# School - Flows and Limits in Kähler Geometry

### List of (confirmed) speakers

### Mini-courses :

Bo Berndtsson , Chalmers University

Hans Joachim Hein , Fordham University

Valentino Tosatti , Northwestern University

Jeff Viaclovsky , Wisconsin University

### Talks :

TBA

### Titles and abstracts

**Bo Berndtsson**(Chalmers University) :*Direct image bundles and variations of complex structures*

Given a smooth proper fibration $p:\mathcal X\to B$ and $L$ a line bundle over $\mathcal X$, the direct image $$ E:= p_*(L) $$ is in many cases a holomorphic vector bundle over $B$. Its fibers are the spaces of holomorphic sections of $L$ over the fibers of $p$, $X_t=p^{-1}(t)$, and they can be given various $L^2$-metrics. In case the fibration is of relative dimension $n$ so that the fibers are compact Riemann surfaces, special cases of this situation can be used to study the variation of complex structures on the fibers $X_t$. (The fibers are all diffeomorphic, but their complex structure varies with $t$, so we can view the family $X_t$ as a family of variations of complex structures on one fixed smooth manifold.) When the relative dimension is higher than one the situation is more complicated and one needs to consider also higher direct images. I will discuss the problems that arise in this connection, with previous work of Siu, Schumacher and To-Yeung and some recent joint work with Xu Wang and Mihai Paun.

**Hans-Joachim Hein**(Fordham University) :*Tangent cones of Calabi-Yau varieties*

It has been known for about 10 years that the classical Calabi-Yau theorem on the existence and uniqueness of Ricci-flat Kahler metrics on smooth complex manifolds with zero first Chern class can be extended to a natural setting of weak Kahler metrics on singular complex varieties. However, until relatively recently nothing was known - even in the simplest nontrivial examples - about the precise asymptotic behavior of these weak Ricci-flat metrics at the singularities of the underlying varieties. I will explain work of Donaldson-Sun, H-Naber and H-Sun that resolves this question in certain cases.

**Valentino Tosatti**(Northwestern University) :*Metric Limits of Calabi-Yau Manifolds*

In this mini-course I will give an introduction to the study of limits of Ricci-flat Kahler metrics on a compact Calabi-Yau manifold when the Kahler class degenerates to the boundary of the Kahler cone. Analytically, the problem is to prove suitable uniform a priori estimates for solutions of a degenerating family complex Monge-Ampère equations, away from some singular set. Geometrically, this can be used to understand the Gromov-Hausdorff limit of these metrics. And if the manifold is projective algebraic and the limiting class is rational, the limits possess an algebraic structure and are obtained from the initial manifold via contraction morphisms from Mori theory.

**Jeff Viaclovsky**(Wisconsin University) : The geometry of SFK ALE metrics

I will discuss some of the basics of scalar-flat Kaehler (SFK) metrics, and focus on the geometry of SFK metrics which are asymptotically locally Euclidean (ALE). These space arise as "bubbles" in the compactness theory of Calabi's extremal Kaehler metrics. I will also present some of the deformation theory of SFK ALE metrics.