Recent zbMATH articles in MSC 32https://zbmath.org/atom/cc/322021-11-25T18:46:10.358925ZWerkzeugA singular mathematical promenadehttps://zbmath.org/1472.000012021-11-25T18:46:10.358925Z"Ghys, Étienne"https://zbmath.org/authors/?q=ai:ghys.etienneAt the first look, one may feel that the book title is a little bit strange. The word singular in the title refers to the concept of singularity of a curve and does not mean a trip made by an individual person. It is a promenade into the mathematical world. The tour is interesting, entertaining and enjoyable, but it may be little bit difficult for those who have insufficient mathematical knowledge. So some mathematical maturity is required to fully appreciate the beauty presented by the author. When you go through the subjects of it you will find it a wonderfully crafted book. The book consists of 30 chapters. Each chapter provides a rich read. Several chapters are fairly independent from the rest of the book. It is a remarkable achievement in terms of its content, structure, and style. In almost all chapters the author shows excellent examples of mathematical exposition and utilize history to enrich a contemporary mathematical investigations. Actually he weaves historical stories in between the combinatorics, complex analysis, and algebraic geometry \dots etc. and does it all in a very readable and remarkable way. The design of the book is amazing: it contains many pictures and illustrations, scanned manuscripts, references, remarks, all written in the right margin of the pages (so one has the information immediately available). The text contains many historical quotations in different languages, with translations, and interesting analysis of the mathematics of our ``classics'' (Newton, Gauss, Hipparchus \dots etc). Hence the book will please any budding or professional mathematician. I can say that, principally, for professional readers, the book is an enjoyable reading due to the versatility of subjects using too many illustrations and remarks that enriched the concepts of the classical notions. In fact most of the material in the book can be regarded as an advanced undergraduate/early graduate level, even there are some material that is significantly more advanced. One very remarkable aspects of the book is the treat of historical matters. Some of the very classical notions such as the fundamental theorem of algebra, the theory of Puisseux series, the linking number of knots, discrete mathematics, operads, resolution of curve singularities, complex singularities, and more, have been discussed and explained in an enlightening way.
The author of the book, Professor Étienne Ghys, Director of Research at the École Normale Superiere de Lyon, is a skilled, gifted versatile expositor mathematician. He wrote his book in a relaxed, informal manner with lots of exclamation marks, figures, supporting computer graphics and illustrations that are mathematically helpful and visually engaging. It is interesting to know that most of illustrations have been produced by Ghys himself and who has waived all copyright and related or neighboring rights which is a good evidence of Ghys's service towards the dissemination of mathematical ideas. Ghys is a prominent researcher, broadly in geometry and dynamics. He was awarded the Clay Award for Dissemination of Mathematics in 2015.
As the author mentioned in his book, the motivation for writing such an interesting book came from a fact brought to his attention by his colleague, Maxim Kontsevich, in 2009 that relates the relative position of the graphs of four real polynomials under certain conditions imposed on the polynomials. So he begins the book with an attractive theorem of Maxim Kontsevich scribbled for him on a Paris metro ticket who stated it in a nice: Theorem. There do not exist four polynomials \(P_1, \dots , P_4 \in R[x] \) with \(P_1(x) < P_2(x) < P_3(x) < P_4(x)\) for all small negative \(x\), and \(P_2(x) < P_4(x) < P_1(x) < P_3(x)\) for small positive \(x\).
In fact Ghys begins his promenade with this attractive theorem. Amazingly, this result basically characterizes what can or cannot happen for crossings, not only for graphs of arbitrary collections of polynomials, but indeed for all real analytic planar curves. Actually the book explores very different questions related to this problem, and follows on different ramifications. Ghys discussed the more general singularities of algebraic curves in the plane, explaining how the concepts were developed historically. I recommend to assign parts of it as an independent studies for both undergraduate and graduate students.Book review of: G.-M. Greuel et al., Singular algebraic curveshttps://zbmath.org/1472.000122021-11-25T18:46:10.358925Z"Degtyarev, Alex"https://zbmath.org/authors/?q=ai:degtyarev.alexReview of [Zbl 1411.14001].The elliptic KZB connection and algebraic de Rham theory for unipotent fundamental groups of elliptic curveshttps://zbmath.org/1472.111832021-11-25T18:46:10.358925Z"Luo, Ma"https://zbmath.org/authors/?q=ai:luo.ma.1Summary: We develop an algebraic de Rham theory for unipotent fundamental groups of once punctured elliptic curves over a field of characteristic zero using the universal elliptic KZB connection of \textit{D. Calaque} et al. [Prog. Math. 269, 165--266 (2009; Zbl 1241.32011)] and \textit{A. Levin} and \textit{G. Racinet} [``Towards multiple elliptic polylogarithms'', Preprint, \url{arXiv:math/0703237}]. We use it to give an explicit version of Tannaka duality for unipotent connections over an elliptic curve with a regular singular point at the identity.The completed finite period map and Galois theory of supercongruenceshttps://zbmath.org/1472.112382021-11-25T18:46:10.358925Z"Rosen, Julian"https://zbmath.org/authors/?q=ai:rosen.julianSummary: A period is a complex number arising as the integral of a rational function with algebraic number coefficients over a region cut out by finitely many inequalities between polynomials with rational coefficients. Although periods are typically transcendental numbers, there is a conjectural Galois theory of periods coming from the theory of motives. This paper formalizes an analogy between a class of periods called multiple zeta values and congruences for rational numbers modulo prime powers (called supercongruences). We construct an analog of the motivic period map in the setting of supercongruences and use it to define a Galois theory of supercongruences. We describe an algorithm using our period map to find and prove supercongruences, and we provide software implementing the algorithm.Characteristic classes of affine varieties and Plücker formulas for affine morphismshttps://zbmath.org/1472.140062021-11-25T18:46:10.358925Z"Esterov, Alexander"https://zbmath.org/authors/?q=ai:esterov.alexander-iSummary: An enumerative problem on a variety \(V\) is usually solved by reduction to intersection theory in the cohomology of a compactification of \(V\). However, if the problem is invariant under a ``nice'' group action on \(V\) (so that \(V\) is spherical), then many authors suggested a better home for intersection theory: the direct limit of the cohomology rings of all equivariant compactifications of \(V\). We call this limit the affine cohomology of \(V\) and construct affine characteristic classes of subvarieties of a complex torus, taking values in the affine cohomology of the torus.{
}This allows us to make the first steps in computing affine Thom polynomials. Classical Thom polynomials count how many fibers of a generic proper map of a smooth variety have a prescribed collection of singularities and our affine version addresses the same question for generic polynomial maps of affine algebraic varieites. This notion is also motivated by developing an intersection-theoretic approach to tropical correspondence theorems: they can be reduced to the computation of affine Thom polynomials, because the fundamental class of a variety in the affine cohomology is encoded by the tropical fan of this variety.{
}The first concrete answer that we obtain is the affine version of what were, historically speaking, the first three Thom polynomials--the Plücker formulas for the degree and the number of cusps and nodes of a projectively dual curve. This, in particular, characterizes toric varieties whose projective dual is a hypersurface, computes the tropical fan of the variety of double tangent hyperplanes to a toric variety, and describes the Newton polytope of the hypersurface of non-Morse polynomials of a given degree. We also make a conjecture on the general form of affine Thom polynomials; a key ingredient is the \(n\)-ary fan, generalizing the secondary polytope.Néron models of intermediate Jacobians associated to moduli spaceshttps://zbmath.org/1472.140102021-11-25T18:46:10.358925Z"Dan, Ananyo"https://zbmath.org/authors/?q=ai:dan.ananyo"Kaur, Inder"https://zbmath.org/authors/?q=ai:kaur.inderGiven a family \(X\) of smooth curves degenerating to a one-nodal curve \(X_0\), one can consider the Gieseker moduli space \(\mathcal{G}_{X_t}(2,\mathcal{L} _t)\) of semistable vector bundles of rank two and a determinant of odd degree which also varies in a family. For every smooth curve \(X_t\) in the family, one can consider the intermediate Jacobian \(J^i(\mathcal{G}_{X_t}(2,\mathcal{L} _t))\), and all these intermediate Jacobians fit together in one analytic family. To extend this at \(t=0\), different Neron models, constructed Clemens, Saito, Schnell, Zucker and Green-Griffiths-Kerr, are available. In this paper, the authors prove that all these Neron models coincide, and, moreover, they give a description of the special fiber and they prove that it is a semi-abelian varieties in some cases. In particular, in the case of the second intermediate Jacobian, the special fiber is isomorphic to the second generalized intermediate Jacobian of \(\mathcal{G}_{X_0}(2,\mathcal{L}_0)\).Hermitian metrics of positive holomorphic sectional curvature on fibrationshttps://zbmath.org/1472.140132021-11-25T18:46:10.358925Z"Chaturvedi, Ananya"https://zbmath.org/authors/?q=ai:chaturvedi.ananya"Heier, Gordon"https://zbmath.org/authors/?q=ai:heier.gordonIn the article under review, the authors adress the construction of Hermitian metrics with positive holomorphic curvature on compact complex manifolds. The ambiant space is actually the total space of a fibration (holomorphic submersion) \(\pi:X\to Y\) and it is rather natural to ask wether the existence of metrics with positive curvature both on \(Y\) and on the fibers of \(\pi\) implies the existence of such a metric on \(X\).
