Showing papers in "Bulletin of The London Mathematical Society in 2022"
TL;DR: In this article , an overview on derived nonhomogeneous Koszul duality over a field, mostly based on the author's memoir, is presented, which is intended to serve as a pedagogical introduction and a summary of the covariant duality between DG-algebras and curved DG-coalgebrAs, as well as between DGmodules and CDG-comodules.
Abstract: This is an overview on derived nonhomogeneous Koszul duality over a field, mostly based on the author’s memoir [51]. The paper is intended to serve as a pedagogical introduction and a summary of the covariant duality between DG-algebras and curved DG-coalgebras, as well as between DG-modules and CDG-comodules. Some personal reminiscences are included as a part of historical
7 citations
TL;DR: In this paper , a different version of the well-known connection between Hopf-Galois structures and skew braces, building on a recent paper of A. Koch and P. J. Truman, is presented.
Abstract: We present a different version of the well-known connection between Hopf--Galois structures and skew braces, building on a recent paper of A. Koch and P. J. Truman. We show that the known results that involve this connection easily carry over to this new perspective, and that new ones naturally appear. As an application, we present new insights on the study of the surjectivity of the Hopf--Galois correspondence, explaining in more detail the role of bi-skew braces in Hopf--Galois theory.
7 citations
TL;DR: For some special window functions, this article showed that the ones over which the Wavelet transform concentrates optimally are exactly the discs with respect to the pseudohyperbolic metric of the upper half space.
Abstract: For some special window functions $\psi_{\beta} \in H^2(\mathbb{C}^+),$ we prove that, over all sets $\Delta \subset \mathbb{C}^+$ of fixed hyperbolic measure $
u(\Delta),$ the ones over which the Wavelet transform $W_{\overline{\psi_{\beta}}}$ with window $\overline{\psi_{\beta}}$ concentrates optimally are exactly the discs with respect to the pseudohyperbolic metric of the upper half space. This answers a question raised by Abreu and D\"orfler. Our techniques make use of a framework recently developed in a previous work by F. Nicola and the second author, but in the hyperbolic context induced by the dilation symmetry of the Wavelet transform. This leads us naturally to use a hyperbolic rearrangement function, as well as the hyperbolic isoperimetric inequality, in our analysis.
6 citations
TL;DR: In this article , the existence of a pluriclosed metric on an Oeljeklaus-Toma manifold X ( K, U ) $X(K, U)$ was characterized in terms of number-theoretical conditions, yielding restrictions on the third Betti number and the Dolbeault cohomology group.
Abstract: Oeljeklaus–Toma (OT) manifolds are higher dimensional analogues of Inoue-Bombieri surfaces and their construction is associated to a finite extension K $K$ of Q $\mathbb {Q}$ and a subgroup of units U $U$ . We characterize the existence of pluriclosed metrics (also known as strongly Kähler with torsion (SKT) metrics) on any OT manifold X ( K , U ) $X(K, U)$ purely in terms of number-theoretical conditions, yielding restrictions on the third Betti number b 3 $b_3$ and the Dolbeault cohomology group H ∂ ¯ 2 , 1 $H^{2,1}_{\overline{\partial }}$ . Combined with the main result in (Dubickas, Results Math. 76 (2021), 78), these numerical conditions render explicit examples of pluriclosed OT manifolds in arbitrary complex dimension. We prove that in complex dimension 4 and type ( 2 , 2 ) $(2, 2)$ , the existence of a pluriclosed metric on X ( K , U ) $X(K, U)$ is entirely topological, namely, it is equivalent to b 3 = 2 $b_3=2$ . Moreover, we provide an explicit example of an OT manifold of complex dimension 4 carrying a pluriclosed metric. Finally, we show that no OT manifold admits balanced metrics, but all of them carry instead locally conformally balanced metrics.
6 citations
TL;DR: In this paper , a connection between the Bogomolov-Tschinkel conjecture and the Ivanov conjecture about virtual homology of mapping class groups was established, and it was shown that every genus g$g$ X$X$ virtually dominates a fixed Riemann surface Y$Y$ of genus at least two if and only if there exists a finite index subgroup Γ.
