(q) g q, then (Bl p M, L ε ) is relatively unstable for all sufficiently small ε > 0. Here g q 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 question 3, one would need to weaken the hypothesis to relative nonpolystability. As suggested in [48], an alternate approach that I am actively exploring is to interpret f p itself as a moment map on the set of T -invariant points p, with respect to a perturbed Kähler form.

5 Next, one can also explore an analog of question 3 for the Calabi flow on blow-ups. Recall that on the level of Kähler potentials, the Calabi flow is given by ϕ t = S t S, ϕ(0) = 0, where S t is the scalar curvature of the metric ω(t) = ω 0 + 1 ϕ(t), and S is the average of S t, and is a topological constant depending only on [ω 0 ] and c 1 (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 ω ε L ε be the family of Kähler metrics on Bl p M considered above. Is there a long-time solution to the Calabi flow starting from ω ε, 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 Kähler manifold X, can one blow up enough number of points so that the resulting Kähler manifold 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 hyperkähler manifolds, 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 surface π : X P 1. We require E to satisfy the following. (1) There exists an ε 0 such that E is stable with respect to the Kähler class [ˆω ε0 ] := π [ω 0 ] + ε[ω X ], where ω 0 is a metric on P 1 and ω X 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 [ˆω ε ] for all ε (0, ε 0 ). Now fix a hermitian metric H 0 on E and let Ξ 0 be the corresponding Chern connection. By Yau s resolution of the Calabi conjecture [60], there exists a unique Ricci flat metric ω ε [ˆω ε ]. Since E is stable with respect to [ˆω ε ], it follows [15] that there exist connections Ξ ε complex gauge equivalent to Ξ 0, and solving the Hermitian-Yang-Mills equation F Ξε ω ε = 0, where F Ξε is the curvature of the connection Ξ ε. Our main theorem is as follows.

6 Theorem 2.1 (D.-Jacob, [24]). For any sequence ε k 0, there exists points b 1, b N P 1 such that for any b P 1 \ {b 1, b N } lim Ξ ε k Eb A b L 2 k 1 (E b,h 0,g 0 ) = 0. Here E b denotes the fiber over b, g 0 is a fixed Riemannian metric on E b, and A b is the flat connection on E b uniquely determined by the holomorphic structure on E. Remark 2.2. (1) We do not need to use gauge transformations in the above theorem. Allowing for gauge transformations, we can in fact obtain smooth convergence. (2) In [33] it is proved the restriction of ε 1 ω ε 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 S 4. (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 Ξ ε 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 Ξ ε Eb to A 0. 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 b 1, b N such that for any compact set K π 1 (P 1 \ {b 1,, b N }), there is a constant such that F Ξε L (K,g 0,H 0 ) C? 2.1. Further questions. A natural question is whether one can improve the fiber-wise convergence to a global convergence on compact subsets of P 1 \ {b 1, b N } for some points b 1, b N P 1. Taking a slightly different perspective, instead of considering connections complex-gauge equivalent to Ξ 0, one can fix the holomorphic structure, and consider Hermitian-Einstein metrics H ε. Question 7. With the same setting as above, can one obtain uniform C 0 or L 1 2 bounds on H ε, where the norms are measured with respect to a fixed Hermitian metric H 0? 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 Kähler-Einstein metrics This section summarizes the work done as a graduate student.

