Recovering the Weinstein Splitting Theorem for Poisson manifolds using the Local Splitting theorem for Lie algebroids

How can I recover the Weinstein Splitting Theorem for Poisson manifolds using the Local Splitting theorem for Lie algebroids? I am using the formulation of the theorem given in https://arxiv.org/pdf/math/0007132.pdf. On Page 13, Fernades gives the following hint

I made some fruitless attempts =(

Wu formula for manifolds with boundary

The classical Wu formula claims that if $ M$ is a smooth closed $ n$ -manifold with fundamental class $ z\in H_n(M;\mathbb{Z}_2)$ , then the total Stiefel-Whitney class $ w(M)$ is equal to $ Sq(v)$ , where $ v=\sum v_i\in H^*(M;\mathbb{Z}_2)$ is the unique cohomology class such that $ $ \langle v\cup x,z\rangle=\langle Sq(x),z\rangle$ $ for all $ x\in H^*(M;\mathbb{Z}_2)$ . Thus, for $ k\ge0$ , $ v_k\cup x=Sq^k(x)$ for all $ x\in H^{n-k}(M;\mathbb{Z}_2)$ , and $ $ w_k(M)=\sum_{i+j=k}Sq^i(v_j).$ $ Here the Poincare duality guarantees the existence and uniqueness of $ v$ .

My question: if $ M$ is a smooth compact $ n$ -manifold with boundary, is there a similar Wu formula? In this case, there is a fundamental class $ z\in H_n(M,\partial M;\mathbb{Z}_2)$ and the relative Poincare duality claims that capping with $ z$ yields duality isomorphisms $ $ D:H^p(M,\partial M;\mathbb{Z}_2)\to H_{n-p}(M;\mathbb{Z}_2)$ $ and $ $ D:H^p(M;\mathbb{Z}_2)\to H_{n-p}(M,\partial M;\mathbb{Z}_2).$ $

Thank you!

Covariant derivative of the Monge-Ampere equation on Kähler manifolds

I am reading D. Joyce book “Compact manifolds with special holonomy” and I have some problems of understanding some computation on page 111, the first line in the proof of Proposition 5.4.6. More specific the following:

Let $ (M,\omega, J)$ be a compact Kähler manifold with Kähler form $ \omega$ and complex structure $ J$ . In holomorphic coordinates $ \omega$ is of the form $ \omega = ig_{\alpha \overline{\beta}}dz^{\alpha} \wedge d\overline{z}^{\beta}$ . Associated to the above data we have the Riemannian metric $ g$ which may be written in holomorphic coordinates as $ g=g_{\alpha \overline{\beta}}(dz^{\alpha}\otimes d\overline{z}^{\beta} + d\overline{z}^{\beta} \otimes dz^{\alpha})$ . Associated to $ g$ let $ \nabla$ be the Levi-Civita connection which also defines a covariant derivative on tensors. For a function $ \phi$ on $ M$ one may compute $ \nabla^{k}\phi$ . For example $ \nabla \phi = (\nabla_{\lambda}\phi)dz^{\lambda} + (\nabla_{\overline{\lambda}}\phi)d\overline{z}^{\lambda}=(\partial_{\lambda}\phi)dz^{\lambda} + (\partial_{\overline{\lambda}}\phi)d\overline{z}^{\lambda}$ (once applied on functions is as the usual $ d$ ) and $ \nabla_{\alpha \beta}\phi = \partial_{\alpha \beta} \phi – \partial_{\gamma}\phi \Gamma^{\gamma}_{\alpha \beta}$ , $ \nabla_{\alpha \overline{\beta}}\phi = \partial_{\alpha \overline{\beta}}\phi$ etc.