Abstract: We establish a connection between the conjecture of Bogomolov–Tschinkel about unramified correspondences and the Ivanov conjecture about the virtual homology of mapping class groups. Given g⩾2$g\geqslant 2$ , we show that every genus g$g$ Riemann surface X$X$ virtually dominates a fixed Riemann surface Y$Y$ of genus at least two if and only if there exists a finite index subgroup Γ
6 citations
TL;DR: Theorem 3.3 as discussed by the authors simplifies the checking of the inductive Alperin weight condition for the remaining open cases of simple groups of Lie type and is strongly related in form to the criterion of the second author for inductive McKay conditions (see Späth, Bull. Lond. Math. Soc. 44 (2012), no. 3, 426-438, 2.12).
Abstract: We give a criterion that simplifies the checking of the inductive Alperin weight condition for the remaining open cases of simple groups of Lie type (see Theorem 3.3). It is strongly related in form to the criterion of the second author for the inductive McKay conditions (see Späth, Bull. Lond. Math. Soc. 44 (2012), no. 3, 426–438, 2.12) that has proved very useful. The proof follows from a Clifford theory for weights intrinsically present in the proof of reduction theorems of the Alperin weight conjecture given by Navarro–Tiep and the second author. We also give a related criterion for the inductive blockwise Alperin weight condition (Theorem 4.5).
5 citations
TL;DR: Symplectic duality as discussed by the authors is a new frontier in representation theory, and it has been studied extensively in quantization, categorification, and enumerative geometry, particularly for affine Grassmannian slices.
Abstract: Symplectic resolutions are an exciting new frontier of research in representation theory. One of the most fascinating aspects of this study is symplectic duality: the observation that these resolutions come in pairs with matching properties. The Coulomb branch construction allows us to produce and study many of these dual pairs. These notes survey much recent work in this area including quantization, categorification, and enumerative geometry. We particularly focus on ADE quiver varieties and affine Grassmannian slices.
5 citations
TL;DR: In this article , it was shown that a hypersurface is K-polystable and not K-stable if it is quasi-smooth, and that such hypersurfaces are not stable if they are smooth.
Abstract: For every integer $a \geq 2$, we relate the K-stability of hypersurfaces in the weighted projective space $\mathbb{P}(1,1,a,a)$ of degree $2a$ with the GIT stability of binary forms of degree $2a$. Moreover, we prove that such a hypersurface is K-polystable and not K-stable if it is quasi-smooth.
4 citations
TL;DR: In particular, this paper showed that the smallest singular value of A + M $A+M$ can be computed in O(n ) $O(sqrt {n})$ for all ε ⩾ 0 $\epsilon \geqslant 0$ , provided only that A $A$ has Ω ( n ) $\Omega (n)$ singular values which are O( n )$O(\sqrt n})$ .
Abstract: Let A $A$ be an n × n $n\times n$ real matrix, and let M $M$ be an n × n $n\times n$ random matrix whose entries are independent and identically distributed sub-Gaussian random variables with mean 0 and variance 1. We make two contributions to the study of s n ( A + M ) $s_n(A+M)$ , the smallest singular value of A + M $A+M$ . (1) We show that for all ε ⩾ 0 $\epsilon \geqslant 0$ , P [ s n ( A + M ) ⩽ ε ] = O ( ε n ) + 2 e − Ω ( n ) , \begin{equation*} \mathbb {P}[s_n(A + M) \leqslant \epsilon ] = O(\epsilon \sqrt {n}) + 2e^{-\Omega (n)}, \end{equation*} provided only that A $A$ has Ω ( n ) $\Omega (n)$ singular values which are O ( n ) $O(\sqrt {n})$ . This extends a well-known result of Rudelson and Vershynin, which requires all singular values of A $A$ to be O ( n ) $O(\sqrt {n})$ . (2) We show that any bound of the form sup ∥ A ∥ ⩽ n C 1 P [ s n ( A + M ) ⩽ n − C 3 ] ⩽ n − C 2 \begin{equation*} \sup _{\Vert A\Vert \leqslant n^{C_1}}\mathbb {P}[s_n(A+M)\leqslant n^{-C_3}] \leqslant n^{-C_2} \end{equation*} must have C 3 = Ω ( C 1 C 2 ) $C_3 = \Omega (C_1 \sqrt {C_2})$ . This complements a result of Tao and Vu, who proved such a bound with C 3 = O ( C 1 C 2 + C 1 + 1 ) $C_3 = O(C_1C_2 + C_1 + 1)$ , and counters their speculation of possibly taking C 3 = O ( C 1 + C 2 ) $C_3 = O(C_1 + C_2)$ .
