Online one-day workshop

15th November 2021, 12:15-3:30pm GMT

Generalized connections and curvature have been approached from different perspectives. How do they compare to each other? What are they useful for? What are the major results? Where is the theory heading?

Speakers: Miquel Cueca, Liana David, Mario García-Fernández, Ryushi Goto, Marco Gualtieri, Madeleine Jotz-Lean

This meeting was held on ZOOM in November 2021, 12:15-3:30pm GMT.

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Organizing committee:

Vicente Cortés (Hamburg)Roberto Rubio (Barcelona)

Ryushi Goto

The Kobayashi-Hitchin correspondence of generalized holomorphic vector bundles over generalized Kahler manifolds of symplectic type

Slides here

Slides here

(12:20-12:35pm GMT)

The notion of an Einstein-Hermitian metric of a generalized holomorphic vector bundle over a generalized Kahler manifold of symplectic type is introduced from the moment map framework. In this talk we show the Kobayashi-Hitchin correspondence, that is, the equivalence of the existence of an Einstein-Hermitian metric and ψ-polystability of a generalized holomorphic vector bundle over a compact generalized Kahler manifold of symplectic type. Poisson modules provide intriguing generalized holomorphic vector bundles and we obtain ψ-stable Poisson modules over complex surfaces which are not stable in the ordinary sense.

The talk is based on papers arXiv:1903.07425 and arXiv:1707.03143..

The notion of an Einstein-Hermitian metric of a generalized holomorphic vector bundle over a generalized Kahler manifold of symplectic type is introduced from the moment map framework. In this talk we show the Kobayashi-Hitchin correspondence, that is, the equivalence of the existence of an Einstein-Hermitian metric and ψ-polystability of a generalized holomorphic vector bundle over a compact generalized Kahler manifold of symplectic type. Poisson modules provide intriguing generalized holomorphic vector bundles and we obtain ψ-stable Poisson modules over complex surfaces which are not stable in the ordinary sense.

The talk is based on papers arXiv:1903.07425 and arXiv:1707.03143..

Miquel Cueca

(12:40-12:55pm GMT)

A metric vector bundle (E, <,>) has an associated grade algebra, the Kaller-Waldmann algebra C(E). When E is a Courant algebroid thenC(E) has a Cartan calculus. We define Courant connections using C(E) and show that the curvature is an endomorphism valued 2 cochaim. This is a joint work with Rajan Mehta.

A metric vector bundle (E, <,>) has an associated grade algebra, the Kaller-Waldmann algebra C(E). When E is a Courant algebroid thenC(E) has a Cartan calculus. We define Courant connections using C(E) and show that the curvature is an endomorphism valued 2 cochaim. This is a joint work with Rajan Mehta.

Madeleine Jotz-Lean

(1:00-1:15pm GMT)

In this short talk I will quickly recall how Dorfman connections are equivalent to linear splittings of the standard Courant algebroids over vector bundles. Then I will compare the notion with the one of generalised connections in Courant algebroids.

In this short talk I will quickly recall how Dorfman connections are equivalent to linear splittings of the standard Courant algebroids over vector bundles. Then I will compare the notion with the one of generalised connections in Courant algebroids.

Mario Garcia-Fernandez

Generalized Hermitian metrics and Chern curvature

(1:30-1:45pm GMT)

A new aspect of generalized connections has recently arised in arXiv:2106.13716 and arXiv:2008.07004, in relation to holomorphic Courant algebroids. In this talk I will describe conditions for a generalized metric to induce a holomorphic Courant algebroid endowed with a Hermitian metric with an interesting Chern connection. If time allows, I will apply this picture to obtain GIT obstructions to the existence of pluriclosed Hermitian metrics with vanishing Bismut Ricci form.

A new aspect of generalized connections has recently arised in arXiv:2106.13716 and arXiv:2008.07004, in relation to holomorphic Courant algebroids. In this talk I will describe conditions for a generalized metric to induce a holomorphic Courant algebroid endowed with a Hermitian metric with an interesting Chern connection. If time allows, I will apply this picture to obtain GIT obstructions to the existence of pluriclosed Hermitian metrics with vanishing Bismut Ricci form.

Liana David

Generalized connections, spinors and T-duality

Slides here

Slides here

(1:50-2:05pm GMT)

I will extend the T-duality for exact Courant algebroids, developed by Cavalcanti and Gualtieri, to the larger setting of transitive Courant algebroids. I will show that the T-duality between two transitive Courant algebroids E and \tilde{E} induces a map between the spaces of sections of their corresponding canonical weighted spinor bundles, which intertwines the canonical Dirac generating operators of E and \tilde{E}. I will present a general existence result for a T-dual of a Courant algebroid, under assumptions generalizing the cohomology integrality conditions in the exact case. This is joint work with Vicente Cortes.

I will extend the T-duality for exact Courant algebroids, developed by Cavalcanti and Gualtieri, to the larger setting of transitive Courant algebroids. I will show that the T-duality between two transitive Courant algebroids E and \tilde{E} induces a map between the spaces of sections of their corresponding canonical weighted spinor bundles, which intertwines the canonical Dirac generating operators of E and \tilde{E}. I will present a general existence result for a T-dual of a Courant algebroid, under assumptions generalizing the cohomology integrality conditions in the exact case. This is joint work with Vicente Cortes.

Marco Gualtieri

Generalized Chern connections and Kahler Ricci Flow

(2:10-2:25pm GMT)

I will explain the relationship between generalized holomorphic bundles and the generalized Kahler Ricci flow.

Discussion

(2:40-3:30pm GMT)

A discussion will follow: we would like to draw connections between all the approaches and put our understanding to test. Please, join us and share your knowledge and insights.

I will explain the relationship between generalized holomorphic bundles and the generalized Kahler Ricci flow.

Discussion

(2:40-3:30pm GMT)

A discussion will follow: we would like to draw connections between all the approaches and put our understanding to test. Please, join us and share your knowledge and insights.