The corresponding question was answered positively by \textit{C.-K. Cheung} [Math. Z. 201, No. 1, 105--119 (1989; Zbl 0648.53037)] in the opposite case of negative curvature. In the positive curvature case, such metrics were constructed by \textit{N. J. Hitchin} [Proc. Symp. Pure Math. 27, Part 2, 65--80 (1975; Zbl 0321.53052)] on Hirzebruch surfaces.
The main result of this article is a positive answer to the above mentioned question. The proof is quite natural although the computations being a bit involved. As explained by the authors, it is not clear if their method can be used either in the semi-positive case or when the map \(\pi\) has singular fibres. Let us make a final remark: the metric cooked up in this article is merely a Hermitian one, even if the data we started with are Kähler.Fano manifolds and stability of tangent bundleshttps://zbmath.org/1472.140442021-11-25T18:46:10.358925Z"Kanemitsu, Akihiro"https://zbmath.org/authors/?q=ai:kanemitsu.akihiroLet \(X\) be a Fano manifold, that is a complex projective manifold such that the anticanonical divisor \(-K_X\) is ample. If the Picard number of \(X\) is one, a widely believed folklore conjecture claims that the tangent bundle \(T_X\) is semistable (in the sense of Mumford-Takemoto). In this paper the author gives a series of counterexamples to this conjecture! \newline The counterexamples are obtained by a family of horospherical varieties classified by \textit{B. Pasquier} [Math. Ann. 344, No. 4, 963--987 (2009; Zbl 1173.14028)]: for these manifolds the action of the group \(\mbox{Aut}^0(X)\) on \(X\) has two orbits, the open orbit \(X^0\) and a closed orbit \(Z\). Moreover the action on the blow-up \(\mbox{Bl}_Z X\) again has two orbits, the open orbit \(X^0\) and the exceptional divisor \(E\). The manifold \(\mbox{Bl}_Z X\) admits a smooth fibration onto a lower-dimensional manifold \(Y\), the push-forward of the relative tangent bundle to \(X\) defines an algebraically integrable foliation \(\mathcal F \subset T_X\). The author shows that this foliation is canonical in the sense that it is the unique algebraically integrable foliation on \(X\) that is \(\mbox{Aut}^0(X)\)-invariant. General arguments show that the stability of \(T_X\) can be verified by computing the slope of the foliation \(\mathcal F\). It turns out that for infinitely many manifolds in Pasquier's list, the subsheaf \(\mathcal F \subset T_X\) destabilises the tangent bundle. The reviewer recommends to any complex geometer to read this beautiful paper.An analytic application of Geometric Invariant Theoryhttps://zbmath.org/1472.140472021-11-25T18:46:10.358925Z"Buchdahl, Nicholas"https://zbmath.org/authors/?q=ai:buchdahl.nicholas-p"Schumacher, Georg"https://zbmath.org/authors/?q=ai:schumacher.georgOne of the mathematical highlights of the 1980s is the establishment of the Hitchin-Kobayashi correspondence. Given a holomorphic vector bundle over a compact Kähler manifold, this correspondence states that the bundle admits a Hermite-Einstein metric if and only if it satisfies the algebro-geometric condition of slope polystability [\textit{S. K. Donaldson}, Proc. Lond. Math. Soc. (3) 50, 1--26 (1985; Zbl 0529.53018); \textit{K. Uhlenbeck} and \textit{S. T. Yau}, Commun. Pure Appl. Math. 39, S257--S293 (1986; Zbl 0615.58045)].
Almost equally important as the statement is the following expectation that arises from this correspondence. The notion of slope stability was introduced in relation to considerations from moduli spaces, and so one should be able to form moduli spaces of holomorphic vector bundles which (equivalently) admit a Hermite-Einstein metric or are slope polystable. This expectation is well understood when the compact Kähler manifold is actually a smooth projective variety through work of many authors, such as \textit{C. T. Simpson} [Publ. Math., Inst. Hautes Étud. Sci. 79, 47--129 (1994; Zbl 0891.14005)] and \textit{D. Greb} et al. [Geom. Topol. 25, No. 4, 1719--1818 (2021; Zbl 07379436)].
The paper under review is a key step towards understanding the story beyond the projective setting. The authors use their prior work, demonstrating which deformations of polystable bundles remain polystable, to construct a moduli space of slope polystable bundles over a compact Kähler manifold. They also construct a natural Kähler metric on the moduli space. The method they employ is to view their prior work as producing ``charts'' on the moduli space; this strategy had previously been used in other contexts, but the work of the authors contains some new features.
The paper is very clearly and carefully written, and should be viewed as an important contribution to the field.Maximally stretched laminations on geometrically finite hyperbolic manifoldshttps://zbmath.org/1472.300172021-11-25T18:46:10.358925Z"Guéritaud, François"https://zbmath.org/authors/?q=ai:gueritaud.francois"Kassel, Fanny"https://zbmath.org/authors/?q=ai:kassel.fannySummary: Let \(\Gamma_0\) be a discrete group. For a pair \((j,\rho)\) of representations of \(\Gamma_0\) into \(\operatorname{PO}(n,1)=\operatorname{Isom}(\mathbb{H}^n)\) with \(j\) geometrically finite, we study the set of \((j,\rho)\)-equivariant Lipschitz maps from the real hyperbolic space \(\mathbb{H}^n\) to itself that have minimal Lipschitz constant. Our main result is the existence of a geodesic lamination that is ``maximally stretched'' by all such maps when the minimal constant is at least \(1\). As an application, we generalize two-dimensional results and constructions of Thurston and extend his asymmetric metric on Teichmüller space to a geometrically finite setting and to higher dimension. Another application is to actions of discrete subgroups \(\Gamma\) of \(\operatorname{PO}(n,1)\times\operatorname{PO}(n,1)\) on \(\operatorname{PO}(n,1)\) by right and left multiplication: we give a double properness criterion for such actions, and prove that for a large class of groups \(\Gamma\) the action remains properly discontinuous after any small deformation of \(\Gamma\) inside \(\operatorname{PO}(n,1)\times\operatorname{PO}(n,1)\).Pluripotential theory on Teichmüller space. I: Pluricomplex Green functionhttps://zbmath.org/1472.300182021-11-25T18:46:10.358925Z"Miyachi, Hideki"https://zbmath.org/authors/?q=ai:miyachi.hidekiIn the paper [Bull. Am. Math. Soc., New Ser. 27, No. 1, 143--147 (1992; Zbl 0766.30016)], \textit{S. L. Krushkal} announced the following result on the pluricomplex Green function on Teichmüller space:
Theorem. Let \(T_{g, m}\) be the Teichmüller space of Riemann surfaces of analytically finite-type \((g,m)\), and let \(d_T\) be the Teichmüller distance on \(T_{g, m}\). Then, the pluricomplex Green function \(g_{T_{g, m}} \) on \(T_{g, m}\) satisfies \[g_{T_{g, m}}(x, y)\log \tanh d_T(x, y)\] for \(x, y \in g_{T_{g, m}}\).
The author has a programm to investigate of the pluripotential theory on Teichmüller space. In this first one of a series of works, the author gave an alternative approach to the Krushkal formula of the pluricomplex Green function on Teichmüller space. In comparison with the original approach by Krushkal, the strategy is more direct here. He first shows that the Teichmüller space carries a natural stratified structure of real-analytic submanifolds defined from the structure of singularities of the initial differentials of the Teichmüller mappings from a given point. Then he gives a description of the Levi form of the pluricomplex Green function using the Thurston symplectic form via Dumas' symplectic structure on the space of holomorphic quadratic differentials [\textit{D. Dumas}, Acta Math. 215, No. 1, 55--126 (2015; Zbl 1334.57020)].A uniqueness property for analytic functions on metric measure spaceshttps://zbmath.org/1472.310162021-11-25T18:46:10.358925Z"Łysik, Grzegorz"https://zbmath.org/authors/?q=ai:lysik.grzegorzLet \((X, \rho)\) be a metric space and let \(\mu\) and \(\nu\) be Borel regular measures that are positive on non-empty open sets and finite on bounded sets. Let \(\Omega\) be an open subset of \(X\). A function \(u \in C(\Omega)\) is said to be \((X, \rho, \mu, \nu)\)-analytic on \(\Omega\) if there exist functions \(u_l \in C(\Omega), l \in {\mathbb{N}}\) and \(\varepsilon \in C(\Omega; {\mathbb{R}}_{+})\) such that \[M_{X}(u; x,R) = \sum_{l=0}^{\infty} u_l(x)R^l\] locally uniformly in \(\{(x,R): x \in \Omega, \, 0 \le R < \varepsilon(x) \}\). Here \(M_{X}\) is the solid mean value function with respect to the measures \(\mu\) and \(\nu\) defined for \(u \in C(\Omega)\), \(x \in \Omega\) and \(0< R <\mbox{ dist}_{\rho}(x, \Omega)\) by \[M_{X}(u; x,R)=\frac{1}{\nu(B_{\rho}(x,R))} \int_{B_{\rho}(x,R)} u(y) \, d\mu(y).\]
In his previous paper [Ann. Acad. Sci. Fenn., Math. 43, No. 1, 475--482 (2018; Zbl 1387.26049)] the author proved that this property characterizes real analytic functions in Euclidean spaces.