4 citations
TL;DR: In this paper , the authors considered the problem of identifiability for mixtures of centered Gaussians from their (exact) moments of degree at most 6$ and showed that the polynomial map is generically one-to-one (up to permutations of $ q_1,\ldots, q_m $ and third roots of unity).
Abstract: We consider the inverse problem for the polynomial map which sends an $m$-tuple of quadratic forms in $n$ variables to the sum of their $d$-th powers. This map captures the moment problem for mixtures of $m$ centered $n$-variate Gaussians. In the first non-trivial case $d = 3$, we show that for any $ n\in \mathbb N $, this map is generically one-to-one (up to permutations of $ q_1,\ldots, q_m $ and third roots of unity) in two ranges: $m\le {n\choose 2} + 1 $ for $n \leq 16$ and $ m\le {n+5 \choose 6}/{n+1 \choose 2}-{n+1 \choose 2}-1$ for $n>16$, thus proving generic identifiability for mixtures of centered Gaussians from their (exact) moments of degree at most $ 6 $. The first result is obtained by studying the explicit geometry of the tangential contact locus of the variety of sums of cubes of quadratic forms at concrete points, while the second result is accomplished using a link between secant non-defectivity with identifiability. The latter approach generalizes also to sums of $ d $-th powers of $k$-forms for $d \geq 3$ and $k \geq 2$.
4 citations
TL;DR: In this paper , the authors consider the Erdős-Rényi evolution of random graphs, where a new uniformly distributed edge is added to the graph in every step, and show that with high probability, the graph becomes rigid in Rd$\mathbb {R}^d$ at the very moment its minimum degree becomes d$d$ .
Abstract: We consider the Erdős–Rényi evolution of random graphs, where a new uniformly distributed edge is added to the graph in every step. For every fixed d⩾1$d\geqslant 1$ , we show that with high probability, the graph becomes rigid in Rd$\mathbb {R}^d$ at the very moment its minimum degree becomes d$d$ , and it becomes globally rigid in Rd$\mathbb {R}^d$ at the very moment its minimum degree becomes d+1$d+1$ .
TL;DR: In this paper , it was proved that the pure paraunitary group over a von Neumann algebra coincides with the structure group of its projection lattice, which is known to be a complete invariant of the OML.
Abstract: It is proved that the pure paraunitary group over a von Neumann algebra coincides with the structure group of its projection lattice. The structure group of an arbitrary orthomodular lattice (OML) is a group with a right invariant lattice order, and as such it is known to be a complete invariant of the OML. The pure paraunitary group PPU(A)$\mbox{PPU}(\mathcal {A})$ of a von Neumann algebra A$\mathcal {A}$ is a normal subgroup of the paraunitary group PU(A)$\mbox{PU}(\mathcal {A})$ with the group U(A)$\mbox{U}(\mathcal {A})$ of unitaries in A$\mathcal {A}$ as cokernel. By a result of Heunen and Reyes, A$\mathcal {A}$ is determined by the action of U(A)$\mbox{U}(\mathcal {A})$ on PPU(A)$\mbox{PPU}(\mathcal {A})$ . In this sense, it follows that the paraunitary group is a complete invariant of any von Neumann algebra.
TL;DR: In this article , the authors review deformation, cohomology and homotopy theories of relative Rota-Baxter (RB$\mathsf {RB}$ ) Lie algebras.
Abstract: In this paper, we review deformation, cohomology and homotopy theories of relative Rota–Baxter ( RB$\mathsf {RB}$ ) Lie algebras, which have attracted quite much interest recently. Using Voronov's higher derived brackets, one can obtain an L∞$L_\infty$ ‐algebra whose Maurer–Cartan elements are relative RB$\mathsf {RB}$ Lie algebras. Then using the twisting method, one can obtain the L∞$L_\infty$ ‐algebra that controls deformations of a relative RB$\mathsf {RB}$ Lie algebra. Meanwhile, the cohomologies of relative RB$\mathsf {RB}$ Lie algebras can also be defined with the help of the twisted L∞$L_\infty$ ‐algebra. Using the controlling algebra approach, one can also introduce the notion of homotopy relative RB$\mathsf {RB}$ Lie algebras with close connection to pre‐Lie ∞$_\infty$ ‐algebras. Finally, we briefly review deformation, cohomology and homotopy theories of relative RB$\mathsf {RB}$ Lie algebras of nonzero weights.
TL;DR: For a positive integer t$t$ , let Ft$F_t$ denote the graph of the t×t$t\times t$ grid as mentioned in this paper , it is shown that there exists a constant C =C(t)$C=C(T)$ such that ex(n,Ft)⩽Cn3/2$\mathrm{ex}(n and F t )−Cn^{ 3/2}$ .
