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Davide Cesare Veniani
Davide Cesare Veniani
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Action on whole cohomology ofby automorphisms of ihs manifolds

For all known deformation types of irreducible holomorphic symplectic manifolds (which I call K3, K3$^n$, Kum$^n$, OG$^3$, OG$^5$, the exponent being half the complex dimension), it is known that the map ${\rm Aut}(X) \rightarrow {\rm O}(H^*(X,\mathbb Z))$${\rm Aut}(X) \rightarrow {\rm Aut}(H^*(X,\mathbb Z))$ is injective.

For K3, K3$^n$ and OG$^5$ this follows from the fact that the map ${\rm Aut}(X) \rightarrow {\rm O}(H^2(X,\mathbb Z))$${\rm Aut}(X) \rightarrow {\rm Aut}(H^2(X,\mathbb Z))$ is injective (Beauville, Mongardi--Rapagnetta). For Kum$^n$, this is a theorem by Oguiso. For OG$^3$, this is known to experts.

Is this known in general?

Action on whole cohomology of automorphisms of ihs manifolds

For all known deformation types of irreducible holomorphic symplectic manifolds (which I call K3, K3$^n$, Kum$^n$, OG$^3$, OG$^5$, the exponent being half the complex dimension), it is known that the map ${\rm Aut}(X) \rightarrow {\rm O}(H^*(X,\mathbb Z))$ is injective.

For K3, K3$^n$ and OG$^5$ this follows from the fact that the map ${\rm Aut}(X) \rightarrow {\rm O}(H^2(X,\mathbb Z))$ is injective (Beauville, Mongardi--Rapagnetta). For Kum$^n$, this is a theorem by Oguiso. For OG$^3$, this is known to experts.

Is this known in general?

Action on cohomology by automorphisms of ihs manifolds

For all known deformation types of irreducible holomorphic symplectic manifolds (which I call K3, K3$^n$, Kum$^n$, OG$^3$, OG$^5$, the exponent being half the complex dimension), it is known that the map ${\rm Aut}(X) \rightarrow {\rm Aut}(H^*(X,\mathbb Z))$ is injective.

For K3, K3$^n$ and OG$^5$ this follows from the fact that the map ${\rm Aut}(X) \rightarrow {\rm Aut}(H^2(X,\mathbb Z))$ is injective (Beauville, Mongardi--Rapagnetta). For Kum$^n$, this is a theorem by Oguiso. For OG$^3$, this is known to experts.

Is this known in general?

Source Link
Davide Cesare Veniani
Davide Cesare Veniani
  • 845
  • 6
  • 15

Action on whole cohomology of automorphisms of ihs manifolds

For all known deformation types of irreducible holomorphic symplectic manifolds (which I call K3, K3$^n$, Kum$^n$, OG$^3$, OG$^5$, the exponent being half the complex dimension), it is known that the map ${\rm Aut}(X) \rightarrow {\rm O}(H^*(X,\mathbb Z))$ is injective.

For K3, K3$^n$ and OG$^5$ this follows from the fact that the map ${\rm Aut}(X) \rightarrow {\rm O}(H^2(X,\mathbb Z))$ is injective (Beauville, Mongardi--Rapagnetta). For Kum$^n$, this is a theorem by Oguiso. For OG$^3$, this is known to experts.

Is this known in general?