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In the case of a genus $g$ surface $\Sigma$, it is well known that $MCG(\Sigma) = \pi_0 \mathit{Diff}^+(\Sigma)$ is generated by Dehn twists, which come from a Kahler degeneration with smooth total space in which $\Sigma$ acquires an $A_1$-singularity. This has a generalization to higher dimension: if $X$ is a $2n$-dimensional symplectic manifold, with a (framed) Lagrangian sphere $L \subset X$, there is a diffeomorphism which preserves the symplectic form defined on the local model $T^* S^n$ as the time-$\pi$ map of a truncated geodesic flow on the cotangent bundle. Like the surface case, generalized Dehn twists also have a singularity-theory description as the monodromy of a Kahler degeneration with smooth total space where $M$ acquires an $A_1$-singularity (see Seidel's " A long exact sequence in Floer homology").

When $n$ is odd, Dehn twists have infinite order in $\pi_0 Symp(X)$ and $\pi_0 \mathit{Diff}^+(X)$ because of Picard-Lefschetz formula for the action on middle homology. When $n=4$, Dehn twists have order 2 in $\pi_0 \mathit{Diff}^+(X)$ (here smooth topology and symplectic topology diverge). For $n>4$ even, Dehn twists have finite order in $\pi_0 \mathit{Diff}^+(X)$ (see Lemma 4.1 in Exotic Iterated Dehn twists). We can even bound that order explicitly as follows from the work of Stevens and Kauffman-Krilov.

The monodromy around a Kahler degeneration to an ADE-singularities are known to yield products of Dehn twists (See Seidel's "Lagrangian 2-spheres can be symplectically knotted" or Smith-Thomas-Yau "Symplectic conifold transitions").

My question is about generalization of that:

1. Are there singularities whose miniversal deformation space is interesting enough so that it yields non-trivial (ideally, infinite order) elements in $\pi_i \mathit{Diff}^+(X)$ with $i \geq 1$?

2. Is there some kind of generalized "Picard-Lefschetz" formula for the action in that case?

In the case of a genus $g$ surface $\Sigma$, it is well known that $MCG(\Sigma) = \pi_0 \operatorname{Diff}^+(\Sigma)$ is generated by Dehn twists, which come from a Kahler degeneration with smooth total space in which $\Sigma$ acquires an $A_1$-singularity. This has a generalization to higher dimension: if $X$ is a $2n$-dimensional symplectic manifold, with a (framed) Lagrangian sphere $L \subset X$, there is a diffeomorphism which preserves the symplectic form defined on the local model $T^* S^n$ as the time-$\pi$ map of a truncated geodesic flow on the cotangent bundle. Like the surface case, generalized Dehn twists also have a singularity-theory description as the monodromy of a Kahler degeneration with smooth total space where $M$ acquires an $A_1$-singularity (see Seidel's " A long exact sequence in Floer homology").

When $n$ is odd, Dehn twists have infinite order in $\pi_0 \operatorname{Symp}(X)$ and $\pi_0 \operatorname{Diff}^+(X)$ because of Picard-Lefschetz formula for the action on middle homology. When $n=4$, Dehn twists have order 2 in $\pi_0 \operatorname{Diff}^+(X)$ (here smooth topology and symplectic topology diverge). For $n>4$ even, Dehn twists have finite order in $\pi_0 \operatorname{Diff}^+(X)$ (see Lemma 4.1 in Exotic Iterated Dehn twists). We can even bound that order explicitly as follows from the work of Stevens and Kauffman-Krilov.

The monodromy around a Kahler degeneration to an ADE-singularities are known to yield products of Dehn twists (See Seidel's "Lagrangian 2-spheres can be symplectically knotted" or Smith-Thomas-Yau "Symplectic conifold transitions").

My question is about generalization of that:

1. Are there singularities whose miniversal deformation space is interesting enough so that it yields non-trivial (ideally, infinite order) elements in $\pi_i \operatorname{Diff}^+(X)$ with $i \geq 1$?