Abstract: For a positive integer t$t$ , let Ft$F_t$ denote the graph of the t×t$t\times t$ grid. Motivated by a 50‐year‐old conjecture of Erdős about Turán numbers of r$r$ ‐degenerate graphs, we prove that there exists a constant C=C(t)$C=C(t)$ such that ex(n,Ft)⩽Cn3/2$\mathrm{ex}(n,F_t)\leqslant Cn^{3/2}$ . This bound is tight up to the value of C$C$ . One of the interesting ingredients of our proof is a novel way of using the tensor power trick.
TL;DR: In this paper , a somewhat sharp uniqueness condition for the fractional Laplace equation is established, and several weighted fractional Sobolev spaces are introduced for Dirichlet type problems.
Abstract: For the fractional Laplace equation, a surprising observation is the non-uniqueness for the basic Dirichlet type problems. In this paper, a somewhat sharp uniqueness condition for the fractional Laplace equation is established. We derive the $L^p$-estimate for fractional Laplacian operators to better understand this phenomena. Several weighted fractional Sobolev spaces appear naturally. We then establish the embedding relations between these spaces. These existence-uniqueness conditions and the spaces we introduce here are intrinsically related to the fractional Laplacian. These are basic properties to the fractional Laplace equations and can be useful in the study of related problems.
TL;DR: For the weighted Bergman spaces on the unit disk, the authors obtained sharp hypercontractive inequalities for $L^p\rightarrow L^q$ for $q\geq 2, and also gave some estimates for $0.
Abstract: \begin{abstract} We obtain sharp $L^p\rightarrow L^q$ hypercontractive inequalities for the weighted Bergman spaces on the unit disk $\mathbb{D}$ with the usual weights \\ $\frac{\alpha-1}{\pi}(1-|z|^2)^{\alpha-2},\alpha>1$ for $q\geq 2,$ thus solving an interesting case of a problem from \cite{JANSON}. We also give some estimates for $0
TL;DR: In this article , local smoothing estimates for the fractional Schrödinger operator with α > 1/α > 1$ were considered and they improved the previously best-known results of Guo-Roos-Yung (arXiv:1710.10988 (2017)) and Rogers-Seeger (J. Für Die Reine und Ang.
Abstract: In this paper, we consider local smoothing estimates for the fractional Schrödinger operator e i t ( − Δ ) α / 2 $e^{it(-\Delta )^{\alpha /2}}$ with α > 1 $\alpha >1$ . Using the k $k$ -broad ‘norm’ estimates of Guth–Hickman–Iliopoulou (Acta Math. 223 (2019), 251–376), we improve the previously best-known results of local smoothing estimates of Guo–Roos–Yung (arXiv:1710.10988 (2017)) and Rogers–Seeger (J. Für Die Reine und Ang. Math. (Crelles Journal), 640 (2010), 47–66).
TL;DR: In this article , a flat vector bundle on an algebraic variety supports two definable structures given by the flat and algebraic coordinates, and these two structures coincide, subject to a condition on the local monodromy at infinity.
Abstract: A flat vector bundle on an algebraic variety supports two natural definable structures given by the flat and algebraic coordinates. In this note we show these two structures coincide, subject to a condition on the local monodromy at infinity which is satisfied for all flat bundles underlying variations of Hodge structures.
TL;DR: The right-angled Artin pro-p$p$ group GΓ$G_{\Gamma }$ associated to a finite simplicial graph Γ$Gamma$ was shown to be a right-angle Artin group in this paper .
Abstract: Let p$p$ be a prime. The right‐angled Artin pro‐ p$p$ group GΓ$G_{\Gamma }$ associated to a finite simplicial graph Γ$\Gamma$ is the pro‐ p$p$ completion of the right‐angled Artin group associated to Γ$\Gamma$ . We prove that the following assertions are equivalent: (i) no induced subgraph of Γ$\Gamma$ is a square or a line with four vertices (a path of length 3); (ii) every closed subgroup of GΓ$G_{\Gamma }$ is itself a right‐angled Artin pro‐ p$p$ group (possibly infinitely generated); (iii) GΓ$G_{\Gamma }$ is a Bloch–Kato pro‐ p$p$ group; (iv) every closed subgroup of GΓ$G_{\Gamma }$ has torsion free Abelianization; (v) GΓ$G_{\Gamma }$ occurs as the maximal pro‐ p$p$ Galois group GK(p)$G_K(p)$ of some field K$K$ containing a primitive p$p$ th root of unity; (vi) GΓ$G_{\Gamma }$ can be constructed from Zp$\mathbb {Z}_p$ by iterating two group theoretic operations, namely, direct products with Zp$\mathbb {Z}_p$ and free pro‐ p$p$ products. This settles in the affirmative a conjecture of Quadrelli and Weigel. Also, we show that the Smoothness Conjecture of De Clercq and Florence holds for right‐angled Artin pro‐ p$p$ groups. Moreover, we prove that GΓ$G_{\Gamma }$ is coherent if and only if each circuit of Γ$\Gamma$ of length greater than three has a chord.
