Sheaves of maximal intersection and multiplicities of stable log maps

A great number of theoretical results are known about log Gromov–Witten invariants (Abramovich and Chen in Asian J Math 18:465–488, 2014; Chen in Ann Math (2) 180:455–521, 2014; Gross and Siebert J Am Math Soc 26: 451–510, 2013), but few calculations are worked out. In this paper we restrict to surfaces and to genus 0 stable log maps of maximal tangency. We ask how various natural components of the moduli space contribute to the log Gromov–Witten invariants. The first such calculation (Gross et al. in Duke Math J 153:297–362, 2010, Proposition 6.1) by Gross–Pandharipande–Siebert deals with multiple covers over rigid curves in the log Calabi–Yau setting. As a natural continuation, in this paper we compute the contributions of non-rigid irreducible curves in the log Calabi–Yau setting and that of the union of two rigid curves in general position. For the former, we construct and study a moduli space of “logarithmic” 1-dimensional sheaves and compare the resulting multiplicity with tropical multiplicity. For the latter, we explicitly describe the components of the moduli space and work out the logarithmic deformation theory in full, which we then compare with the deformation theory of the analogous relative stable maps.

Isomonodromic Laplace transform with coalescing eigenvalues and confluence of Fuchsian singularities

We consider a Pfaffian system expressing isomonodromy of an irregular system of Okubo type, depending on complex deformation parameters $$u=(u_1,\ldots ,u_n)$$ u = ( u 1 , … , u n ) , which are eigenvalues of the leading matrix at the irregular singularity. At the same time, we consider a Pfaffian system of non-normalized Schlesinger-type expressing isomonodromy of a Fuchsian system, whose poles are the deformation parameters $$u_1,\ldots ,u_n$$ u 1 , … , u n . The parameters vary in a polydisc containing a coalescence locus for the eigenvalues of the leading matrix of the irregular system, corresponding to confluence of the Fuchsian singularities. We construct isomonodromic selected and singular vector solutions of the Fuchsian Pfaffian system together with their isomonodromic connection coefficients, so extending a result of Balser et al. (I SIAM J Math Anal 12(5): 691–721, 1981) and Guzzetti (Funkcial Ekvac 59(3): 383–433, 2016) to the isomonodromic case, including confluence of singularities. Then, we introduce an isomonodromic Laplace transform of the selected and singular vector solutions, allowing to obtain isomonodromic fundamental solutions for the irregular system, and their Stokes matrices expressed in terms of connection coefficients. These facts, in addition to extending (Balser et al. in I SIAM J Math Anal 12(5): 691–721, 1981; Guzzetti in Funkcial Ekvac 59(3): 383–433, 2016) to the isomonodromic case (with coalescences/confluences), allow to prove by means of Laplace transform the main result of Cotti et al. (Duke Math J arXiv:1706.04808, 2017), namely the analytic theory of non-generic isomonodromic deformations of the irregular system with coalescing eigenvalues.

Geodesic planes in geometrically finite acylindrical -manifolds

Let M be a geometrically finite acylindrical hyperbolic $3$ -manifold and let $M^*$ denote the interior of the convex core of M. We show that any geodesic plane in $M^*$ is either closed or dense, and that there are only countably many closed geodesic planes in $M^*$ . These results were obtained by McMullen, Mohammadi and Oh [Geodesic planes in hyperbolic 3-manifolds. Invent. Math.209 (2017), 425–461; Geodesic planes in the convex core of an acylindrical 3-manifold. Duke Math. J., to appear, Preprint, 2018, arXiv:1802.03853] when M is convex cocompact. As a corollary, we obtain that when M covers an arithmetic hyperbolic $3$ -manifold $M_0$ , the topological behavior of a geodesic plane in $M^*$ is governed by that of the corresponding plane in $M_0$ . We construct a counterexample of this phenomenon when $M_0$ is non-arithmetic.