7 3.1. Complex Monge-Ampère equations. Complex Monge-Ampère equations play an important role in the study of Kähler-Einstein metrics. Let (M, ω) be a Kähler manifold and consider the following equation for ϕ L (X, ω) (5) (ω + 1 ϕ) n = e γϕ Ω N j=1 s j 2(1 β, ω ϕ := ω + 1 ϕ > 0, j) h j where Ω is a smooth volume form, s j is a section of a line bundle L j and h j is a Hermitian metric on L j. Suppose the divisor D j cut-out by s j is smooth. For technical reasons additionally we also assume that the divisor D = (1 β j )D j is Kawamata log-terminal (that is, β j (0, 1)) and simple normal crossing. The Ricci curvature of ω ϕ then satisfies Ric(ω ϕ ) = γϕ + [D] + χ, where χ is some smooth form (depending on Ω and h j s), and [D] is the current of integration along D. A local model for ω ϕ is a metric with cone singularities. Recall that a flat cone metric on C, in polar coordinates, is given by ds 2 = dr 2 + β 2 r 2 dθ 2 and the corresponding Kähler form in complex coordinates is 1 z 2β. A Kähler current 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 1 z j 2(1 βj) dz j d z j + N 1 dz j d z 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-Păun [34]. With Jian Song [20], I was able to provide a much shorter proof of this. Theorem 3.1. Any solution ω ϕ to equation (5) is quasi-isometric to a cone metric along D. Our idea of the proof was to approximate ω ϕ by smooth metrics ω η 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 ω ϕ be a solution of (5), and let d ϕ be the induced distance function on the open manifold X \ D. Then there exist uniform constants A, Λ 1, and a sequence ω η [ω] of smooth Kähler metrics such that (1) Ric(ω η ) > Aω η ; diam(x, ω η ) < Λ. (2) (X, ω η ) converges to (X, d) in the Gromov-Hausdorff sense, where as above, (X, d) is the metric completion of (X \ D, d ϕ ). (3) X reg = X \ D, where X reg is the regular set in the sense of Cheeger-Colding consisting of points whose tangent cones are isometric to C n. In particular X reg 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 X reg is open and D is of real co-dimension two, combining with the results in [14] one can immediately conclude that X reg is convex.

8 Theorem 3.3 (D., [21]). (X \D, d ϕ ) 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 Kähler-Einstein metrics along possibly non-smooth simple normal crossing divisors. 3.2. Connecting toric manifolds by Kähler-Einstein metrics. Recall that a toric manifold X of dimension n is a complex manifold with a Hamiltonian action of the torus, T n (S 1 ) n, which induces a holomorphic action of (C ) n, with a free, open and dense orbit X 0 X. A toric manifold is determined by a Delzant polytope P R n, and in fact X 0 P T n, where the identification to the first factor is given by a moment map for the action of T n. Conversely, if we fix a Kähler class 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. Figure 1. The polytopes for P 2 and Bl p(p 2 ) 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 pair if D = a j D j with a j [0, 1) and (K X + D) is an ample class. Here K X is the anticanonical class of X. If (X, D) is a toric log-fano pair, then a Kähler current ω (K X + D) is said to be a toric conical Kähler-Ricci soliton if there exists a toric holomorphic vector field ξ H 0 (X, T 1,0 X) solving (6) Ric(ω) = ω + L ξ ω + [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-Ricci soliton, 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 = N j=1 (1 β j)d j. (1) Then (X, D) admits a toric conical Kähler Ricci soliton. Moreover, if we set P = {x R n l j (x) = v j x + β j }, then the vector field ξ is given by ξ = n i=1 c iz i solution to xe c x dx = 0 and (z 1,, z n ) are the standard coordinates on (C ) n X 0 X. P z i where c = (c 1,, c n ) is the unique

9 Figure 2. The degeneration of Bl pp 2 to P 2 at the level of the polytopes, or equivalently at the level of the Kähler classes (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-Einstein metric 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 X 0 and X 1 be two n-dimensional toric manifolds. Then, there exist a family {(X t, ω t )} t [0,1] of n-dimensional toric manifolds X t with toric conical KE metrics ω t for t [0, 1], such that (X t, ω t ) 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 Kähler manifold 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 ɛ[e] for small ɛ > 0? (2) Denoting the conical KE metrics by ω ɛ, does ( X, ω ɛ ) (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 Kähler metrics and Hermitian-Yang-Mills connections. I have succeeded 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 techniques from complex Monge-Ampère equations, 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 L 2 -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]. References [1] Abramovich, D., Karu, K., Matsuhi, K. and Wlodarczyk, J. Torification and factorization of birational maps, Jour. of AMS, Vol. 15, no. 3 (2002) 531 572.