This paper considers \((X, \rho, \mu, \nu)\)-analytic functions in asymmetric normed spaces and in proper locally uniquely geodesic asymmetric metric spaces, where in addition, for any ball \(B(x,R)=B_{\rho}(x,R)\) with small enough radius \(R\) and \(0 < c < 1\), the mapping that maps the point \(y \in B(x,R)\) to the unique point \(\widetilde{y}\) which lies on the geodesic connecting \(y\) and \(x\) and satisfies \(\rho(x, \widetilde{y}) = c\rho(x, y)\) is a homeomorphism.
The main result proved by the author in this paper is the following uniqueness property: if \((X,\rho)\) is a proper locally uniquely geodesic asymmetric metric space as above or an asymmetric strictly convex normed space over \({\mathbb{R}}^n\) and \(u\) is an \((X, \rho, \mu, \nu)\)-analytic function that vanishes on a non-empty open subset of the connected set \(X\), then \(u\) vanishes on the whole set \(X\).A short proof of the symmetric determinantal representation of polynomialshttps://zbmath.org/1472.320012021-11-25T18:46:10.358925Z"Stefan, Anthony"https://zbmath.org/authors/?q=ai:stefan.anthony"Welters, Aaron"https://zbmath.org/authors/?q=ai:welters.aaron-tA recent theorem [\textit{J. W. Helton} et al., J. Funct. Anal. 240, No. 1, 105--191 (2006; Zbl 1135.47005)] asserts that a real, multivariate polynomial can be written as the determinant of an affine pencil of real, symmetric matrices. The proof by Helton et al. was derived from an elaborate construct of non-commutative algebra, in its turn inspired by control system theory. The note by Stefan and Welters offers an elementary proof of the same result. The authors ingeniously exploit Schur complement identities, allowing a generalization of the main result to certain fields of finite characteristic. The strong algorithmic flavor of the proof may appeal to wider groups of scientists touching numerical matrix analysis in their studies.Global variants of Hartogs' theoremhttps://zbmath.org/1472.320022021-11-25T18:46:10.358925Z"Bochnak, Jacek"https://zbmath.org/authors/?q=ai:bochnak.jacek"Kucharz, Wojciech"https://zbmath.org/authors/?q=ai:kucharz.wojciechThis is a very interesting paper. The classical Hartogs theorem about separately analytic complex functions being analytic finds here a very fine generalization, which is also global. Let's have a look at the theorems proved in the paper:
Theorem 1.1. Let \(X=X_1\times\cdots\times X_n\) be the product of \(n\) complex algebraic manifolds and let \(f:U\to C\) be a function defined on an open subset \(U\) of \(X\). Assume that for each nonsingular algebraic curve \(C\subset X\), parallel to one of the factors of \(X\), the restriction \(f|U\cap C\) is a holomorphic function. Then \(f\) is a holomorphic function.
Theorem 1.2. Let \(X=X_1\times\cdots\times X_n\) be the product of \(n\) complex algebraic manifolds and let \(f:U\to C\) be a function defined on an open subset \(U\) of \(X\). Assume that for each nonsingular algebraic curve \(C\subset X\), parallel to one of the factors of \(X\), the restriction \(f|U\cap C\) is a Nash function. Then \(f\) is a Nash function.
Theorems 1.1 and 1.2 have a suitable analogue for regular functions.
Definition. Let \(X\) be a complex algebraic manifold. A function \(f:U\to C\), defined on an open subset \(U\) of \(X\), is said to be regular if there exists a rational function \(R\) on \(X\) such that \(U\subset X\smallsetminus\mathrm{Pole}(R)\) and \(f=R|U\), where Pole\((R)\) stands for the polar set of \(R\). Clearly, any regular function on \(U\) is a Nash function.
Theorem 1.3. Let \(X=X_1\times\cdots\times X_n\) be the product of \(n\) complex algebraic manifolds and let \(f:U\to C\) be a function defined on an open subset \(U\) of \(X\). Assume that for each nonsingular algebraic curve \(C\subset X\), parallel to one of the factors of \(X\), the restriction \(f|U\cap C\) is a regular function. Then \(f\) is a regular function.
The definition of a curve being ``parallel'' goes as follows: We say that a subset \(A\) of \(X\) is parallel to the \(i\)-th factor of \(X\) if \(\pi_j(A)\) consists of one point for each \(j=i\).
The paper is very well written, which is typical for these authors, very well organized, very clear, but the methods used are not easy.
For instance, in Proposition 2.3 below the Hironaka desingularization theorem is to be used. Not surprising, given the strength of the result, but worth noticing.
Proposition 2.3. Let \(X\) be a complex algebraic manifold and let \(f:U\to C\) be a function defined on an open subset \(U\) of \(X\). Assume that for each nonsingular algebraic curve \(C\subset X\) the restriction \(f|U\cap C\) is a holomorphic function. Then \(f\) is a holomorphic function.
The proof begins: According to Hironaka's theorem on resolution of singularities [\textit{H. Hironaka}, Ann. Math. (2) 79, 109--203 (1964; Zbl 0122.38603)], we may assume that the manifold \(X\) is projective. Hironaka's theorem cannot be avoided here.
The paper is virtually self contained, as the results used are well quoted and easy to find, even if difficult in themselves.
Each tool that needs adjusting (like Noether's normalization lemma presented here as Lemma 1.2) is adjusted and a full proof is given. This makes the paper very pleasant for the reader.
The results seem very useful. It is often easier to examine functions on algebraic curves (cf. [\textit{J. Kollár} et al., Math. Ann. 370, No. 1--2, 39--69 (2018; Zbl 1407.14056)]).Hedgehogs in higher dimensions and their applicationshttps://zbmath.org/1472.320032021-11-25T18:46:10.358925Z"Lyubich, Mikhail"https://zbmath.org/authors/?q=ai:lyubich.mikhail"Radu, Remus"https://zbmath.org/authors/?q=ai:radu.remus"Tanase, Raluca"https://zbmath.org/authors/?q=ai:tanase.ralucaSummary: In this paper we study the dynamics of germs of holomorphic diffeomorphisms of \((\mathbb{C}^n,0)\) with a fixed point at the origin with exactly one neutral eigenvalue. We prove that the map on any local center manifold of \(0\) is quasiconformally conjugate to a holomorphic map and use this to transport results from one complex dimension to higher dimensions.
For the entire collection see [Zbl 1446.37001].Further remarks on the higher dimensional Suita conjecturehttps://zbmath.org/1472.320042021-11-25T18:46:10.358925Z"Balakumar, G. P."https://zbmath.org/authors/?q=ai:balakumar.g-p"Borah, Diganta"https://zbmath.org/authors/?q=ai:borah.diganta"Mahajan, Prachi"https://zbmath.org/authors/?q=ai:mahajan.prachi"Verma, Kaushal"https://zbmath.org/authors/?q=ai:verma.kaushalWhen \(D\) is a domain in \(\mathbb{C}^n\), the biholomorphic invariant \( F_D^k(z) \) at a point \(z\) in \(D\) is the product of the Bergman kernel function on the diagonal at \(z\) and the Lebesgue measure of the Kobayashi indicatrix at \(z\). The authors have previously studied this invariant [Proc. Am. Math. Soc. 147, No. 8, 3401--3411 (2019; Zbl 1435.32011)], which was introduced by \textit{Z. Błocki} [Lect. Notes Math. 2116, 53--63 (2014; Zbl 1321.32003)].
