Volume 2020, Issue 760

We initiate the study of average intersection theory in real Grassmannians. We define the expected degree edeg⁡G⁢(k,n){\operatorname{edeg}G(k,n)} of the real Grassmannian G⁢(k,n){G(k,n)} as the average number of real k -planes meeting nontrivially k⁢(n-k){k(n-k)} random subspaces of ℝn{\mathbb{R}^{n}}, all of dimension n-k{n-k}, where these subspaces are sampled uniformly and independently from G⁢(n-k,n){G(n-k,n)}. We express edeg⁡G⁢(k,n){\operatorname{edeg}G(k,n)} in terms of the volume of an invariant convex body in the tangent space to the Grassmannian, and prove that for fixed k≥2{k\geq 2} and n→∞{n\to\infty}, edeg⁡G⁢(k,n)=deg⁡Gℂ⁢(k,n)12⁢εk+o⁢(1),\operatorname{edeg}G(k,n)=\deg G_{\mathbb{C}}(k,n)^{\frac{1}{2}\varepsilon_{k}% +o(1)}, where deg⁡Gℂ⁢(k,n){\deg G_{\mathbb{C}}(k,n)} denotes the degree of the corresponding complex Grassmannian and εk{\varepsilon_{k}} is monotonically decreasing with limk→∞⁡εk=1{\lim_{k\to\infty}\varepsilon_{k}=1}. In the case of the Grassmannian of lines, we prove the finer asymptotic edeg⁡G⁢(2,n+1)=83⁢π5/2⁢n⁢(π24)n⁢(1+𝒪⁢(n-1)).\operatorname{edeg}G(2,n+1)=\frac{8}{3\pi^{5/2}\sqrt{n}}\biggl{(}\frac{\pi^{2}% }{4}\biggr{)}^{n}(1+\mathcal{O}(n^{-1})). The expected degree turns out to be the key quantity governing questions of the random enumerative geometry of flats. We associate with a semialgebraic set X⊆ℝ⁢Pn-1{X\subseteq\mathbb{R}\mathrm{P}^{n-1}} of dimension n-k-1{n-k-1} its Chow hypersurface Z⁢(X)⊆G⁢(k,n){Z(X)\subseteq G(k,n)}, consisting of the k -planes A in ℝn{\mathbb{R}^{n}} whose projectivization intersects X . Denoting N:=k⁢(n-k){N:=k(n-k)}, we show that 𝔼⁢#⁢(g1⁢Z⁢(X1)∩⋯∩gN⁢Z⁢(XN))=edeg⁡G⁢(k,n)⋅∏i=1N|Xi||ℝ⁢Pm|,\mathbb{E}\#(g_{1}Z(X_{1})\cap\cdots\cap g_{N}Z(X_{N}))=\operatorname{edeg}G(k% ,n)\cdot\prod_{i=1}^{N}\frac{|X_{i}|}{|\mathbb{R}\mathrm{P}^{m}|}, where each Xi{X_{i}} is of dimension m=n-k-1{m=n-k-1}, the expectation is taken with respect to independent uniformly distributed g1,…,gm∈O⁢(n){g_{1},\dots,g_{m}\in O(n)} and |Xi|{|X_{i}|} denotes the m -dimensional volume of Xi{X_{i}}.

We study the cohomological Hall algebra Y♭{\operatorname{Y}\nolimits^{\flat}} of a Lagrangian substack Λ♭{\Lambda^{\flat}} of the moduli stack of representations of the preprojective algebra of an arbitrary quiver Q , and their actions on the cohomology of Nakajima quiver varieties. We prove that Y♭{\operatorname{Y}\nolimits^{\flat}} is pure and we compute its Poincaré polynomials in terms of (nilpotent) Kac polynomials. We also provide a family of algebra generators. We conjecture that Y♭{\operatorname{Y}\nolimits^{\flat}} is equal, after a suitable extension of scalars, to the Yangian 𝕐{\mathbb{Y}} introduced by Maulik and Okounkov. As a corollary, we prove a variant of Okounkov’s conjecture, which is a generalization of the Kac conjecture relating the constant term of Kac polynomials to root multiplicities of Kac–Moody algebras.

We study the intersection of two copies of Gr⁢(2,5){\mathrm{Gr}(2,5)} embedded in ℙ9{{{\mathbb{P}}}^{9}}, and the intersection of the two projectively dual Grassmannians in the dual projective space. These intersections are deformation equivalent, derived equivalent Calabi–Yau threefolds. We prove that generically they are not birational. As a consequence, we obtain a counterexample to the birational Torelli problem for Calabi–Yau threefolds. We also show that these threefolds give a new pair of varieties whose classes in the Grothendieck ring of varieties are not equal, but whose difference is annihilated by a power of the class of the affine line. Our proof of non-birationality involves a detailed study of the moduli stack of Calabi–Yau threefolds of the above type, which may be of independent interest.

We generalize the maximal time existence of Kähler–Ricci flow in [ G. Tian and Z. Zhang , On the Kähler–Ricci flow on projective manifolds of general type, Chin. Ann. Math. Ser. B 27 (2006), no. 2, 179–192] and [ J. Song and G. Tian , The Kähler–Ricci flow through singularities, Invent. Math. 207 (2017), no. 2, 519–595] to the conical case. Furthermore, if the log canonical bundle KM+(1-β)⁢[D]{K_{M}+(1-\beta)[D]} is big or big and nef, we can examine the limit behaviors of such conical Kähler–Ricci flow. Moreover, these results still hold when D is a simple normal crossing divisor.

In this paper, we reprove a global converse theorem of Cogdell and Piatetski-Shapiro using purely global methods.

In this paper we prove that groups as in the title are convex cocompact in the mapping class group.

We study the covariant model structure on dendroidal spaces, and establish direct relations to the homotopy theory of algebras over a simplicial operad as well as to the homotopy theory of special Γ-spaces. As an important tool in the latter comparison, we present a sharpening of the classical Barratt–Priddy–Quillen theorem.

We give a direct, explicit and self-contained construction of a local Lie groupoid integrating a given Lie algebroid which only depends on the choice of a spray vector field lifting the underlying anchor map. This construction leads to a complete account of local Lie theory and, in particular, to a finite-dimensional proof of the fact that the category of germs of local Lie groupoids is equivalent to that of Lie algebroids.