10 [2] Orbifold compactness for spaces of Riemannian metrics and applications, Math. Ann. 331 (2005), no. 4, 739-778. [3] Arezzo, C. and Pacard, F. Blowing up and desingularizing constant scalar curvature Kähler manifolds, Acta Math., 196 (2006) n.o. 2, 179-228, [4] Arezzo, C., Pacard, F. and Singer, M. Extremal metrics on blowups, Duke Math. J. 157 (2011), no. 1, 1 51. [5] Aubin, T. Équations du type Monge-Ampère sur les variétés Kählériennes compactes, Bull. Sci. Math. (2) 102 (1978), no. 1, 63 95. [6] Berman, R.J. and Berndtsson, B. Real Monge-Ampére equations and Kähler-Ricci Solitons on toric log-fano varieties, Ann. Fac. Sci. Toulouse Math. (6) 22 (2013), no. 4, 649?711. [7] Robert J. Berman and David Witt Nystrom. Complex optimal transport and the pluripotential theory of kähler-ricci solitons. arxiv:1401.8264. [8] Brendle, S. Ricci flat Kähler metrics with edge singularities, Int. Math. Res. Not. IMRN 2013, no. 24, 5727?5766. [9] Calabi, E. Extremal Khler metrics, Seminar on Differential Geometry, pp. 259 290, Ann. of Math. Stud., 102, Princeton Univ. Press, Princeton, N.J., 1982. [10] Chen, X., Donaldson, S. and Sun, S. Kähler-Einstein metrics on Fano manifolds, I - III, J. Amer. Math. Soc. 28 (2015), no. 1, 183-278. [11] Chen, X. and He, W. On the Calabi flow., Amer. J. Math. 130 (2008), no. 2, 539-570. [12] Cheeger, J. and Colding, T. On the structure of spaces with Ricci curvature bounded below., J. Differential Geom. I in 46 (1997), no. 3, 406-480, II in 54 (2000), no. 1, 13-35 and III in 54 (2000), no. 1, 37-74. [13] Cheeger, J and Tian, G. Curvature and injectivity radius estimates for Einstein 4-manifolds, J. Amer. Math. Soc. 19 (2006), no. 2, 487?525. [14] Colding, T. and Naber, A. Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications, Ann. of Math. (2) 176 (2012), no. 2, 1173-1229. [15] Donaldson, S.K. Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. London Math. Soc. (3) 50 (1985), 1-26. [16] Donaldson, S.K. Scalar curvature and stability of toric varieties, J. Differential Geom. 62 (2002), no. 2, 289-349. [17] Donaldson, S.K. and Sun, S. Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry, arxiv:1206.2609. [18] S. Dostoglou and D. Salamon, Self-dual instantons and holomorphic curves, Ann. Math. 139 (1994), no. 3, 581-640. [19] Datar, V., Guo, B., Song, J. and Wang, X. Connecting toric manifolds by conical Kähler-Einstein metrics, arxiv:1308.6781, accepted in Adv. of Math. [20] Datar, V. and Song, J. A remark on Kähler metrics with conical singularities along a simple normal crossing divisor, Bull. Lond. Math. Soc. 47 (2015), no. 6, 1010-1013. [21] Datar, V. On convexity of the regular set of conical Kähler-Einstein metrics, Math. Res. Lett. 23 (2016), no. 1, 105 126. [22] Datar, V. and Szekelyhidi, G. Kähler-Einstein metrics along the smooth continuity method, Geom. Funct. Anal. (GAFA) 26 (2016), no. 4, 975-1010. [23] Datar, V. Expansions of solutions to extremal metric type equations on blow-ups of csck surfaces, arxiv:1708.01024, submitted [24] Datar, V. and Jacob, A. Hermitian-Yang-Mills connections on collapsing elliptically fibered K3 surfaces, arxiv:1710.03898, submitted. [25] Datar, V. A remark on relatively unstable blow-ups, in preparation. [26] Delcroix, T. K-stability of Fano spherical varieties, arxiv:1608.01852. [27] R. Dervan. Uniform stability of twisted constant scalar curvature Kähler metrics. arxiv:1412.0648. [28] Friedman, R., Morgan, J., and Witten, E. Vector bundles and F theory. Comm. Math. Phys. 187 (1997), no. 3, 679-743. [29] J. Fu, Limiting Behavior of a Class of Hermitian-Yang-Mills Metrics, arxiv:1203.2986. [30] K. Fukaya, Multivalued Morse theory, asymptotic analysis and mirror symmetry. Proc. Sympos. Pure Math., 73, Amer. Math. Soc., Providence, RI, (2005) 205-278. [31] Futaki, A. and Mabuchi, T. Bilinear forms and extremal Kähler vector fields associated with Kähler classes. Math. Ann. 301 (1995), no. 2, 199 210.