The first main theorem states that if \(D\) is bounded, pseudoconvex, and \(h\)-extendible at a boundary point \(p\) with local model \(D_\infty\), then the nontangential limit of \(F_D^k\) at \(p\) equals the value of \(F_{D_\infty}^k\) at the interior point \( (-1, 0, \ldots, 0)\). The second main result addresses bounded, strongly pseudoconvex polyhedral domains in \(\mathbb{C}^2\) with piecewise smooth boundary. The authors classify the possible limiting behaviors of \(F_D^k\) at singular boundary points by adapting the scaling method that \textit{K.-T. Kim} and \textit{J. Yu} applied to analyze the boundary behavior of the holomorphic sectional curvature of the Bergman metric for such domains [Pac. J. Math. 176, No. 1, 141--163 (1996; Zbl 0886.32020)].Moduli space of meromorphic differentials with marked horizontal separatriceshttps://zbmath.org/1472.320052021-11-25T18:46:10.358925Z"Boissy, Corentin"https://zbmath.org/authors/?q=ai:boissy.corentinA (compact) translation surface is a pair \((X, \omega)\), where \(X\) is a (compact) Riemann surface and \(\omega\) is a holomorphic 1-form on the surface. Locally integrating the form defines a flat metric on the surface, with conical singularities. If one only asks the form \(\omega\) to be \emph{meromorphic}, obtaining a non-compact translation surface with infinite area (if the poles of \(\omega\) are not all simple).
The study of such objects is motivated by the fact that they appear naturally when dealing with compactifications of the moduli space of translation surfaces. More precisely, if a sequence \((X_n,\omega_n)\) converges to the boundary of the moduli space in the Deligne-Mumford compactification, then the thick components of \(X_n\), appropriately rescaled, converge to meromorphic differentials, see, e.g., [\textit{A. Eskin} et al., Publ. Math., Inst. Hautes Étud. Sci. 120, 207--333 (2014; Zbl 1305.32007)].
In this article, the author studies the topology of the moduli spaces of translation surfaces with poles equipped with an extra combinatorial data: the choice, for each singularity of (an equivalence class of) horizontal separatrix, denoted \(\mathcal{H}^{\text{hor}}\). Here, by horizontal separatrix we mean either an horizontal geodesic line ending (or beginning) at a conical singularity or an horizontal geodesic going to infinity in the flat metric if the singularity is a non simple pole.
The main result of the paper is a complete characterization of the connected components of \(\mathcal{H}^{\text{hor}}\). Similarly to the case of compact translation surfaces, proven in [\textit{M. Kontsevich} and \textit{A. Zorich}, Invent. Math. 153, No. 3, 631--678 (2003; Zbl 1087.32010)], in the general case of genus greater than 1 and underlining surfaces not belonging to the hyperelliptic component, there are at most 2 connected components, classified but an invariant \(\operatorname{Sp}\) which is a generalization of the classical \(\operatorname{Arf}\). In the hyperelliptic case the extra symmetry of the underlining surface yields more components. Finally, the genus-0 case is the most complicated one, depending also on the combinatorics of the singularities.
The main result is obtained by reducing the problem to the study of some cyclic group coming for the forgetful map from \(\mathcal{H}^{\text{hor}}\) to \(\mathcal{H}^{\text{sing}}\), which is the space of translation surfaces with poles in which singularities have given names. The latter space has the same connected components of the moduli space of translation with poles, which were classified by the author in [Comment. Math. Helv. 90, No. 2, 255--286 (2015; Zbl 1323.30060)].
Generalizing two local surgeries introduced in the article by Kontsevich and Zorich [loc. cit.], called \emph{breaking up a zero} and \emph{bubbling a handle}, so that they can be performed also on poles, one constructs some important elements in the cyclic groups coming from the covering. Using some topological analysis, one then proceeds to show that these elements always exist. If the above elements generate the whole cyclic group, then the corresponding moduli space is connected. If they generate a finite index subgroup, then the index gives the number of connected components. Depending on the genus and on whether or not we are in the hyperelliptic case, components are distinguished by some topological invariant, related to the classical \(\operatorname{Arf}\) invariant and to the parity of the spin structure.Topics on Teichmüller spaceshttps://zbmath.org/1472.320062021-11-25T18:46:10.358925Z"Seppälä, Mika"https://zbmath.org/authors/?q=ai:seppala.mika(no abstract)Growth of the Weil-Petersson inradius of moduli spacehttps://zbmath.org/1472.320072021-11-25T18:46:10.358925Z"Wu, Yunhui"https://zbmath.org/authors/?q=ai:wu.yunhuiFor a surface \(S_{g,n}\) of genus \(g\) with \(n\) punctures satisfying \(3g-3+n>0\), let Teich(\(S_{g,n}\)) be the Teichmüller space of \(S_{g,n}\) with the Weil-Petersson metric and let \(\mathcal{M}_{g,n}\) be the moduli space of \(S_{g,n}\) defined as the quotient of Teich\((S_{g,n})\) by the mapping class group Mod(\(S_{g,n}\)). The author studies the asymptotic behavior of the inradius of \(\mathcal{M}_{g,n}\) either as \(g\to \infty\) or \(n\to\infty\). It is proved that for all \(n\ge 0\) and \(g\ge 2\), the inradius of \(\mathcal{M}_{g,n}\) is comparable to \(\sqrt{\ln g}\) by a constant independent of \(g\); for all \(g\geq 0\) and \(n\geq 4\), it is comparable to 1 by a constant independent of \(n\). Here, the inradius is defined by the maximum of the Weil-Petersson distances dist\({}_{wp}(X, \partial\overline{\mathcal{M}}_{g,n})\) among all \(X\in \mathcal{M}_{g,n}\). To prove these results, the author considers the systole function \(\ell_{sys}\) on Teich(\(S_{g,n}\)) and gives a key theorem which establishes Lipschitz continuity of the square root of \(\ell_{sys}\) with respect to the Weil-Petersson distance, where the Lipschitz constant can be taken independently of \(g\) and \(n\). For the proof of Lipschitz continuity the author uses a thin-thick decomposition of the Weil-Petersson geodesics connecting two points in Teich(\(S_{g,n}\)) and estimates the norm of the gradient of the square root of geodesic length functions. \par Let \(\mathcal{M}_{g,n}^{\geqslant \epsilon}\) denote the \(\epsilon\)-thick part of of \(\mathcal{M}_{g,n}\). The moduli space \(\mathcal{M}_{g,n}\) is foliated by \(\partial\mathcal{M}_{g,n}^{\geqslant \epsilon}\) for all \(\epsilon\). The author shows that for any \(s>t\geq 0\) the Weil-Petersson distance between \(\partial\mathcal{M}_{g,n}^{\geqslant s}\) and \(\partial\mathcal{M}_{g,n}^{\geqslant t}\) is comparable to \(\sqrt{s}-\sqrt{t}\) by a constant independent of \(g\) and \(n\). \par Another interesting result in this paper is that for a closed surface \(S_g\) the author shows the asymptotic behavior of the Weil-Petersson volume of geodesic balls as \(g\to \infty\), where the geodesic balls have a finite radius and are away from the boundary of the completion of Teich(\(S_g\)).Corrigendum to: ``Maximal open radius for Strebel point''https://zbmath.org/1472.320082021-11-25T18:46:10.358925Z"Yao, Guowu"https://zbmath.org/authors/?q=ai:yao.guowuA correction of a minor error to the proof of Theorem 2 in [ibid. 178, No. 2, 311--324 (2015; Zbl 1329.32006), lines 17--20 on page 323] is given.Orthogonality of divisorial Zariski decompositions for classes with volume zerohttps://zbmath.org/1472.320092021-11-25T18:46:10.358925Z"Tosatti, Valentino"https://zbmath.org/authors/?q=ai:tosatti.valentinoConsider the following statement:
Conjecture: Let \((X, \omega)\) be a compact Kähler manifold, and \(\alpha\) a pseudoeffective \((1,1)\) class. Then \[ \langle \alpha^{n-1} \rangle \cdot \alpha = \mathrm{Vol}(\alpha), \] where \(\textrm{Vol}(\alpha)\) is the volume of the class \(\alpha\) and \(\langle \cdot \rangle\) is the moving intersection product of classes in the sense of Boucksom.
The above is known as the orthogonality conjecture for divisorial Zariski decompositions, which was observed by \textit{S. Boucksom} et al. [J. Algebr. Geom. 22, No. 2, 201--248 (2013; Zbl 1267.32017); J. Algebr. Geom. 18, No. 2, 279--308 (2009; Zbl 1162.14003)] and is equivalent to the weak transcendental Morse inequalities, the \(C^1\) differentiability of the volume function on the big cone, and the ``cone duality'' conjecture, i.e., \textit{the dual cone of the pseudoeffective cone is the movable cone}.