Finiteness and infiniteness results for Torelli groups of (hyper-)Kähler manifolds

The Torelli group $$\mathcal T(X)$$ T ( X ) of a closed smooth manifold X is the subgroup of the mapping class group $$\pi _0(\mathrm {Diff}^+(X))$$ π 0 ( Diff + ( X ) ) consisting of elements which act trivially on the integral cohomology of X. In this note we give counterexamples to Theorem 3.4 by Verbitsky (Duke Math J 162(15):2929–2986, 2013) which states that the Torelli group of simply connected Kähler manifolds of complex dimension $$\ge 3$$ ≥ 3 is finite. This is done by constructing under some mild conditions homomorphisms $$J: \mathcal T(X) \rightarrow H^3(X;\mathbb Q)$$ J : T ( X ) → H 3 ( X ; Q ) and showing that for certain Kähler manifolds this map is non-trivial. We also give a counterexample to Theorem 3.5 (iv) in (Verbitsky in Duke Math J 162(15):2929–2986, 2013) where Verbitsky claims that the Torelli group of hyperkähler manifolds are finite. These examples are detected by the action of diffeomorphsims on $$\pi _4(X)$$ π 4 ( X ) . Finally we confirm the finiteness result for the special case of the hyperkähler manifold $$K^{[2]}$$ K [ 2 ] .

An Algebraic Brascamp–Lieb Inequality

The Brascamp–Lieb inequalities are a very general class of classical multilinear inequalities, well-known examples of which being Hölder’s inequality, Young’s convolution inequality, and the Loomis–Whitney inequality. Conventionally, a Brascamp–Lieb inequality is defined as a multilinear Lebesgue bound on the product of the pullbacks of a collection of functions $$f_j\in L^{q_j}(\mathbb {R}^{n_j})$$ f j ∈ L q j ( R n j ) , for $$j=1,\ldots ,m$$ j = 1 , … , m , under some corresponding linear maps $$B_j$$ B j . This regime is now fairly well understood (Bennett et al. in Geom Funct Anal 17(5):1343–1415, 2008), and moving forward there has been interest in nonlinear generalisations, where $$B_j$$ B j is now taken to belong to some suitable class of nonlinear maps. While there has been great recent progress on the question of local nonlinear Brascamp–Lieb inequalities (Bennett et al. in Duke Math J 169(17):3291–3338, 2020), there has been relatively little regarding global results; this paper represents some progress along this line of enquiry. We prove a global nonlinear Brascamp–Lieb inequality for ‘quasialgebraic’ maps, a class that encompasses polynomial and rational maps, as a consequence of the multilinear Kakeya-type inequalities of Zhang and Zorin-Kranich. We incorporate a natural affine-invariant weight that both compensates for local degeneracies and yields a constant with minimal dependence on the underlying maps. We then show that this inequality generalises Young’s convolution inequality on algebraic groups with suboptimal constant.

Isomorphisms between determinantal point processes with translation-invariant kernels and Poisson point processes

We prove the Bernoulli property for determinantal point processes on $ \mathbb{R}^d $ with translation-invariant kernels. For the determinantal point processes on $ \mathbb{Z}^d $ with translation-invariant kernels, the Bernoulli property was proved by Lyons and Steif [Stationary determinantal processes: phase multiplicity, bernoullicity, and domination. Duke Math. J.120 (2003), 515–575] and Shirai and Takahashi [Random point fields associated with certain Fredholm determinants II: fermion shifts and their ergodic properties. Ann. Probab.31 (2003), 1533–1564]. We prove its continuum version. For this purpose, we also prove the Bernoulli property for the tree representations of the determinantal point processes.