TL;DR: The authors describe Lagrangian fibrations by Prym varieties on holomorphic symplectic varieties that deform to compactifications of Sp $\mathrm{Sp}$ -Hitchin systems.
Abstract: We describe certain Lagrangian fibrations by Prym varieties on holomorphic symplectic varieties that deform to compactifications of Sp $\mathrm{Sp}$ -Hitchin systems.
TL;DR: A subset of a Banach space is called equilateral if the distances between any two of its distinct elements are the same as mentioned in this paper , and it is proved that there exist nonseparable Banach spaces (in fact of density continuum) with no infinite equilateral subset.
Abstract: A subset of a Banach space is called equilateral if the distances between any two of its distinct elements are the same. It is proved that there exist nonseparable Banach spaces (in fact of density continuum) with no infinite equilateral subset. These examples are strictly convex renormings of ℓ 1 ( [ 0 , 1 ] ) $\ell _1([0,1])$ . A wider class of renormings of ℓ 1 ( [ 0 , 1 ] ) $\ell _1([0,1])$ which admit no uncountable equilateral sets is also considered.
TL;DR: Bregman and Clay as discussed by the authors showed that right-angled Artin groups with geometric dimension 2 have vanishing minimal volume entropy, and they extended this characterization to higher dimensions by extending it to higher dimensional groups.
Abstract: Bregman and Clay recently characterized which right‐angled Artin groups with geometric dimension 2 have vanishing minimal volume entropy. In this note, we extend this characterization to higher dimensions.
TL;DR: In this article , Castryck, Cluckers, Dittmann and Nguyen proved uniform upper bounds on the number of points of bounded height on affine affine curves.
Abstract: In this article, we prove several new uniform upper bounds on the number of points of bounded height on varieties over F q [ t ] $\mathbb {F}_q[t]$ . For projective curves, we prove the analogue of Walsh' result with polynomial dependence on q $q$ and the degree d $d$ of the curve. For affine curves, this yields an improvement to bounds by Sedunova, and Cluckers, Forey and Loeser. In higher dimensions, we prove a version of dimension growth for hypersurfaces of degree d ⩾ 64 $d\geqslant 64$ , building on work by Castryck, Cluckers, Dittmann and Nguyen in characteristic zero. These bounds depend polynomially on q $q$ and d $d$ , and it is this dependence which simplifies the treatment of the dimension growth conjecture.
TL;DR: In this article , a short alternative proof of an estimate obtained by Mantel, Muratov and Simon in (Arch Rational Mech. Anal. 239 (2021), 219-299) regarding the rigidity of degree ± 1$\pm 1$ conformal maps of S2$\mathbb {S}^2$ , that is, its Möbius transformations, is presented.
Abstract: In this note, we present a short alternative proof of an estimate obtained by Mantel, Muratov and Simon in (Arch Rational Mech. Anal. 239 (2021), 219–299) regarding the rigidity of degree ±1$\pm 1$ conformal maps of S2$\mathbb {S}^2$ , that is, its Möbius transformations.
TL;DR: In this article , it was shown that any absolute minimizer of a supremal functional defined by a C2$C^2$ quasiconvex Hamiltonian f(x,s,p)$f(x s,p,p), allowing variable dependence, is a viscosity solution to the Aronsson equation.