This was proven for \(X\) projective in [\textit{S. Boucksom} et al., J. Algebr. Geom. 22, No. 2, 201--248 (2013; Zbl 1267.32017); \textit{D. W. Nyström}, J. Am. Math. Soc. 32, No. 3, 675--689 (2019; Zbl 1429.32031)], and formulated as a conjecture on arbitrary compact Kähler manifolds in [\textit{S. Boucksom} et al., J. Algebr. Geom. 22, No. 2, 201--248 (2013; Zbl 1267.32017)]. The main result of this note is a proof of the orthogonality conjecture on arbitrary compact Kähler manifolds for pseudoeffective \((1,1)\) classes that are assumed to have volume zero.An application of Cartan's equivalence method to Hirschowitz's conjecture on the formal principlehttps://zbmath.org/1472.320102021-11-25T18:46:10.358925Z"Hwang, Jun-Muk"https://zbmath.org/authors/?q=ai:hwang.jun-mukLet \(A\) be a compact complex submanifold of a complex manifold \(X\). The submanifold \(A \subset X\) is said to \textit{satisfy the formal principle} if given a compact submanifold \(\tilde{A}\) of a complex manifold \(\tilde{X}\), a formal isomorphism \(\psi: (A/X)_{\infty} \rightarrow (\tilde{A}/\tilde{X})_{\infty}\) between the formal neighbourhoods, and a positiver integer \(l\), we can find a biholomorphism \[ \Psi: (A/X)_{\mathcal{O}} \rightarrow (\tilde{A}/\tilde{X})_{\mathcal{O}} \] such that \(\Psi|_{(A/X)_l} = \psi|_{(A/X)_l}\). Here we have written \((A/X)_{l}\) for the \(l\)-th order neighbourhood and \((A/X)_{\mathcal{O}}\) for the germ of Euclidean neighbourhoods of \(A\) in \(X\).
In order to compare the germ of \(A \subset X\) with the germ of the zero section of the normal bundle \(N_{A/X}\) (a question considered by many authors, e.g., [\textit{M. Abate} et al., Adv. Math. 220, No. 2, 620--656 (2009; Zbl 1161.32011)]) the author introduces the following terminology: A vector bundle \(W\) on a compact complex manifold \(A\) is said to satisfy the formal principle if the zero section \(0_A \subset W\) satisfies the formal principle in the classical sense.
A conjecture of \textit{A. Hirschowitz} [Ann. Math. (2) 113, 501--514 (1981; Zbl 0421.32029)] states the following:
Conjecture 1.3. Let \(A \subset X\) be an unobstructed compact submanifold of a complex manifold. Assume that the normal bundle \(N_{A/X}\) is globally generated, i.e., the sequence \[ 0 \rightarrow H^0(A,N_{A/X} \otimes m_x) \rightarrow H^0(A,N_{A/X}) \rightarrow N_{A/X,x} \rightarrow 0, \] where \(m_x\) is the maximal ideal at \(x \in A\), is exact at every \(x \in A.\) Then \(A \subset X\) satisfies the formal principle.
This in turn predicts the following:
Conjecture 1.4. A globally generated vector bundle on a compact complex manifold satisfies the formal principle.
In this paper the author obtains new results in the direction of Conjectures 1.3 and 1.4 by viewing families of submanifolds on a complex manifold as a geometric structure in the sense of Cartan (equivalence method). The main result states that if the sections of the normal bundle separate points in the setting of Conjecture 1.3, then the formal principle holds for sufficiently general deformations of \(A\) in \(X\). More precisely, one main novelty of the paper is to prove statements in terms of the Douady space, for example the main result is of the following form: ``Let \(X\) be a complex manifold and let \(K\) be a subset of the Douady space satisfying certain hypotheses (here omitted for brevity). Then there exists a nowhere-dense subset \(S\) of \(K\) such that the submanifolds corresponding to any point of \(K \setminus S\) satisfies the formal principle''.
The author proceeds to give a number of applications of this result regarding globally generated vector bundles that satisfy the formal principle (towards Conjecture 1.4). In particular Conjecture 1.4 is proven for Fano manifolds.
As a further application improvements to Cartan-Fubini type extension theorems are discussed, cf. [\textit{J.-M. Hwang} and \textit{N. Mok}, J. Math. Pures Appl. (9) 80, No. 6, 563--575 (2001; Zbl 1033.32013)], as part of a program to replace difficult-to-check transcendental conditions by algebraic conditions.Positivity of direct images of fiberwise Ricci-flat metrics on Calabi-Yau fibrationshttps://zbmath.org/1472.320112021-11-25T18:46:10.358925Z"Braun, Matthias"https://zbmath.org/authors/?q=ai:braun.matthias"Choi, Young-Jun"https://zbmath.org/authors/?q=ai:choi.young-jun"Schumacher, Georg"https://zbmath.org/authors/?q=ai:schumacher.georgThe setup of the paper is as follows: let \(p:X\to Y\) be a proper holomorphic submersion of relative dimension \(n\) between complex manifolds such that \(X\) admits a Kähler metric \(\omega\) and for all \(y\in Y\), the fiber \(X_y\) is a Calabi-Yau manifold, i.e., \(c_1(X_y)=0\in H^2(X_y, \mathbb R)\). By Yau's solution of the Calabi conjecture [\textit{S.-T. Yau}, Commun. Pure Appl. Math. 31, 339--411 (1978; Zbl 0369.53059)], there exists a unique smooth \((1,1)\)-form \(\omega_{\mathrm{KE}}=\omega+dd^c \varphi\) such that for all \(y\in Y\), one has
\begin{enumerate}
\item[\((i)\)] \(\omega_{\mathrm{KE}}|_{X_y}\) is a Kähler Ricci-flat metric on \(X_y\),
\item[\((ii)\)] \(\int_{X_y} \varphi \, \omega_{\mathrm {KE}}|_{X_y}=0\).
\end{enumerate}
The main result is that the fiberwise integral \(p_*\omega_{\mathrm{KE}}^{n+1}\) is a semipositive \((1,1)\)-form on \(Y\). \\
In the case of families of canonically polarized manifolds, the third author [Invent. Math. 190, No. 1, 1--56 (2012; Zbl 1258.32005)] proved that the relative Kähler-Einstein metric is semipositive on the total space of the fibration \(X\), but this needs not be the case for families of Calabi-Yau manifolds as was observed by [\textit{J. Cao} et al., ``Variation of singular Kähler-Einstein metrics: Kodaira dimension zero'', Preprint, \url{arXiv:1908.08087}].
The proof of the theorem goes roughly as follows. First, one reduces to the case where \(Y=\mathbb D\) is the unit disk in \(\mathbb C\), and one introduces the geodesic curvature \(c(\varphi)\) of \(\omega_{\mathrm{KE}}\) by \(\omega_{\mathrm{KE}}^{n+1}=c(\varphi) \omega_{\mathrm{KE}}^{n}\wedge idt\wedge d\bar t\). The first step consists in approximating \(\varphi\) by a family of smooth functions \(\varphi_\varepsilon\) obtained as solutions of the fiberwise Monge-Ampère equation \((\omega+dd^c \varphi_\varepsilon)^n=e^{\varepsilon \varphi_\varepsilon} \omega_{\mathrm{KE}}^{n}\) on each \(X_t\). Set \(\omega_{\mathrm{KE},\varepsilon}:=\omega+dd^c \varphi_\varepsilon\). The main point of this approximation is that Schumacher's formula for the geodesic curvature can be written as
\[
-\Delta_{\omega_{\mathrm{KE},\varepsilon}} c(\varphi_\varepsilon)+\varepsilon c(\varphi_\varepsilon)=A_\varepsilon+B_\varepsilon
\]
where \(A_\varepsilon\ge 0\) and \(\int_{X_t} B_\varepsilon \omega_{\mathrm{KE},\varepsilon}^n=0\) from which the conclusion \(\int_{X_t} c(\varphi_\varepsilon) \omega_{\mathrm{KE},\varepsilon}^n \ge 0\) follows, and the main result as well by letting \(\varepsilon\) got to zero.Cohomologies on almost complex manifolds and the \(\partial \overline{\partial} \)-lemmahttps://zbmath.org/1472.320122021-11-25T18:46:10.358925Z"Chan, Ki Fung"https://zbmath.org/authors/?q=ai:chan.ki-fung"Karigiannis, Spiro"https://zbmath.org/authors/?q=ai:karigiannis.spiro"Tsang, Chi Cheuk"https://zbmath.org/authors/?q=ai:tsang.chi-cheukThe authors define and study three natural chain complexes associated to an almost complex manifold \((M,J)\), and their cohomology.
The basic operator in the construction of the complexes is the \textit{algebraic derivation} \(\iota_K\) associated to a vector-valued \(k\) form \(K\) on \(M\), defined by sending a form \(\alpha\) to \(K^j \wedge (\iota_{e_j} \alpha)\), where \(K = K^j e_j\) in a local frame \(\{e_j\}\), and \(\iota_{e_j}\) is the usual interior product. One then forms the \textit{Nijenhuis-Lie derivation} \(\mathcal{L}_K\) by taking the graded commutator \([\iota_K, d]\) with the exterior derivative operator.
