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Anon
Anon
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Suppose $C \to (S,0)$ is a family of curves (with at most nodal singularities) parametrized by a pointed curve $(S,0)$, that is proper and flat. Let $u:C \to X/G$ be a family of maps to a quotient stack ($X$ is a complex non-singular variety and $G$ is a reductive group). Given that the equivariant homology class $(u_s)_*[C] \in H^2_G(X)$$(u_s)_*[C_s] \in H^2_G(X)$ is constant for $s \in S \backslash \{0\}$, does $(u_0)_*[C]$$(u_0)_*[C_0]$ also represent the same homology class? Is this obvious?

Suppose $C \to (S,0)$ is a family of curves (with at most nodal singularities) parametrized by a pointed curve $(S,0)$, that is proper and flat. Let $u:C \to X/G$ be a family of maps to a quotient stack ($X$ is a complex non-singular variety and $G$ is a reductive group). Given that the equivariant homology class $(u_s)_*[C] \in H^2_G(X)$ is constant for $s \in S \backslash \{0\}$, does $(u_0)_*[C]$ also represent the same homology class? Is this obvious?

Suppose $C \to (S,0)$ is a family of curves (with at most nodal singularities) parametrized by a pointed curve $(S,0)$, that is proper and flat. Let $u:C \to X/G$ be a family of maps to a quotient stack ($X$ is a complex non-singular variety and $G$ is a reductive group). Given that the equivariant homology class $(u_s)_*[C_s] \in H^2_G(X)$ is constant for $s \in S \backslash \{0\}$, does $(u_0)_*[C_0]$ also represent the same homology class? Is this obvious?

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Anon
Anon
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Suppose $C \to (S,0)$ is a family of curves (with at most nodal singularities) parametrized by a pointed curve $(S,0)$, that is proper and flat. Let $u:C \to X/G$ isbe a family of maps to a quotient stack ($X$ is a complex non-singular variety and $G$ is a reductive group). Given that the equivariant homology class $u_s^*[C] \in H^2_G(X)$$(u_s)_*[C] \in H^2_G(X)$ is constant for $s \in S \backslash \{0\}$, does $u_0^*[C]$$(u_0)_*[C]$ also represent the same homology class? Is this obvious?

Suppose $C \to (S,0)$ is a family of curves (with at most nodal singularities) parametrized by a pointed curve $(S,0)$, and $u:C \to X/G$ is a family of maps to a quotient stack ($X$ is a complex non-singular variety and $G$ is a reductive group). Given that the equivariant homology class $u_s^*[C] \in H^2_G(X)$ is constant for $s \in S \backslash \{0\}$, does $u_0^*[C]$ also represent the same homology class? Is this obvious?

Suppose $C \to (S,0)$ is a family of curves (with at most nodal singularities) parametrized by a pointed curve $(S,0)$, that is proper and flat. Let $u:C \to X/G$ be a family of maps to a quotient stack ($X$ is a complex non-singular variety and $G$ is a reductive group). Given that the equivariant homology class $(u_s)_*[C] \in H^2_G(X)$ is constant for $s \in S \backslash \{0\}$, does $(u_0)_*[C]$ also represent the same homology class? Is this obvious?

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Anon
Anon
  • 778
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  • 11

Is equivariant homology class preserved in the limit?

Suppose $C \to (S,0)$ is a family of curves (with at most nodal singularities) parametrized by a pointed curve $(S,0)$, and $u:C \to X/G$ is a family of maps to a quotient stack ($X$ is a complex non-singular variety and $G$ is a reductive group). Given that the equivariant homology class $u_s^*[C] \in H^2_G(X)$ is constant for $s \in S \backslash \{0\}$, does $u_0^*[C]$ also represent the same homology class? Is this obvious?