## Final Report Summary - CAMEGEST (Extremal Kaehler metrics and geometric stability)

Since the birth of modern geometry in the nineteenth century with the work of Gauss and Riemann, the problem of equipping a given regular set ("manifold") with the metric of "as constant as possible" curvature have become the Holy Graal in Geometry. In its complex incarnation, this problem asks whether a Kahler manifold has, and in this case how many, a canonical Kahler metric. Since the pioneering work of Calabi, the existence problem for Kahler-Einstein (and more generally constant scalar curvature or extremal Kahler) metrics has attracted considerable attention.

In this circle of ideas the most fascinating unsolved problem is represented by the so-called Yau-Tian-Donaldson conjecture, which predicts the equivalence between the K-polystability of a polarized manifold and the existence of a constant scalar curvature (or more generally extremal) Kahler metric in the polarization class.

A large part of current research in Kahler geometry fits in the circle of ideas revolving around the aforementioned conjecture, and it is directly or indirectly related to that.

In the last three years a number of exciting results has been proved, along research lines initiated by Tian and Donaldson, pursuing a proof of the YTD conjecture. Very recently a proof in the important special case of Kahler-Einstein metrics has been given by Tian and Chen-Donaldson-Sun independently.

In fact one of the key tools in this fundamental result is the use of metrics with conical singularities, and in fact part of our research of the last year has focused on these themes even if not part of the original project.

Indeed, in a joint work with C. Arezzo and G. La Nave, we proved a regularity result for solutions of complex Monge-Ampere equations with singular behaviour along a smooth divisor. This in turn allows to identify some sufficient geometric conditions, following a suggestion of S. Donaldson, which guarantee the boundedness of the Riemann tensor of conical metrics.

Let us briefly describe the other results we obtained:

- In a joint work with G. Tian has been proved a stability theorem for the Calabi flow (defined on the space of the complex structures compatible with a given symplectic form) near an extremal Kahler metric. This result has been used to extend the Chen-Tian uniqueness theorem for extremal Kahler metric to the case when the complex structure is deformed. More precisely has been proved that given two smooth special degenerations of the same polarized Kahler manifold, if the central fibers are extremal Kahler and they realize the infimum of Calabi Energy, then they are biholomorphic. This generalizes a recent result on uniqueness of constant scalar curvature Kahler metrics under deformation of the complex structure due to Chen-Sun. The paper containing these results is in preparation.

- In a joint work with C. Arezzo and G. La Nave, the study of the Calabi flow has been approached using symplectic reduction formalism. In this context a number of natural geometric flows in the space of Kahler potentials including the geodesic equation and the pseudo Calabi flow (as defined by Chen and Zheng) have been studied and new sets of challenging and interesting Partial Differential Equations have identified.

.

- Using the momentum construction of Hwang and Singer, some evidences have been found for the idea that the symplectic reduction of an extremal Kahler metric with respect to the extremal torus action should be extremal.

- In a joint work with C. Arezzo and G. La Nave, applying Mumford's semistable reduction theorem and results of Ross-Thomas, has been proved that K-semistability of a polarized manifold can be checked just on smooth test configurations with simple normal crossing central fiber. As application has been produced many examples of K-unstable polarized manifolds.

- In a joint work with F. Zuddas, applying results of Mabuchi, it has been proved that on a projective bundle over a smooth curve the existence of a constant scalar curvature Kahler metric is equivalent to asymptotic Chow stability. On the other hand has been produced examples of constant scalar curvature polarized surfaces that are asymptotically Chow unstable.

With the sponsorship of this research project the following publications have been completed. Those submitted for publication to refereed journals but not yet accepted will be attached at the end of the "Dissemination" section below.

1) C. Arezzo, A. Della Vedova. " On the K -stability of complete intersections in polarized manifolds". Adv. Math. 226 (2011), no. 6, 4796-4815.

2) C. Arezzo, A. Della Vedova, G. La Nave. "Singularities and K-semistability". Int. Math. Res. Not. IMRN 2012, no. 4, 849-869.

3) A. Della Vedova, F. Zuddas. "Scalar curvature and asymptotic Chow stability of projective bundles and blowups". Trans. Amer. Math. Soc. 364 (2012), no. 12, 6495-6511.

4) C. Arezzo, A. Della Vedova, G. La Nave. "Geometric Flows and Kahler Reduction", to appear on Journal of Symplectic Geometry (available at http://arxiv.org/abs/1304.5728).

5) C. Arezzo, A. Della Vedova, G. La Nave. "On the curvature of conical Kahler-Einstein metrics", submitted to Compositio Mathematica.

6) A. Della Vedova. " Berezin-Toeplitz quantization of the Laplace operator", preprint.

The results obtained during the period of this project have been reported at the following important international conferences and research institutions:

02/2011: "Extremal metrics: evolution equations and stability". CIRM, Marseille (FR).

06/2011: "Complex Analysis and Geometry - XX‚". CIRM, Levico (IT).

12/2011: "Complex Geometry Working Seminar". Princeton University, NJ (USA).

01/2012: "Geometry Seminar". ICTP, Trieste (IT).

02/2012: "Algebraic Topology and Physics Seminar". Simons Center for Geometry and Physics, Stony Brook NY (USA).

02-05/2012: BICMR (Beijing, CN) on the invitation of prof. G. Tian.

06/2012: "Conference and school on Geometric Analysis". ICTP, Trieste (IT).

10/2012: "The 18th International Symposium on Complex Geometry (Sugadaira symposium)", Sugadaira (JP).