On an almost complex manifold, we have two natural vector-valued forms: \(J\) itself, a vector-valued one-form, and its Nijenhuis tensor \(N\), a vector-valued two-form. One notes that \((\mathcal{L}_J)^2 = -\mathcal{L}_N\); by the Newlander-Nirenberg theorem, it follows that \(\mathcal{L}_J\) squares to zero if and only if \(J\) is induced by holomorphic charts. In general, neither \(\mathcal{L}_J\) nor \(\mathcal{L}_N\) square to zero. However, the authors notice that \([d, \mathcal{L}_J] = [d, \mathcal{L}_N] = [\mathcal{L}_J, \mathcal{L}_N]\), and hence any of the three operators \(d, \mathcal{L}_J, \mathcal{L}_N\) maps the kernel of any other of the three operators to itself. Hence we obtain three chain complexes:
\begin{itemize}
\item \((\ker \mathcal{L}_J, d)\), whose cohomology the authors call the \textit{\(J\)-cohomology} of \((M,J)\),
\item \((\ker \mathcal{L}_N, d)\), whose cohomology is called the \textit{\(N\)-cohomology}, and
\item \((\ker \mathcal{L}_N , \mathcal{L}_J)\), whose cohomology is called the \textit{\(J\)-twisted \(N\)-cohomology}.
\end{itemize}
If \(J\) is integrable, then clearly the \(N\)-cohomology reduces to the de Rham cohomology, and furthermore \(\mathcal{L}_J = -d^c = -J^{-1}dJ\), so the \(J\)-twisted \(N\)-cohomology is the \(d^c\)-cohomology, isomorphic to de Rham cohomology.
These complexes are natural with respect to pseudoholomorphic maps of almost complex manifolds; hence these cohomologies give invariants of almost complex manifolds which are furthermore practically computable, as the authors go on to demonstrate. As an example of distinguishing diffeomorphic but not isomorphic almost complex manifolds, the authors employ the \(N\)-cohomology to distinguish several non-isomorphic almost complex structures on \(S^1 \times \mathbb{R}^3\).
The remainder of the paper is spent studying the \(J\)-cohomology, which gives a new invariant even in the integrable case. Among other results, the authors prove the \(J\)-cohomology of a compact complex manifold is finite-dimensional, and that in the nonintegrable compact case, it is finite-dimensional in zeroth, first, and top degrees.
The authors introduce a \(d\mathcal{L}_J\)-lemma for almost complex manifolds, which is equivalent to the much studied \(\partial \bar{\partial}\)-lemma (or equivalently the \(dd^c\)-lemma) in the integrable case. An almost complex manifold satisfies the \(d\mathcal{L}_J\)-lemma if and only if the natural map from \(J\)-cohomology to de Rham cohomology is an isomorphism. As an illustration of the applicability of this notion, the authors give a new proof that Hopf surfaces and the Iwasawa manifold do not satisfy the \(\partial \bar{\partial}\)-lemma (for Hopf surfaces, one would usually argue this by, say, observing that the \(\partial \bar{\partial}\)-lemma implies degeneration of the Hodge-de Rham spectral sequence, which further implies that the first Betti number must be even; for the Iwasawa manifold, one can show it admits a non-trivial triple Massey product, violating the rational homotopy theoretic formality that is guaranteed for \(\partial \bar{\partial}\)-manifolds by work of Deligne-Griffiths-Morgan-Sullivan). The authors compute that the rank of the first \(J\)-cohomology of Hopf surfaces and the Iwasawa manifold is strictly smaller than the corresponding first Betti number, thus showing the \(d\mathcal{L}_J\)-lemma is not satisfied.
The authors then study in detail the \(J\)-cohomology of a family of almost complex structures on the four-dimensional torus, and end with some potential future directions for investigation.Computing regular meromorphic differential forms via Saito's logarithmic residueshttps://zbmath.org/1472.320132021-11-25T18:46:10.358925Z"Tajima, Shinichi"https://zbmath.org/authors/?q=ai:tajima.shinichi"Nabeshima, Katsusuke"https://zbmath.org/authors/?q=ai:nabeshima.katsusukeThe concept of regular meromorphic differential forms was introduced independently by \textit{E. Kunz} [Manuscr. Math. 15, 91--108 (1975; Zbl 0299.14013)] and \textit{D. Barlet} [C. R. Acad. Sci., Paris, Sér. A 282, 579--582 (1976; Zbl 0323.32006)]. Somewhat later the reviewer [Adv. Sov. Math. 1, 211--246 (1990; Zbl 0731.32005)] proved that in the hypersurface case such forms naturally appear as the image of the logarithmic residue introduced by \textit{K. Saito} [J. Fac. Sci., Univ. Tokyo, Sect. I A 27, 265--291 (1980; Zbl 0496.32007)]. The authors of the paper under review present an algorithm for computing regular meromorphic differential forms using Saito's residue and torsion differentials of the modules of regular holomorphic forms (see [the rewiever, Complex Variables, Theory Appl. 50, No. 7--11, 777--802 (2005; Zbl 1083.32024)]. Then their constructions are applied for explicit computations of the Gauss-Manin connection in the case of isolated hypersurface singularities. As an example, the cases of functions of two and three variables are analyzed in detail.Erratum to: ``Divisionally free arrangements of hyperplanes''https://zbmath.org/1472.320142021-11-25T18:46:10.358925Z"Abe, Takuro"https://zbmath.org/authors/?q=ai:abe.takuroFrom the text: The aim of this note is to correct the statement and the proof of Theorem 6.2 in the author's paper [ibid. 204, No. 1, 317--346 (2016; Zbl 1341.32023)], which is not correct as it was stated. All the other results in the paper are correct as they were stated.On CR-structures and the general quadratic structurehttps://zbmath.org/1472.320152021-11-25T18:46:10.358925Z"Khan, Mohammad Nazrul Islam"https://zbmath.org/authors/?q=ai:khan.mohd-nazrul-islam"Das, Lovejoy S."https://zbmath.org/authors/?q=ai:das.lovejoy-s-kSummary: The object of the present paper is to determine the relationship between CR-structure and the general quadratic structure and find some basic results. We discuss integrability conditions and prove certain theorems on CR-structure and the general quadratic structure.Regular multi-types and the Bloom conjecturehttps://zbmath.org/1472.320162021-11-25T18:46:10.358925Z"Huang, Xiaojun"https://zbmath.org/authors/?q=ai:huang.xiaojun"Yin, Wanke"https://zbmath.org/authors/?q=ai:yin.wankeIn a fundamental paper [J. Differ. Geom. 6, 523--542 (1972; Zbl 0256.35060)], \textit{J. J. Kohn} first introduced a finite-type condition for a smooth hypersurface in \(\mathbb{C}^2\) to investigate the subellipticity of \(\overline{\partial}\)-Neumann operators over a bounded weakly pseudoconvex domain in \(\mathbb{C}^2\). Kohn's finite-type condition has various generalizations to higher dimensions by many mathematicians such as Bloom-Graham, Bloom, D'Angelo, Catlin and Sibony, etc. These conditions played important roles in answering fundamental problems in several complex variables. For an integer \(s\) with \(1\leq s \leq n-1\) and a point \(p\) on a smooth hypersurface \(M\subset {\mathbb C}^n\) with \(n\ge 2\), \textit{T. Bloom} [Trans. Am. Math. Soc. 263, 515--529 (1981; Zbl 0459.32005)] defined three notions of type taking values in \([2,\infty]\): regular contact type \(a^{(s)}(p)\), iterated commutator type \(t^{(s)}(p)\) and Levi-form type \(c^{(s)}(p)\). Bloom conjectured that these three invariants, though defined very differently, are the same when \(M\) is pseudoconvex. This famous problem is now known as the Bloom conjecture. In the same paper, Bloom proved \(a^{(1)}=c^{(1)}\) for a hypersurface in \(\mathbb{C}^3\). Earlier, \textit{T. Bloom} and \textit{I. Graham} [J. Differ. Geom. 12, 171--182 (1977; Zbl 0436.32013)] and \textit{T. Bloom} [in: Several complex variables, Proc. int. Conf., Cortona/Italy 1976--77, 14--22 (1978; Zbl 0421.32022)] have already achieved the conjecture for \(s=n-1\).
The paper under review confirms the Bloom conjecture for \(s=n-2\), which is the first affirmative result on the Bloom conjecture where the pseudoconvexity is fundamentally needed. Notice that the previous theorem of Bloom-Graham for \(s=n-1\) holds for any smooth hypersurfaces and there are many examples for \(s=n-2\) where the Bloom conjecture fails for non-pseudoconvex hypersurfaces. As a corollary, the two authors provide a complete solution of the conjecture for a smooth hypersurface in \(\mathbb{C}^3\), making the first important progress 40 years after Bloom proposed his conjecture.