In the first sentence of the proof of proposition 5.4.6 Joyce considers the equation $ \det(g_{\alpha \overline{\beta}} + \partial_{\alpha \overline{\beta}}\phi) = e^{f}\det(g_{\alpha \overline{\beta}})$ , where $ f:M\rightarrow \mathbb{R}$ is a smooth function on $ M$ . After taking the $ \log$ of this equation he obtains $ \log[\det(g_{\alpha \overline{\beta}} + \partial_{\alpha \overline{\beta}}\phi)] – \log[\det(g_{\alpha \overline{\beta}} )] = f$ which is obviously a globaly defined equality of functions on $ M$ . Now he takes the covariant derivative $ \nabla$ of this equation and obtains $ \nabla_{\overline{\lambda}}f = g’^{\mu \overline{\nu}}\nabla_{\overline{\lambda} \mu \overline{\nu}}\phi$ where $ g’^{\mu \overline{\nu}}$ is the inverse of the metric $ g’_{\alpha \overline{\beta}} = g_{\alpha \overline{\beta}} + \partial_{\alpha \overline{\beta}}\phi$ (which he assumes to exists). This last step (when taking the covariant derivative) I do not understant.

In my computation I have the following: When taking the covariant derivative $ \nabla_{\overline{\lambda}}$ of the equation $ \log[\det(g_{\alpha \overline{\beta}} + \partial_{\alpha \overline{\beta}}\phi)] – \log[\det(g_{\alpha \overline{\beta}} )] = f$ and using the formula for the derivative of the determinant I obtain $ g’^{\alpha \overline{\beta}}(\partial_{\overline{\lambda}}g_{\alpha \overline{\beta}} + \partial_{\overline{\lambda} \alpha \overline{\beta}}\phi) – g^{\alpha \overline{\beta}}(\partial_{\overline{\lambda}}g_{\alpha \overline{\beta}}) = \partial_{\overline{\lambda}}f = \nabla_{\overline{\lambda}}f$ . This is obviously different to his formula. Moreover the term $ \nabla_{\overline{\lambda}\mu \overline{\nu}}\phi$ contains not only derivatives of order $ 3$ of $ \phi$ but it also contains a term with second derivatives of $ \phi$ .

My question is: Where is my mistake? Have I understood something wrong?

Can we define topological quantum field theories on Calabi-Yau manifolds?

Calabi Yau manifolds are Kähler manifolds with vanishing first Chern class. According to the conjecture of E. Calabi , for a Kähler manifold M , if $ c_1 (M) = 0 $ , then M would admit a Ricci-flat Kähler metric . Explicitly, $ Ric(g) = -i\partial\bar{\partial}log(det(g))=0 $

Or, $ \partial\bar{\partial}log(det(g))=0$ . This means that this is a combination of holomorphic and anti-holomorohic fucntions since both holomorphic and anti-holomorphic are present multiplicatively . Then we conclude that $ log(det(g))=\bar{f}(x)+f(x)$

$ det(g) = \bar{\omega}(x)\omega(x)$ where the Kähler metric is given by $ g=i\partial\bar{\partial}K$ , $ K$ is the Kähler potential.

Now a Schwarz type topological quantum field theory satisfies the condition that the correlation function of the observables of the theory must be independent of the metric which encodes the geometry of background manifold of the theory i.e. $ \frac{\delta}{\delta g^{\mu\nu}}\langle O_{1}….O_{\alpha}\rangle = \frac{\int_{}^{}[D\phi]O_1….O_\alpha e^{-iS[\phi]}}{\int_{}^{}[D\phi]e^{-iS}} = 0 $

If the metric is Hermitian , then its called a Hermitian quantum field theory . Now if we a consider that the metric is Kähler , what additional conditions should we impose on the theory so that the ambient manifold of the theory is Calabi-Yau?

The explicit form of the metric of Calabi-Yau space is still unknown, is that a big restriction against defining such a theory ?

Volume growth of balls in manifolds with bounded geometry

Suppose $ (M,g)$ is a Riemannian manifold. Let us say that $ M$ has bounded geometry if its injectivity radius is uniformly bounded below by a constant $ \epsilon>0$ , and the curvature tensor and all its derivatives are uniformly bounded over $ M$ .