Curved Rickard complexes and link homologies

Rickard complexes in the context of categorified quantum groups can be used to construct braid group actions. We define and study certain natural deformations of these complexes which we call curved Rickard complexes. One application is to obtain deformations of link homologies which generalize those of Batson–Seed [3] [J. Batson and C. Seed, A link-splitting spectral sequence in Khovanov homology, Duke Math. J. 164 2015, 5, 801–841] and Gorsky–Hogancamp [E. Gorsky and M. Hogancamp, Hilbert schemes and y-ification of Khovanov–Rozansky homology, preprint 2017] to arbitrary representations/partitions. Another is to relate the deformed homology defined algebro-geometrically in [S. Cautis and J. Kamnitzer, Knot homology via derived categories of coherent sheaves IV, colored links, Quantum Topol. 8 2017, 2, 381–411] to categorified quantum groups (this was the original motivation for this paper).

Toric degenerations of cluster varieties and cluster duality

We introduce the notion of a $Y$-pattern with coefficients and its geometric counterpart: an $\mathcal {X}$-cluster variety with coefficients. We use these constructions to build a flat degeneration of every skew-symmetrizable specially completed $\mathcal {X}$-cluster variety $\widehat {\mathcal {X} }$ to the toric variety associated to its g-fan. Moreover, we show that the fibers of this family are stratified in a natural way, with strata the specially completed $\mathcal {X}$-varieties encoded by $\operatorname {Star}(\tau )$ for each cone $\tau$ of the $\mathbf {g}$-fan. These strata degenerate to the associated toric strata of the central fiber. We further show that the family is cluster dual to $\mathcal {A}_{\mathrm {prin}}$ of Gross, Hacking, Keel and Kontsevich [Canonical bases for cluster algebras, J. Amer. Math. Soc. 31 (2018), 497–608], and the fibers cluster dual to $\mathcal {A} _t$. Finally, we give two applications. First, we use our construction to identify the toric degeneration of Grassmannians from Rietsch and Williams [Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians, Duke Math. J. 168 (2019), 3437–3527] with the Gross–Hacking–Keel–Kontsevich degeneration in the case of $\operatorname {Gr}_2(\mathbb {C} ^{5})$. Next, we use it to link cluster duality to Batyrev–Borisov duality of Gorenstein toric Fanos in the context of mirror symmetry.

When the sieve works II

For a set of primes {\mathcal{P}}, let {\Psi(x;\mathcal{P})} be the number of positive integers {n\leq x} all of whose prime factors lie in {\mathcal{P}}. In this paper we classify the sets of primes {\mathcal{P}} such that {\Psi(x;\mathcal{P})} is within a constant factor of its expected value. This task was recently initiated by Granville, Koukoulopoulos and Matomäki [A. Granville, D. Koukoulopoulos and K. Matomäki, When the sieve works, Duke Math. J. 164 2015, 10, 1935–1969] and their main conjecture is proved in this paper. In particular, our main theorem implies that, if not too many large primes are sieved out in the sense that\sum_{\begin{subarray}{c}p\in\mathcal{P}\\ x^{1/v}<p\leq x^{1/u}\end{subarray}}\frac{1}{p}\geq\frac{1+\varepsilon}{u},for some {\varepsilon>0} and {v\geq u\geq 1}, then\Psi(x;\mathcal{P})\gg_{\varepsilon,v}x\prod_{\begin{subarray}{c}p\leq x\\ p\notin\mathcal{P}\end{subarray}}\bigg{(}1-\frac{1}{p}\bigg{)}.

UNIFORM ASYMPTOTIC FORMULAS FOR RESTRICTED BIPARTITE PARTITIONS

In this paper, we investigate $\unicode[STIX]{x1D70B}(m,n)$, the number of partitions of the bipartite number$(m,n)$ into steadily decreasing parts, introduced by Carlitz [‘A problem in partitions’, Duke Math. J.30 (1963), 203–213]. We give a relation between $\unicode[STIX]{x1D70B}(m,n)$ and the crank statistic $M(m,n)$ for integer partitions. Using this relation, we establish some uniform asymptotic formulas for $\unicode[STIX]{x1D70B}(m,n)$.