A major difficulty of working on the Bloom conjecture is to find a good use of the pseudoconvexity. The authors use these conditions in a very clever way to validate the Hopf lemma and to obtain a uniqueness of solutions of a complex linear partial differential equation with real part plurisubharmonic functions. It is also interesting to notice the important role played by the Euler vector field in their proof.On holomorphic extendability and the strong maximum principle for CR functionshttps://zbmath.org/1472.320172021-11-25T18:46:10.358925Z"Berhanu, S."https://zbmath.org/authors/?q=ai:berhanu.shiferawThe paper investigates necessary and sufficient conditions for a \(\mathcal{C}^\infty\) surface \(\mathcal{M}\) in \(\mathbb{C}^n\) to satisfy a strong maximum principle for CR-functions.
Definition. \(\mathcal{M}\) satisfies the weak maximum principle if \(\forall U\subset \mathcal{M}\) connected open set and \(\forall h\) CR-function in \(U\), either \(h\) is constant or \(|h|\) has no weak local maximum in \(U\).
Definition. \(\mathcal{M}\) satisfies the strong maximum principle if \(\forall U\subset \mathcal{M}\) connected open set and \(\forall h\) CR-function in \(U\), \(|h|\) has no strict maximum in \(U\).
While the weak maximum principle is well understood, the same is not true for the strong one. In the paper various results (for the general case or specific cases) are proven:
\begin{itemize}
\item If \(\mathcal{M}\) satisfies the extendability property (i.e., all CR-functions on \(\mathcal{M}\) extend to holomorphic functions), then \(\mathcal{M}\) satisfies the strong maximum principle.
\item (nearly a viceversa) If \(\mathcal{M}\) satisfies the strong maximum principle, then there is a dense subset \(\Sigma\subset\mathcal{M}\) such that CR-functions on \(\mathcal{M}\) extend to a neighbourhood of \(\Sigma\).
\item For real analytic tubes, the strong maximum principle is equivalent to analytic hypoellipticity.
\item If \(\mathcal{M}\) is of hypersurface type and every locally integrable CR distribution defined on an open subset of \(\mathcal M\) is smooth, then \(\mathcal M\) satisfies the strong maximum principle.
\end{itemize}
The paper also devotes some space to counterexamples, e.g., an example of a hypersurface \(\mathcal{M}\subset\mathbb{C}^6\) satisfying the strong maximum principle for the restrictions of holomorphic functions, but not for CR-functions. An example like this was already known [the author, Contemp. Math. 205, 1--13 (1997; Zbl 0901.32015)], but the example here is simpler.The stationary disc method in the unique jet determination of CR automorphismshttps://zbmath.org/1472.320182021-11-25T18:46:10.358925Z"Bertrand, Florian"https://zbmath.org/authors/?q=ai:bertrand.florianFinite jet determination of holomorphic maps of real manifolds has gained notable attention during the recent decades. Among them, we have in particular the impressive result of \textit{S. S. Chern} and \textit{J. Moser} in their celebrated work [Acta Math. 133, 219--271 (1974; Zbl 0302.32015)] showing that every holomorphic automorphism of a certain real-analytic nondegenerate hypersurface in an arbitrary complex space, which preserves some nondegenerate point \(p\) is uniquely determined by its jets of order two at this point. One finds a large amount of considerable works on the finite jet determination of holomorphic maps in the real analytic setting.
In the smooth case, most results on this issue rely on the method of complete differential systems initiated by Cartan and Chern-Moser. But in the case of \textit{finitely smooth} manifolds, it seems that the method of stationary discs is the only known way to treat the problem of finite jet determination. Attached to a given real submanifold \(M\subset\mathbb{C}^N\), the stationary discs are actually a family of analytic invariant objects which admit a lift with a pole of order at most 1 at the origin. The main idea behind attaching analytic stationary discs to real submanifolds is a boundary value problem, namely a nonlinear problem of Riemann-Hilbert type.
The paper under review studies the finite jet determination of holomorphic automorphisms of some specific, but interesting, (degenerate and nondegenerate) kinds of real submanifolds in arbitrary complex spaces by constructing the attached stationary discs and analyzing their corresponding geometric features.Adiabatic limit and the Frölicher spectral sequencehttps://zbmath.org/1472.320192021-11-25T18:46:10.358925Z"Popovici, Dan"https://zbmath.org/authors/?q=ai:popovici.danIn complex geometry, it is well known that the Frölicher spectral sequence of a compact Kähler manifold degenerates at \(E_1\) page (in particular it degenerates at \(E_2\) page). Since the Kähler condition is quite restrictive for compact complex manifolds of dimension at least 3, it is natural to seek other metric conditions which ensure the \(E_2\)-degeneration of the Frölicher spectral sequence.
Let \(X\) be a compact complex manifold with a Hermitian metric \(\omega\). In this article, the author gives a sufficient metric condition for degeneration at \(E_2\), which roughly says that the torsion of \(\omega\) is ``small''. One of the new ideas is to consider the rescalings of \(\omega\) and \(\partial\), which is an adaption of the adiabatic limit construction associated with a Riemann foliation (see, e.g., [\textit{E. Witten}, Commun. Math. Phys. 100, 197--229 (1985; Zbl 0581.58038)]) to the case of the splitting \(d=\partial +\overline{\partial}\). It seems interesting to point out that similar ideas also appeared in the setting of non-abelian Hodge theory, see [\textit{C. Simpson}, Mixed twistor structures, arXiv preprint alg-geom/9705006, 1997] and Theorem 2.2.4 in [\textit{C. Sabbah}, Polarizable twistor \(\mathcal{D}\)-modules. Paris: Sociéteé Mathématique de France (2005; Zbl 1085.32014)].
Moreover, using a variant of the Efremov-Shubin variational principle, along with the pesudodifferential Laplacian in [the author, Int. J. Math. 27, No. 14, Article ID 1650111, 31 p. (2016; Zbl 1365.53067)] and Demaily's Bochner-Kodaira-Nakano formula for Hermitian metrics, the author finds a formula for the dimensions of the vector spaces on each page of the Frölicher spectral sequence in terms of of the number of small eigenvalues of the rescaled Laplacian. This formula is of independent interest, and is inspired by the analogous result for foliations proven in [\textit{J. A. Álvarez López} and \textit{Y. A. Kordyukov}, Geom. Funct. Anal. 10, No. 5, 977--1027 (2000; Zbl 0965.57024)].Corrigendum to: ``Viscosity solutions to degenerate complex Monge-Ampère equations''https://zbmath.org/1472.320202021-11-25T18:46:10.358925Z"Eyssidieux, Philippe"https://zbmath.org/authors/?q=ai:eyssidieux.philippe"Guedj, Vincent"https://zbmath.org/authors/?q=ai:guedj.vincent"Zeriahi, Ahmed"https://zbmath.org/authors/?q=ai:zeriahi.ahmedSummary: The proof of the comparison principle in our article [ibid. 64, No. 8, 1059--1094 (2011; Zbl 1227.32042)] is not complete. We provide here an alternative proof, valid in the ample locus of any big cohomology class, and discuss the resulting modifications.Parabolic complex Monge-Ampère equations on compact Kähler manifoldshttps://zbmath.org/1472.320212021-11-25T18:46:10.358925Z"Picard, Sebastien"https://zbmath.org/authors/?q=ai:picard.sebastien"Zhang, Xiangwen"https://zbmath.org/authors/?q=ai:zhang.xiangwenThe authors consider the parabolic Monge-Ampère equation
\[
\partial_t u =F(e^{-f}\det (\delta^i_j + \nabla^i\nabla_j u)),
\]
with
\[
\omega + i \partial \bar \partial u(x,t)>0,
\]
where \((X, \omega)\) is a compact Kähler manifold, \(f\in C^{\infty}(X, \mathbb{R})\) is a given function, \(F: \mathbb{R}_+ \rightarrow \mathbb{R}\) is a smooth strictly increasing function, and \(u\) is the unkown solution. Under the smooth initial data \(u(x,0)=u_0(x)\), the long-time existence and convergence of the parabolic equation are studied.
The main theorem (Theorem 1.1) is proved without any concavity (or convexity) assumption on the speed function \(F\), which includes the Kähler-Ricci flow (\(F(\rho)=\log \rho\)), the inverse Monge-Ampère flow (\(F(\rho)=1-\rho^{-1}\)), conformally Kähler Anomaly flow (\(F(\rho)=\rho\)), and the modified version of the Anomaly flow with zero slope parameter (\(F(\rho)=\rho^a\)) as special cases.
