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Register a new userOrbits in $(mathbb{P}^r)^n$ and equivariant quantum cohomology
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We compute the $GL_{r+1}$-equivariant Chow class of the $GL_{r+1}$-orbit closure of any point $(x_1, ldots, x_n) in (mathbb{P}^r)^n$ in terms of the rank polytope of the matroid represented by $x_1, ldots, x_n in mathbb{P}^r$. Using these classes and generalizations involving point configurations in higher dimensional projective spaces, we define for each $dtimes n$ matrix $M$ an $n$-ary operation $[M]_hbar$ on the small equivariant quantum cohomology ring of $mathbb{P}^r$, which is the $n$-ary quantum product when $M$ is an invertible matrix. We prove that $M mapsto [M]_hbar$ is a valuative matroid polytope association. Like the quantum product, these operations satisfy recursive properties encoding solutions to enumerative problems involving point configurations of given moduli in a relative setting. As an application, we compute the number of line sections with given moduli of a general degree $2r+1$ hypersurface in $mathbb{P}^r$, generalizing the known case of quintic plane curves.
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This note contains a generalization to $p>2$ of the authors previous calculations of the coefficients of $(mathbb{Z}/2)^n$-equivariant ordinary cohomology with coefficients in the constant $mathbb{Z}/2$-Mackey functor. The algberaic results by S.Kriz allow us to calculate the coefficients of the geometric fixed point spectrum $Phi^{(mathbb{Z}/p)^n}Hmathbb{Z}/p$, and more generally, the $mathbb{Z}$-graded coefficients of the localization of $Hmathbb{Z}/p_{(mathbb{Z}/p)^n}$ by inverting any chosen set of embeddings $S^0rightarrow S^{alpha_i}$ where $alpha_i$ are non-trivial irreducible representations. We also calculate the $RO(G)^+$-graded coefficients of $Hmathbb{Z}/p_{(mathbb{Z}/p)^n}$, which means the cohomology of a point indexed by an actual (not virtual) representation. (This is the non-derived part, which has a nice algebraic description.)
We propose a new theory of (non-split) P^n-functors. These are F: A -> B for which the adjunction monad RF is a repeated extension of Id_A by powers of an autoequivalence H and three conditions are satisfied: the monad condition, the adjoints condition, and the highest degree term condition. This unifies and extends the two earlier notions of spherical functors and split P^n-functors. We construct the P-twist of such F and prove it to be an autoequivalence. We then give a criterion for F to be a P^n-functor which is stronger than the definition but much easier to check in practice. It involves only two conditions: the strong monad condition and the weak adjoints condition. For split P^n-functors, we prove Segals conjecture on their relation to spherical functors. Finally, we give four examples of non-split P^n-functors: spherical functors, extensions by zero, cyclic covers, and family P-twists. For the latter, we show the P-twist to be the derived monodromy of associated Mukai flop, the so-called `flop-flop = twist formula.
Let $p(cdot): mathbb R^nto(0,infty)$ be a variable exponent function satisfying the globally log-Holder continuous condition. In this article, the authors first obtain a decomposition for any distribution of the variable weak Hardy space into good and bad parts and then prove the following real interpolation theorem between the variable Hardy space $H^{p(cdot)}(mathbb R^n)$ and the space $L^{infty}(mathbb R^n)$: begin{equation*} (H^{p(cdot)}(mathbb R^n),L^{infty}(mathbb R^n))_{theta,infty} =W!H^{p(cdot)/(1-theta)}(mathbb R^n),quad thetain(0,1), end{equation*} where $W!H^{p(cdot)/(1-theta)}(mathbb R^n)$ denotes the variable weak Hardy space. As an application, the variable weak Hardy space $W!H^{p(cdot)}(mathbb R^n)$ with $p_-:=mathopmathrm{ess,inf}_{xinrn}p(x)in(1,infty)$ is proved to coincide with the variable Lebesgue space $W!L^{p(cdot)}(mathbb R^n)$.
A projectively normal Calabi-Yau threefold $X subseteq mathbb{P}^n$ has an ideal $I_X$ which is arithmetically Gorenstein, of Castelnuovo-Mumford regularity four. Such ideals have been intensively studied when $I_X$ is a complete intersection, as well as in the case where $X$ is codimension three. In the latter case, the Buchsbaum-Eisenbud theorem shows that $I_X$ is given by the Pfaffians of a skew-symmetric matrix. A number of recent papers study the situation when $I_X$ has codimension four. We prove there are 16 possible betti tables for an arithmetically Gorenstein ideal $I$ with $mathrm{codim}(I)=4=mathrm{reg}(I)$, and that exactly 8 of these occur for smooth irreducible nondegenerate threefolds. We investigate the situation in codimension five or more, obtaining examples of $X$ with $h^{p,q}(X)$ not among those appearing for $I_X$ of lower codimension or as complete intersections in toric Fano varieties. A key tool in our approach is the use of inverse systems to identify possible betti tables for $X$.
We introduce a Bredon motivic cohomology theory for smooth schemes defined over a field and equipped with an action by a finite group. These cohomology groups are defined for finite dimensional representations as the hypercohomology of complexes of equivariant correspondences in the equivariant Nisnevich topology. We generalize the theory of presheaves with transfers to the equivariant setting and prove a Cancellation Theorem.
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