[32] Gross, M. and Wilson, P.M.H. Large complex structure limits of K3 surfaces, J. Differential Geom. 55 (2000), no. 3, 475 546. [33] Gross, M., Tosatti, V. and Zhang, Y. Collapsing of abelian fibered Calabi-Yau manifolds. Duke Math. J. 162 (2013), no. 3, 517 551. [34] Guenancia, H. and Păun, M. Conic singularities metrics with prescribed Ricci curvature: The case of general cone angles along normal crossing divisors, J. Differential Geom. 103 (2016), no. 1, 15?57. [35] Ilten, N. and Süss, H. K-stability for Fano manifolds with torus action of complexity 1, Duke Math. J. 166 (2017), no. 1, 177 204. [36] Jeffres, T., Mazzeo, R. and Rubinstein, Y. Kähler-Einstein metrics with edge singularities, Ann. of Math. (2) 183 (2016), no. 1, 95-176. [37] Legendre, E. Toric Kähler-Einstein metrics and convex compact polytopes, arxiv:1112.3239. [38] Li, C. Greatest lower bounds on Ricci curvature for toric Fano manifolds Adv. Math. 226 (2011), no. 6, 4921 4932. [39] T. Nishinou, Convergence of Hermitian-Yang-Mills Connections on Kähler Surfaces and mirror symmetry, arxiv:math/0301324. [40] Phong, D., Song, J., Sturm, J., Wang, X. The Ricci flow on the sphere with marked points, arxiv:1407.1118. [41] Song, J. and Tian, G. The Kähler-Ricci flow on surfaces of positive Kodaira dimension, Invent. Math. 170 (2007), no. 3, 609-653. [42] Song, J. and Wang, X. The greatest Ricci lower bound, conical Einstein metrics and the Chern number inequality Geom. Topol. 20 (2016), no. 1, 49-102. [43] Song, J. Ricci flow and birational surgery, arxiv: 1304.2607. [44] Song, J. Riemannian geometry of Kähler-Einstein currents II: an analytic proof of Kawamata s base point free theorem, arxiv:1409.8374 [45] Stoppa, J. and Székelyhidi, G. Relative K-stability of extremal metrics, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 4, 899 909 [46] Szekelyhidi, G. Greatest lower bounds on the Ricci curvature of Fano manifolds, Compositio Math. 147 (2011), 319 331. [47] Székelyhidi, G. Extremal metrics and K-stability, Bull. Lond. Math. Soc. 39 (2007), no. 1, 76-84. [48] Székelyhidi, G. Blowing up extremal Kähler manifolds II, Invent. Math. 200 (2015), no. 3, 925 977. [49] Székelyhidi, G. Extremal Kähler metrics, Proceedings of the ICM 2014. [50] Tian, G. On Calabi s conjecture for complex surfaces with positive first Chern class, Invent. Math. 101 (1990), no. 1, 101-172. [51] Tian, G. Kähler-Einstein metrics with positive scalar curvature, Invent. Math. 137 (1997), 1 37. [52] Tian, G. and Viaclovsky, J. Moduli spaces of critical Riemannian metrics in dimension four, Adv. Math. 196 (2005), no. 2, 346-372. [53] Tian, G. K-stability and Kähler-Einstein metrics, Comm. Pure Appl. Math. 68 (2015), no. 7, 1085 1156. [54] Tosatti, V. Adiabatic limits of Ricci-flat Kähler metrics, J. Differential Geom. 84 (2010), no.2, 427-453. [55] V. Tosatti and Y. Zhang, Collapsing hyperkhler manifolds. (Preprint) arxiv:1705.03299 [math.dg]. [56] Taubes, C The existence of anti-self dual conformal structures, J. Differential Geom. 36 (1992) 163-253. [57] Wang, X. and Zhu, X. Kähler-Ricci solitons on toric manifolds with positive first Chern class, Adv. Math. 188 (2004), no. 1, 87 103. [58] K. Uhlenbeck and S.-T. Yau, On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Comm. Pure and Appl. Math. 39-S (1986), 257-293. [59] C. Vafa, Extending mirror conjecture to Calabi-Yau with bundles, Commun. Contemp. Math., 1 (1999), 65-70. [60] Yau, S.-T. On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I, Comm. Pure Appl. Math. 31 (1978), no. 3, 339 411. [61] Yau, S.-T. Open problems in geometry. Differential geometry: partial differential equations on manifolds, (Los Angeles, CA, 1990), 1 28, Proc. Sympos. Pure Math., 54, Part 1, Amer. Math. Soc., Providence, RI, 1993. 11