Question: If $ M$ has bounded geometry, does it follow that volume growth of balls on $ M$ is at most exponential? More precisely, can one find constants $ C_1$ and $ C_2$ such that, for every $ x\in M$ and $ r>0$ ,

$ $ \text{Vol}(B_r(x))\leq C_1 e^{C_2 r}.$ $

Holomorphically convex manifolds and Bergman complete manifolds

Suppose $ X$ is a complex manifold which admits the Bergman metric. Suppose moreover that the Bergman metric of $ X$ is complete.

It is known that if $ X$ is (biholomorphic to) a bounded domain in some $ \mathbb C^n$ , then it is holomorphically convex, and hence Stein.

In 1962, S. Kobayashi asked in this paper whether in general a Bergman complete complex manifold is necessarily holomorphically convex.

I was wandering what is the state of the art about this.

My motivation comes in particular from the Shafarevich conjecture: the universal cover of a complex projective manifold should always be holomorphically convex.

Now, suppose you are given a complex projective manifold $ X$ whose universal cover $ \tilde X$ admits the Bergman metric $ ds_{\tilde X}^2$ . Then, by invariance under the deck transformation group, it descends to a Kähler metric on $ X$ which is compact and hence $ \tilde X$ is Bergman complete.

Thus, at least in this case, and affirmative answer to Kobayashi’s question would confirm the Shafarevich conjecture in this (very) particular case.

Thanks in advance.

Volume of balls in homogeneous manifolds

Let $ X=G/H$ be a homogeneous manifold, where $ G$ and $ H$ are connected Lie groups and assume there is given a $ G$ -invariant Riemannian metric on $ X$ . Let $ B(R)$ be the closed ball of radius $ R>0$ around the base point $ eH$ and let $ b(R)$ denote its volume. Is it rue that $ $ \lim_{\varepsilon\to 0}\ \limsup_{R\to\infty}\ \frac{b(R+\varepsilon)}{b(R)}=1? $ $ The idea somehow being that volume growth is largest with constant negative curvature in which case it is exponential and thus satisfies our claim.

The totally geodesic manifolds of 3-hyperbolic hypersurface

I am considering a geodesic γ∈H3(−1)γ∈H3(−1), g=4∑i=1(dξi)21−∑i=1(ξi)2. g=4∑i=1(dξi)21−∑i=1(ξi)2. Now I want to make a small perturbation of the geodesic γγ and then get the curve γ˜γ~, for any two different points of the new curve obtained, how to find the totally geodesic manifolds passing through these two points so that they do not intersect and are orthogonal to the new curve? Question: should I give a specific perturbation form?

Embeddings of Flag manifolds

Consider the flag manifold $ \mathbb{F}(a_1,\dots,a_k)$ parametrizing flags of type $ F^{a_1}\subseteq\dots\subseteq F^{a_k}\subseteq V$ in a vector spaces $ V$ of dimension $ n+1$ , where $ F^{a_i}$ is a sub-vector space of dimension $ a_i$ .

Then $ \mathbb{F}(a_1,\dots,a_k)$ embeds in the product of Grassmannians $ G(a_1,V)\times\dots\times G(a_k,V)$ which in turn embeds in $ \mathbb{P}^{N_1}\times\dots\times\mathbb{P}^{N_k}$ via the product of the Pl\”ucker emebeddings. Now we can embedd $ \mathbb{P}^{N_1}\times\dots\times\mathbb{P}^{N_k}$ in a projective space $ \mathbb{P}^N$ via the Segre embedding.

Finally, we get an embedding $ \mathbb{F}(a_1,\dots,a_k)\hookrightarrow\mathbb{P}^N$ . Is this embedding the minimal rational homogeneous embedding of $ \mathbb{F}(a_1,\dots,a_k)$ ?