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Ved V. Datar Research Statement 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 K¨ahlermanifold M is a complex manifold with a closed, real, positive (1; 1) form, called the K¨ahlerform or the K¨ahlermetric. The K¨ahlermetric 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 K¨ahlerclass. An extremal K¨ahlermetric [9] is a critical point of the Calabi functional Z n 2 ! Ca(!) = s! M n! as ! varies in a fixed K¨ahlerclass. Here s! denotes the scalar curvature of the Riemannian metric associated to the K¨ahlerform !. The Euler-Lagrange equation is 1;0 @¯r s! = 0; 1;0 where r s! is the (1; 0) part of the gradient. An especially important special case is that of constant scalar curvature K¨ahler(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. K¨ahler-Einsteinmetrics on Fano manifolds. A special class of cscK metrics are K¨ahler-Einstein (KE) metrics, that is K¨ahlermetrics ! whose Ricci form satisfies (1) Ric(!) = λω 1 2 for some λ = −1; 0; 1. Since Ric(!) 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 K¨ahlermanifolds 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 G´abor Sz´ekelyhidi we proved an equivariant version of conjecture1 for KE metrics. −1 Theorem 1.1 (D.-Sz´ekelyhidi [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 K¨ahler- 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 K¨ahler-Ricci solitons to the so-called modified G-equivariant K-stability introduced in [27,7]. For a smooth form α 2 c1(M), and a holomorphic vector field v on M, we say that (M; (1 − t)α; v) admits a K¨ahler-Ricci soliton if there exists an ! 2 c1(M) solving (2) Ric(!) = t! + (1 − t)α + Lv!; where Lv denotes the Lie derivative. Theorem 1.2 (D.-Sz´ekelyhidi [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 K¨ahler-Ricci soliton for all t < s. In addition if (M; v) is K-stable, then (M; v) admits a K¨ahler-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 Sz¨ekelyhidi. Recall that for a Fano manifold M we define R(M) := sup ft j (2) has a solution with v = 0g: t2(0;1) In [46] Sz´ekelyhidi 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 K¨ahler-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 K¨ahlersurfaces 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 K¨ahlersurfaces with uniform bounds on the total volume, scalar curvature and some 2 mild topological constrains (say bounds on c1 and c2)? Note that on K¨ahlersurfaces, 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 H¨ormander'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 K¨ahlersurfaces, I wish to study the following question. Question 2. Can one extend the "-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 K¨ahlermetric in the class c1(L) for some ample line bundle L. Let π : BlpM ! M be the blow-up at p 2 M with exceptional divisor E. If we ∗ 2 let L" = π L − " E, when does (BlpM; L") 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] Sz´ekelyhidi 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 Sz´ekelyhidi in [48]. To state our main theorem, we set up some notation.
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