Abstract: Given a C2$C^2$ family of vector fields X1,…,Xm$X_1,\ldots ,X_m$ which induces a continuous Carnot–Carathéodory distance, we show that any absolute minimizer of a supremal functional defined by a C2$C^2$ quasiconvex Hamiltonian f(x,s,p)$f(x,s,p)$ , allowing s$s$ ‐variable dependence, is a viscosity solution to the Aronsson equation −∑i=1mXi(f(x,u(x),Xu(x)))∂f∂pi(x,u(x),Xu(x))=0,$$\begin{equation*} \hspace*{10pt}-\sum _{i=1}^m X_i(f(x,u(x),Xu(x)))\frac{\partial f}{\partial p_i} (x,u(x),Xu(x))=0,\hspace*{-10pt} \end{equation*}$$
TL;DR: In this paper , the authors studied the relationship between the categorical entropy of the twist and cotwist functors along a spherical functor and proved that the twist functor coincides with that of the cot wist functor if the essential image of the right adjoint functor contains a split-generator.
Abstract: We study the relationship between the categorical entropy of the twist and cotwist functors along a spherical functor. In particular, we prove the categorical entropy of the twist functor coincides with that of the cotwist functor if the essential image of the right adjoint functor of the spherical functor contains a split-generator. We also see our results to generalize the computations of the categorical entropy of spherical twists and P $\mathbb {P}$ -twists by Ouchi and Fan. As an application, we apply our results to the Gromov–Yomdin-type conjecture by Kikuta–Takahashi.
TL;DR: In this article , the interplay between t-structures in the bounded derived category of finitely presented modules and the unbounded derived categories of all modules over a coherent ring A $A$ using homotopy colimits was explored.
Abstract: We explore the interplay between t-structures in the bounded derived category of finitely presented modules and the unbounded derived category of all modules over a coherent ring A $A$ using homotopy colimits. More precisely, we show that every intermediate t-structure in D b ( mod ( A ) ) $D^b(\operatorname{mod}(A))$ can be lifted to a compactly generated t-structure in D ( Mod ( A ) ) $D(\operatorname{Mod}(A))$ , by closing the aisle and the coaisle of the t-structure under directed homotopy colimits. Conversely, we provide necessary and sufficient conditions for a compactly generated t-structure in D ( Mod ( A ) ) $D(\operatorname{Mod}(A))$ to restrict to an intermediate t-structure in D b ( mod ( A ) ) $D^b(\operatorname{mod}(A))$ , thus describing which t-structures can be obtained via lifting. We apply our results to the special case of HRS-t-structures. Finally, we discuss various applications to silting theory in the context of finite dimensional algebras.
TL;DR: A new approach has been recently developed to study the arithmetic of hyperelliptic curves over local fields of odd residue characteristic via combinatorial data associated to the roots of f $f$ as mentioned in this paper .
Abstract: A new approach has been recently developed to study the arithmetic of hyperelliptic curves y 2 = f ( x ) $y^2=f(x)$ over local fields of odd residue characteristic via combinatorial data associated to the roots of f $f$ . Since its introduction, numerous papers have used this machinery of ‘cluster pictures’ to compute a plethora of arithmetic invariants associated to these curves. The purpose of this user's guide is to summarise and centralise all of these results in a self-contained fashion, complemented by an abundance of examples.
TL;DR: In this article , the decay properties of the Mizohata-takeuchi conjecture in the case of convex curves were investigated and the authors obtained new estimates for exponentially flat curves.
Abstract: Suppose $S$ is a smooth compact hypersurface in $\Bbb R^n$ and $\sigma$ is an appropriate measure on $S$. If $Ef= \hat{fd\sigma}$ is the extension operator associated with $(S,\sigma)$, then the Mizohata-Takeuchi conjecture asserts that $\int |Ef(x)|^2 w(x) dx \leq C (\sup_T w(T)) \| f \|_{L^2(\sigma)}^2$ for all functions $f \in L^2(\sigma)$ and weights $w : \Bbb R^n \to [0,\infty)$, where the $\sup$ is taken over all tubes $T$ in $\Bbb R^n$ of cross-section 1, and $w(T)= \int_T w(x) dx$. This paper investigates how far we can go in proving the Mizohata-Takeuchi conjecture in $\Bbb R^2$ if we only take the decay properties of $\hat{\sigma}$ into consideration. As a consequence of our results, we obtain new estimates for a class of convex curves that include exponentially flat ones such as $(t,e^{-1/t^m})$, $0 \leq t \leq c_m$, $m \in \Bbb N$.
TL;DR: In this article , it was shown that the log smooth deformations of a proper log smooth saturated log Calabi-Yau space are unobstructed, and the same authors also proved the same result for the case of a saturated log-calabi-yau space.
Abstract: We prove that the log smooth deformations of a proper log smooth saturated log Calabi–Yau space are unobstructed.