Volume 2023, Issue 794

We prove that the perfect loop functor LX of a quasi-projective scheme X over a local non-archimedean field k satisfies arc-descent, strengthening a result of Drinfeld. Then we prove that for an unramified reductive group G , the map L ⁢ G → L ⁢ ( G / B ) {LG\rightarrow L(G/B)} is a v -surjection. This gives a mixed characteristic version (for v -topology) of an equal characteristic result (in étale topology) of Bouthier–Česnavičius. In the second part of the article, we use the above results to introduce a well-behaved notion of Deligne–Lusztig spaces X w ⁢ ( b ) {X_{w}(b)} attached to unramified p -adic reductive groups. We show that in various cases these sheaves are ind-representable, thus partially solving a question of Boyarchenko. Finally, we show that the natural covering spaces X ˙ w ˙ ⁢ ( b ) {\dot{X}_{\dot{w}}(b)} are pro-étale torsors over clopen subsets of X w ⁢ ( b ) {X_{w}(b)} , and analyze some examples.

For an invertible quasihomogeneous polynomial 𝒘 {{\boldsymbol{w}}} we prove an all-genus mirror theorem relating two cohomological field theories of Landau–Ginzburg type. On the B -side it is the Saito–Givental theory for a specific choice of a primitive form. On the A -side, it is the matrix factorization CohFT for the dual singularity 𝒘 T {{\boldsymbol{w}}^{T}} with the maximal diagonal symmetry group.

We consider ancient solutions to the mean curvature flow in R n + 1 \mathbb{R}^{n+1} ( n ≥ 3 n\geq 3 ) that are weakly convex, uniformly two-convex, and satisfy two pointwise derivative estimates | ∇ ⁡ A | ≤ γ 1 ⁢ | H | 2 \lvert\nabla A\rvert\leq\gamma_{1}\lvert H\rvert^{2} , | ∇ 2 ⁡ A | ≤ γ 2 ⁢ | H | 3 \lvert\nabla^{2}A\rvert\leq\gamma_{2}\lvert H\rvert^{3} . We show that such solutions are noncollapsed. As an application, in arbitrary codimension, we consider compact 𝑛-dimensional ( n ≥ 5 n\geq 5 ) solutions to the mean curvature flow in R N \mathbb{R}^{N} that satisfy the pinching condition | A | 2 < c ⁢ | H | 2 \lvert A\rvert^{2}<c\lvert H\rvert^{2} for a suitable constant c = c ⁢ ( n ) c=c(n) . We conclude that any blow-up model at the first singular time must be a codimension one shrinking sphere, shrinking cylinder, or translating bowl soliton.

It is well known that all Borel subgroups of a linear algebraic group are conjugate. Berest, Eshmatov, and Eshmatov have shown that this result also holds for the automorphism group Aut ⁢ ( 𝔸 2 ) {{\mathrm{Aut}}({\mathbb{A}}^{2})} of the affine plane. In this paper, we describe all Borel subgroups of the complex Cremona group Bir ⁢ ( ℙ 2 ) {{\rm Bir}({\mathbb{P}}^{2})} up to conjugation, proving in particular that they are not necessarily conjugate. In principle, this fact answers a question of Popov. More precisely, we prove that Bir ⁢ ( ℙ 2 ) {{\rm Bir}({\mathbb{P}}^{2})} admits Borel subgroups of any rank r ∈ { 0 , 1 , 2 } {r\in\{0,1,2\}} and that all Borel subgroups of rank r ∈ { 1 , 2 } {r\in\{1,2\}} are conjugate. In rank 0, there is a one-to-one correspondence between conjugacy classes of Borel subgroups of rank 0 and hyperelliptic curves of genus ℊ ≥ 1 {\mathcal{g}\geq 1} . Hence, the conjugacy class of a rank 0 Borel subgroup admits two invariants: a discrete one, the genus ℊ {\mathcal{g}} , and a continuous one, corresponding to the coarse moduli space of hyperelliptic curves of genus ℊ {\mathcal{g}} . This moduli space is of dimension 2 ⁢ ℊ - 1 {2\mathcal{g}-1} .

We study abelian envelopes for pseudo-tensor categories with the property that every object in the envelope is a quotient of an object in the pseudo-tensor category. We establish an intrinsic criterion on pseudo-tensor categories for the existence of an abelian envelope satisfying this quotient property. This allows us to interpret the extension of scalars and Deligne tensor product of tensor categories as abelian envelopes, and to enlarge the class of tensor categories for which all extensions of scalars and tensor products are known to remain tensor categories. For an affine group scheme G , we show that pseudo-tensor subcategories of 𝖱𝖾𝗉 ⁡ G {\operatorname{{\mathsf{Rep}}}G} have abelian envelopes with the quotient property, and we study many other such examples. This leads us to conjecture that all abelian envelopes satisfy the quotient property.

We prove that any weakly non-collapsed RCD space is actually non-collapsed, up to a renormalization of the measure. This confirms a conjecture raised by De Philippis and the second named author in full generality. One of the auxiliary results of independent interest that we obtain is about the link between the properties tr ⁡ ( Hess ⁡ f ) = Δ ⁢ f \operatorname{tr}(\operatorname{Hess}f)=\Delta f on U ⊆ X U\subseteq{\mathsf{X}} for every 𝑓 sufficiently regular, m = c ⁢ H n \mathfrak{m}=c\mathscr{H}^{n} on U ⊆ X U\subseteq{\mathsf{X}} for some c > 0 c>0 , where U ⊆ X U\subseteq{\mathsf{X}} is open and 𝖷 is a – possibly collapsed – RCD space of essential dimension 𝑛.

In a series of papers, Bamler [5, 4, 6] further developed the high-dimensional theory of Hamilton’s Ricci flow to include new monotonicity formulas, a completely general compactness theorem, and a long-sought partial regularity theory analogous to Cheeger–Colding theory. In this paper we give an application of his theory to lower bounds for the scalar curvatures of singularity models for Ricci flow. In the case of 4-dimensional non-Ricci-flat steady soliton singularity models, we obtain as a consequence a quadratic decay lower bound for the scalar curvature.

We construct a smooth complex projective rational surface with infinitely many mutually non-isomorphic real forms. This gives the first definite answer to a long-standing open question if a smooth complex projective rational surface has only finitely many non-isomorphic real forms or not.

Closed subschemes in projective space with a fixed Hilbert polynomial are parametrized by a Hilbert scheme. We classify the smooth ones. We identify numerical conditions on a polynomial that completely determine when the Hilbert scheme is smooth. We also reinterpret these smooth Hilbert schemes as generalized partial flag varieties and describe the subschemes being parametrized.