Workshop

Let $X$ denote the Gromov-Hausdorff limit of a noncollapsing sequence of riemannian manifolds $(M^n_i,g_i)$, with uniformly bounded Ricci curvature.  Early workers conjectured (circa 1990) that $X$ is a smooth manifold off a closed subset of Hausdorff codimension $4$.  We will explain a proof of this conjecture. This is joint work with Aaron Naber.

Little is currently known about the global properties of the G_2 moduli space of a closed 7-manifold, ie the space of Riemannian metrics with holonomy G_2 modulo diffeomorphisms. A holonomy G_2 metric has an associated G_2-structure, and I will define a Z/48 valued homotopy invariant of a G_2-structure in terms of 8-dimensional coboundaries, and a Z-valued refinement in terms of eta invariants. I will describe examples of manifolds with holonomy G_2 metrics where these invariants are amenable to computation and can be used to prove that the moduli space is disconnected. This is joint work with Diarmuid Crowley and Sebastian Goette

we introduce the L^p Mabuchi-Finsler structures on the space of Kahler metrics and present the resulting path length geometries. All the metric completions are geodesic metric spaces that can be characterized using elements of finite energy pluripotential theory and we focus on three particular cases of interest. Using convexity in the L^\infty geometry, one may be able to obtain C^0 estimates for variational PDE problems arising in Kahler geometry. The L^2 geometry is non-positively curved and seems to be the ideal context to study metric properties of gradient flows of various convex functionals, including the Calabi flow and the J-flow. The L^1 geometry enjoys a compactness property and has metric growth prescribed by the J functional, making it a useful tool in the study of energy properness questions, leading  to the recent resolution of Tian's related conjectures

Based on the compactness of the moduli of non-collapsed Calabi-Yau spaces with mild singularities, we set up a structure theory for polarized K\"ahler Ricci flows with proper geometric bounds. Our theory is a generalization of the structure theory of non-collapsed K\"ahler Einstein manifolds. As applications, we prove the Hamilton-Tian conjecture and the partial-C0

-conjecture of Tian

Consider a Riemannian manifold with bounded Ricci curvature |Ric|\leq n-1 and the noncollapsing lower volume bound Vol(B_1(p))>v>0.  The first main result of this paper is to prove the previously conjectured L^2 curvature bound \fint_{B_1}|\Rm|^2 < C(n,v).  In order to prove this, we will need to first show the following structural result for limits.  Namely, if (M^n_j,d_j,p_j) -> (X,d,p) is a GH-limit of noncollapsed manifolds with bounded Ricci curvature, then the singular set S(X) is n-4 rectifiable with the uniform Hausdorff measure estimates H^{n-4}(S(X)\cap B_1)<C(n,v), which in particular proves the n-4-finiteness conjecture of Cheeger-Colding.  We will see as a consequence of the proof that for n-4 a.e. x\in S(X) that the tangent cone of X at x is unique and isometric to R^{n-4}xC(S^3/G_x) for some G_x\subseteq O(4) which acts freely away from the origin.  The proofs involve several new estimates on spaces with bounded Ricci curvature. This is joint work with Wenshuai Jiang

Donaldson and Thomas (1998) proposed that it should be possible to define interesting invariants "counting" moduli spaces of "Spin(7) instantons" on a Calabi-Yau 4-fold Y (or a compact Spin(7)-manifold), which can be viewed as a complexified version of Donaldson invariants of real 4-manifolds, just as DT invariants of Calabi-Yau 3-folds are a complexified version of Casson invariants of real 3-manifolds.

  Carrying out this programme using gauge theory is very difficult and currently unsolved, because one needs to somehow compactify the moduli spaces of Spin(7) instantons by including singular solutions, and have a nice enough geometric structure on this compactification to define a virtual cycle.

  We outline an alternative approach using (derived) algebraic geometry in which these difficulties do not occur. If a complex vector bundle E --> Y with c_1(E) . [\omega]^3=0 admits a holomorphic vector bundle structure, then just as topological spaces, the moduli spaces of Spin(7) instantons and of Hermitian-Yang-Mills connections coincide, though as non-reduced schemes (in a C-infinity sense) the HYM moduli space is a proper subscheme, as the HYM conditions impose more equations. In fact, the HYM equations are overdetermined elliptic in 4 complex dimensions, so one does not expect to be able to define a virtual cycle for the HYM moduli space.

  In the Derived Algebraic Geometry of Toen-Vezzosi, suppose we are given a derived moduli scheme M of coherent sheaves (or complexes of coherent sheaves) on Y. Then by work of Pantev-Toen-Vaquie-Vessozi, M has a "-2-shifted symplectic structure". Thus the shifted symplectic Darboux Theorem of Bussi-Brav-Joyce gives Zariski local models for M. 

  We will explain how to use these local models to give the underlying complex analytic topological space M* of M the structure of a "derived smooth manifold" M*, whose real virtual dimension is half the expected real virtual dimension of M (!). Heuristically speaking, this is because there is a "Lagrangian fibration" M --> M*, and the dimension of the base of a Lagrangian fibration is half the dimension of the symplectic manifold.

  Now compact, oriented derived manifolds admit virtual cycles, in homology or bordism. This means that if we are given a proper derived moduli scheme M of coherent sheaves on Y of complex virtual dimension d, plus an "orientation" (similar to orientations for instanton moduli spaces in Donaldson theory), we can define a virtual cycle [M*]_{virt} in the homology H_d(M*;Z). This is what we need to define Donaldson-Thomas style invariants "counting" (semi)stable coherent sheaves on a Calabi-Yau 4-fold Y, which will be unchanged under deformations of the complex structure of Y.

  Here is how this deals with the issue of compactification of moduli spaces of gauge-theoretic Spin(7) instantons. The moduli spaces of (say) stable algebraic vector bundles, and of HYM connections, agree by Hitchin-Kobayashi; and the moduli spaces of HYM connections and Spin(7) instantons agree as above. However, algebraic geometry often gives us compactifications of moduli spaces of stable algebraic vector bundles, as moduli spaces of stable torsion-free coherent sheaves, essentially for free. Our algebraic geometry approach does not care about the difference between moduli of vector bundles and moduli of coherent sheaves, but in gauge theory the difference is crucial.

  This is joint work with Dennis Borisov, and relies on earlier joint work with Bussi and Brav. It is also related to papers by Cao and Leung.

The structure of projective algebraic manifolds is to a large extent governed by the geometry of its cones of divisors or curves. In the case of divisors, two cones are of primary importance: the cone of ample divisors and the cone of effective divisors (and the closure of these cones as well). These cones have natural transcendental analogues on any compact Kähler manifold, namely the cone of Kähler classes (called the Kähler cone) and the cone of pseudoeffective (1,1)-classes (called the pseudoeffective cone).

 A conjecture of Boucksom-Demailly-Paun-Peternell says that the pseudoeffective cone is dual to the cone of movable classes. I will discuss my recent proof of the conjecture in the case when the manifold is projective

I will talk about joint work with Tosatti and Zhang on Gromov-Hausdorff collapse of abelian fibred Calabi-yau manifolds. One considers a Calabi-Yau manifold M with a holomorphic map M->N with general fibres being abelian varieties. Given a suitably chosen sequence of Ricci-flat metrics so that the volume of the fibres goes to zero, we show that the Kahler form on M converges to the pull-back of a form on N. Stronger results if the dimension of N is 1 imply that in fact M converges in the Gromov-Hausdorff sense to N. This leads to an extension of an old collapsing result of myself and Wilson in the K3 case with no assumptions on the types of singular fibres