11/2012: "Workshop MACK5", Roma (IT).

06/2013: "Complex Analysis and Geometry - XXI". CIRM, Levico (IT).

Four-months stay at BICMR (Bejing, China) was spent in order to continue a joint work with G. Tian (on leaving from Princeton University at that time).

In this circle of ideas the most fascinating unsolved problem is represented by the so-called Yau-Tian-Donaldson conjecture, which predicts the equivalence between the K-polystability of a polarized manifold and the existence of a constant scalar curvature (or more generally extremal) Kahler metric in the polarization class.

A large part of current research in Kahler geometry fits in the circle of ideas revolving around the aforementioned conjecture, and it is directly or indirectly related to that.

In the last three years a number of exciting results has been proved, along research lines initiated by Tian and Donaldson, pursuing a proof of the YTD conjecture. Very recently a proof in the important special case of Kahler-Einstein metrics has been given by Tian and Chen-Donaldson-Sun independently.

In fact one of the key tools in this fundamental result is the use of metrics with conical singularities, and in fact part of our research of the last year has focused on these themes even if not part of the original project.

Indeed, in a joint work with C. Arezzo and G. La Nave, we proved a regularity result for solutions of complex Monge-Ampere equations with singular behaviour along a smooth divisor. This in turn allows to identify some sufficient geometric conditions, following a suggestion of S. Donaldson, which guarantee the boundedness of the Riemann tensor of conical metrics.

Let us briefly describe the other results we obtained:

- In a joint work with G. Tian has been proved a stability theorem for the Calabi flow (defined on the space of the complex structures compatible with a given symplectic form) near an extremal Kahler metric. This result has been used to extend the Chen-Tian uniqueness theorem for extremal Kahler metric to the case when the complex structure is deformed. More precisely has been proved that given two smooth special degenerations of the same polarized Kahler manifold, if the central fibers are extremal Kahler and they realize the infimum of Calabi Energy, then they are biholomorphic. This generalizes a recent result on uniqueness of constant scalar curvature Kahler metrics under deformation of the complex structure due to Chen-Sun. The paper containing these results is in preparation.

- In a joint work with C. Arezzo and G. La Nave, the study of the Calabi flow has been approached using symplectic reduction formalism. In this context a number of natural geometric flows in the space of Kahler potentials including the geodesic equation and the pseudo Calabi flow (as defined by Chen and Zheng) have been studied and new sets of challenging and interesting Partial Differential Equations have identified.

.

- Using the momentum construction of Hwang and Singer, some evidences have been found for the idea that the symplectic reduction of an extremal Kahler metric with respect to the extremal torus action should be extremal.

- In a joint work with C. Arezzo and G. La Nave, applying Mumford's semistable reduction theorem and results of Ross-Thomas, has been proved that K-semistability of a polarized manifold can be checked just on smooth test configurations with simple normal crossing central fiber. As application has been produced many examples of K-unstable polarized manifolds.

- In a joint work with F. Zuddas, applying results of Mabuchi, it has been proved that on a projective bundle over a smooth curve the existence of a constant scalar curvature Kahler metric is equivalent to asymptotic Chow stability. On the other hand has been produced examples of constant scalar curvature polarized surfaces that are asymptotically Chow unstable.

With the sponsorship of this research project the following publications have been completed. Those submitted for publication to refereed journals but not yet accepted will be attached at the end of the "Dissemination" section below.

1) C. Arezzo, A. Della Vedova. " On the K -stability of complete intersections in polarized manifolds". Adv. Math. 226 (2011), no. 6, 4796-4815.

2) C. Arezzo, A. Della Vedova, G. La Nave. "Singularities and K-semistability". Int. Math. Res. Not. IMRN 2012, no. 4, 849-869.

3) A. Della Vedova, F. Zuddas. "Scalar curvature and asymptotic Chow stability of projective bundles and blowups". Trans. Amer. Math. Soc. 364 (2012), no. 12, 6495-6511.

4) C. Arezzo, A. Della Vedova, G. La Nave. "Geometric Flows and Kahler Reduction", to appear on Journal of Symplectic Geometry (available at http://arxiv.org/abs/1304.5728).

5) C. Arezzo, A. Della Vedova, G. La Nave. "On the curvature of conical Kahler-Einstein metrics", submitted to Compositio Mathematica.

6) A. Della Vedova. " Berezin-Toeplitz quantization of the Laplace operator", preprint.

The results obtained during the period of this project have been reported at the following important international conferences and research institutions:

02/2011: "Extremal metrics: evolution equations and stability". CIRM, Marseille (FR).

06/2011: "Complex Analysis and Geometry - XX‚". CIRM, Levico (IT).

12/2011: "Complex Geometry Working Seminar". Princeton University, NJ (USA).

01/2012: "Geometry Seminar". ICTP, Trieste (IT).

02/2012: "Algebraic Topology and Physics Seminar". Simons Center for Geometry and Physics, Stony Brook NY (USA).

02-05/2012: BICMR (Beijing, CN) on the invitation of prof. G. Tian.

06/2012: "Conference and school on Geometric Analysis". ICTP, Trieste (IT).

10/2012: "The 18th International Symposium on Complex Geometry (Sugadaira symposium)", Sugadaira (JP).

11/2012: "Workshop MACK5", Roma (IT).

06/2013: "Complex Analysis and Geometry - XXI". CIRM, Levico (IT).

Four-months stay at BICMR (Bejing, China) was spent in order to continue a joint work with G. Tian (on leaving from Princeton University at that time).