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3. Connecting Equivariant Notions and Derived Algebraic Geometry

    1. Problem 3.1.

      [M. A. Hill] What is an {eĢtale} map of Green or Tambara functors?
        • Problem 3.2.

          [D. Nardin] What are the explicit generators of $Pic(Sp^G)$?
            •     Wall, Roberts, Damon, Gusein-Zade, and others have defined and studied the local $G$-degree for an equivariant polynomial map $f: \R^n \rightarrow \R^n$ where $\R^n$ is given an action of a group $G$. Their work is motivated by singularity theory, and they show that the invariants encode interesting information, but equivariant homotopy theory does not appear explicitly. Perhaps the notions of degree appearing in these works are connected to the degree in equivariant homotopy theory. We pose several questions along these lines.

              Problem 3.3.

              [J. Kass via K. Wickelgren] (1) Suppose we are given a linear representation of a finite group $G$ on $\C^n$ and a $G$-equivariant polynomial function $f : \C^n \rightarrow \C^n$ with an isolated zero at the origin. Then the local algebra \[ Q(f) := \C [\![ x_1, \dots, x_n ]\!]/(f_1, \dots, f_n)\] of $f$ is a $\C$-algebra with $G$-action. Is this $G$-representation equal to the permutation representation associated to the local degree in equivariant homotopy theory?

              Recall the local degree is an element of the Burnside ring, hence has an associated virtual permutation representation. Palamodov has shown that the rank of $Q(f)$ is the topological local degree.

              (2) Suppose that, in (1), we replace the complex numbers $\C$ with the real numbers $\R$. Then $Q(f)$ carries a distinguished G-invariant symmetric bilinear pairing $\beta$. (See e.g. Eisenbud-Levine.)

              Quite generally, any finite dimensional real $G$-representation with $G$-invariant symmetric bilinear form $(V, \beta)$ decomposes in an essentially unique manner as an orthogonal direct sum \[ V = V_+ \oplus V_- \oplus V_0\] where $\beta$ is positive definite on $V_+$, negative definite on $V_-$, and zero on $V_0$. Call the virtual representation $[V_+] - [V_-]$ the $G$-signature of $V$. Is this virtual $G$-representation of $( Q(f), \beta)$ equal to the permutation representation associated to the local degree in equivariant homotopy theory?

              Note: Eisenbud-Levine and Khimshiashvili have proven that the signature of $\beta$ equals the topological local degree of $f$.

              (3) Suppose now that $f : \C^2 \rightarrow \C^2$ is the gradient of a $G$-invariant equation for a plane curve singularity $0 \in X \subset \C^2$. Then $G$ acts on the analytic branches of the singularity. Is this $G$-set equal to the local degree of $f$ in equivariant homotopy theory?
                • Long Term

                  Problem 3.4.

                  [A. Blumberg] How best should we be making sense of equivariant derived algebraic geometry? (e.g. What is Spec? What site to use? What sheaf? etc.)

                      Cite this as: AimPL: Equivariant derived algebraic geometry, available at http://aimpl.org/equideralggeom.