2. Is there some kind of generalized "Picard-Lefschetz" formula for the action in that case?

In the case of a genus $g$ surface $\Sigma$, it is well known that $MCG(\Sigma) = \pi_0 Diff^+(\Sigma)$ is generated by Dehn twists, which come from a Kahler degeneration with smooth total space in which $\Sigma$ acquires an $A_1$-singularity. This has a generalization to higher dimension: if $X$ is a $2n$-dimensional symplectic manifold, with a (framed) Lagrangian sphere $L \subset X$, there is a diffeomorphism which preserves the symplectic form defined on the local model $T^* S^n$ as the time-$\pi$ map of a truncated geodesic flow on the cotangent bundle. Like the surface case, generalized Dehn twists also have a singularity-theory description as the monodromy of a Kahler degeneration with smooth total space where $M$ acquires an $A_1$-singularity (see Seidel's " A long exact sequence in Floer homology").

When $n$ is odd, Dehn twists have infinite order in $\pi_0 Symp(X)$ and $\pi_0 Diff^+(X)$ because of Picard-Lefschetz formula for the action on middle homology. When $n=4$, Dehn twists have order 2 in $\pi_0 Diff^+(X)$ (here smooth topology and symplectic topology diverge). For $n>4$ even, Dehn twists have finite order in $\pi_0 Diff^+(X)$ (see Lemma 4.1 in Exotic Iterated Dehn twists). We can even bound that order explicitly as follows from the work of Stevens and Kauffman-Krilov.

The monodromy around a Kahler degeneration to an ADE-singularities are known to yield products of Dehn twists (See Seidel's "Lagrangian 2-spheres can be symplectically knotted" or Smith-Thomas-Yau "Symplectic conifold transitions").

My question is about generalization of that:

1. Are there singularities whose miniversal deformation space is interesting enough so that it yields non-trivial (ideally, infinite order) elements in $\pi_i Diff(X)$ with $i \geq 1$?

2. Is there some kind of generalized "Picard-Lefschetz" formula for the action in that case?

In the case of a genus $g$ surface $\Sigma$, it is well known that $MCG(\Sigma) = \pi_0 \mathit{Diff}^+(\Sigma)$ is generated by Dehn twists, which come from a Kahler degeneration with smooth total space in which $\Sigma$ acquires an $A_1$-singularity. This has a generalization to higher dimension: if $X$ is a $2n$-dimensional symplectic manifold, with a (framed) Lagrangian sphere $L \subset X$, there is a diffeomorphism which preserves the symplectic form defined on the local model $T^* S^n$ as the time-$\pi$ map of a truncated geodesic flow on the cotangent bundle. Like the surface case, generalized Dehn twists also have a singularity-theory description as the monodromy of a Kahler degeneration with smooth total space where $M$ acquires an $A_1$-singularity (see Seidel's " A long exact sequence in Floer homology").

When $n$ is odd, Dehn twists have infinite order in $\pi_0 Symp(X)$ and $\pi_0 \mathit{Diff}^+(X)$ because of Picard-Lefschetz formula for the action on middle homology. When $n=4$, Dehn twists have order 2 in $\pi_0 \mathit{Diff}^+(X)$ (here smooth topology and symplectic topology diverge). For $n>4$ even, Dehn twists have finite order in $\pi_0 \mathit{Diff}^+(X)$ (see Lemma 4.1 in Exotic Iterated Dehn twists). We can even bound that order explicitly as follows from the work of Stevens and Kauffman-Krilov.

The monodromy around a Kahler degeneration to an ADE-singularities are known to yield products of Dehn twists (See Seidel's "Lagrangian 2-spheres can be symplectically knotted" or Smith-Thomas-Yau "Symplectic conifold transitions").

My question is about generalization of that:

1. Are there singularities whose miniversal deformation space is interesting enough so that it yields non-trivial (ideally, infinite order) elements in $\pi_i \mathit{Diff}^+(X)$ with $i \geq 1$?

2. Is there some kind of generalized "Picard-Lefschetz" formula for the action in that case?