For the entire collection see [Zbl 1454.00057].Convergence of two-dimensional hypergeometric series for algebraic functionshttps://zbmath.org/1472.330092021-11-25T18:46:10.358925Z"Cherepanskiy, A. N."https://zbmath.org/authors/?q=ai:cherepanskiy.a-n"Tsikh, A. K."https://zbmath.org/authors/?q=ai:tsikh.avgust-k|tsikh.august-kSummary: Description of convergence domains for multiple power series is a quite difficult problem. In 1889 J.Horn showed that the case of hypergeomteric series is more favourable. He found a parameterization formula for surfaces of conjugative radii of such series. But until recently almost nothing was known about the description of convergence domains in terms of functional inequalities \(\rho_j(|a_1|,\dots,|a_m|)< 0\) relatively moduli \(|a_i|\) of series variables. In this paper we give a such description for hypergeometric series representing solutions to tetranomial algebraic equations. In our study we use the remarkable observation by M. Kapranov (1991) consisting in the fact that the Horn's formulae give a parameterization of discriminant locus for a corresponding A-discriminant. We prove that usually the considered convergence domains are determined by a single or two inequalities \(\rho(|a_t|,|a_s|)\lessgtr 0\), where \(\rho\) is a reduced discriminant.Integrable systems, multicomponent twisted Heisenberg-Virasoro algebra and its central extensionshttps://zbmath.org/1472.352812021-11-25T18:46:10.358925Z"Wu, Yemo"https://zbmath.org/authors/?q=ai:wu.yemo"Xu, Xiurong"https://zbmath.org/authors/?q=ai:xu.xiurong"Zuo, Dafeng"https://zbmath.org/authors/?q=ai:zuo.dafengSummary: Let \(\mathscr{D}_N\) be the multicomponent twisted Heisenberg-Virasoro algebra. We compute the second continuous cohomology group with coefficients in \(\mathbb{C}\) and study the bihamiltonian Euler equations associated to \(\mathscr{D}_N\) and its central extensions.Global Schrödinger map flows to Kähler manifolds with small data in critical Sobolev spaces: high dimensionshttps://zbmath.org/1472.353262021-11-25T18:46:10.358925Z"Li, Ze"https://zbmath.org/authors/?q=ai:li.zeSummary: In this paper, we prove that the Schrödinger map flows from \(\mathbb{R}^d\) with \(d \geq 3\) to compact Kähler manifolds with small initial data in critical Sobolev spaces are global. This is a companion work of the authors' previous paper [``Global Schrödinger map flows to Kähler manifolds with small data in critical Sobolev spaces: energy critical case'', Preprint. \url{arXiv:1811.10924}] where the energy critical case \(d = 2\) was solved. In the first part of this paper, for heat flows from \(\mathbb{R}^d\) \((d \geq 3)\) to Riemannian manifolds with small data in critical Sobolev spaces, we prove the decay estimates of moving frame dependent quantities in the caloric gauge setting, which is of independent interest and may be applied to other problems. In the second part, with a key bootstrap-iteration scheme in our previous work [loc. cit.], we apply these decay estimates to the study of Schrödinger map flows by choosing caloric gauge. This work with our previous work solves the open problem raised by \textit{D. Tataru} [Am. J. Math. 123, No. 1, 37--77 (2001; Zbl 0979.35100)].Periodic subvarieties of semiabelian varieties and annihilators of irreducible representationshttps://zbmath.org/1472.370942021-11-25T18:46:10.358925Z"Bell, Jason P."https://zbmath.org/authors/?q=ai:bell.jason-p"Ghioca, Dragos"https://zbmath.org/authors/?q=ai:ghioca.dragosSummary: Let \(G\) be a semiabelian variety defined over a field of characteristic 0, endowed with an endomorphism \({\Phi}\). We prove there is no proper subvariety \(Y \subset G\) which intersects the orbit of each periodic point of \(G\) under the action of \({\Phi}\). As an application, we are able to give a topological characterization of the annihilator ideals of irreducible representations in certain skew polynomial algebras.Shape holomorphy of the Calderón projector for the Laplacian in \(\mathbb{R}^2\)https://zbmath.org/1472.450112021-11-25T18:46:10.358925Z"Henríquez, Fernando"https://zbmath.org/authors/?q=ai:henriquez.fernando"Schwab, Christoph"https://zbmath.org/authors/?q=ai:schwab.christophThe authors establish the holomorphic dependence of the Calderón projector for the Laplace equation on a collection of sufficiently smooth Jordan curves in the Cartesian Euclidean plan. To be precise, they establish holomorphy of the domain-to-operator map associated to the Calderón projector.Composition operators from \(p\)-Bloch space to \(q\)-Bloch space on the fourth Cartan-Hartogs domainshttps://zbmath.org/1472.470202021-11-25T18:46:10.358925Z"Su, Jianbing"https://zbmath.org/authors/?q=ai:su.jianbing"Zhang, Chao"https://zbmath.org/authors/?q=ai:zhang.chao.2Summary: We obtain new generalized Hua's inequality corresponding to \(Y_{\text{IV}}(N, n; K)\), where \(Y_{\text{IV}}(N, n; K)\) denotes the fourth Cartan-Hartogs domain in \(\mathbb{C}^{N + n}\). Furthermore, we introduce the weighted Bloch spaces on \(Y_{\text{IV}}(N, n; K)\) and apply our inequality to study the boundedness and compactness of composition operator \(C_\phi\) from \(\beta^p(Y_{\text{IV}}(N, n; K))\) to \(\beta^q(Y_{\text{IV}}(N, n; K))\) for \(p \geq 0\) and \(q \geq 0\).CR-harmonic mapshttps://zbmath.org/1472.530782021-11-25T18:46:10.358925Z"Dietrich, Gautier"https://zbmath.org/authors/?q=ai:dietrich.gautierSummary: We develop the notion of renormalized energy in Cauchy-Riemann (CR) geometry for maps from a strictly pseudoconvex pseudo-Hermitian manifold to a Riemannian manifold. This energy is a CR invariant functional whose critical points, which we call CR-harmonic maps, satisfy a CR covariant partial differential equation. The corresponding operator coincides on functions with the CR Paneitz operator.A new proof of a conjecture on nonpositive Ricci curved compact Kähler-Einstein surfaceshttps://zbmath.org/1472.530832021-11-25T18:46:10.358925Z"Guan, Zhuang-Dan Daniel"https://zbmath.org/authors/?q=ai:guan.zhuang-dan-danielSummary: In an earlier paper, we gave a proof of the conjecture of the pinching of the bisectional curvature mentioned in those two papers of \textit{Y. Hong} et al. [Acta Math. Sin. 31, No. 5, 595--602 (1988; Zbl 0678.53060); Sci. China, Math. 54, No. 12, 2627--2634 (2011; Zbl 1259.53067)]. Moreover, we proved that any compact Kähler-Einstein surface \(M\) is a quotient of the complex two-dimensional unit ball or the complex two-dimensional plane if (1) \(M\) has a nonpositive Einstein constant, and (2) at each point, the average holomorphic sectional curvature is closer to the minimal than to the maximal. Following Siu and Yang, we used a minimal holomorphic sectional curvature direction argument, which made it easier for the experts in this direction to understand our proof. On this note, we use a maximal holomorphic sectional curvature direction argument, which is shorter and easier for the readers who are new in this direction.A moment map interpretation of the Ricci form, Kähler-Einstein structures, and Teichmüller spaceshttps://zbmath.org/1472.530922021-11-25T18:46:10.358925Z"García-Prada, Oscar"https://zbmath.org/authors/?q=ai:garcia-prada.oscar"Salamon, Dietmar"https://zbmath.org/authors/?q=ai:salamon.dietmar-aSummary: This paper surveys the role of moment maps in Kähler geometry. The first section discusses the Ricci form as a moment map and then moves on to moment map interpretations of the Kähler-Einstein condition and the scalar curvature (Quillen-Fujiki-Donaldson). The second section examines the ramifications of these results for various Teichmüller spaces and their Weil-Petersson symplectic forms and explains how these arise naturally from the construction of symplectic quotients. The third section discusses a symplectic form introduced by Donaldson on the space of Fano complex structures.
For the entire collection see [Zbl 1461.37002].A finiteness theorem for holonomic DQ-modules on Poisson manifoldshttps://zbmath.org/1472.530972021-11-25T18:46:10.358925Z"Kashiwara, Masaki"https://zbmath.org/authors/?q=ai:kashiwara.masaki"Schapira, Pierre"https://zbmath.org/authors/?q=ai:schapira.pierreSummary: On a complex symplectic manifold, we prove a finiteness result for the global sections of solutions of holonomic DQ-modules in two cases: (a) by assuming that there exists a Poisson compactification, (b) in the algebraic case. This extends our previous result in which the symplectic manifold was compact. The main tool is a finiteness theorem for \(\mathbb{R}\)-constructible sheaves on a real analytic manifold in a nonproper situation.Lengths of closed geodesics on random surfaces of large genushttps://zbmath.org/1472.570282021-11-25T18:46:10.358925Z"Mirzakhani, Mariam"https://zbmath.org/authors/?q=ai:mirzakhani.maryam"Petri, Bram"https://zbmath.org/authors/?q=ai:petri.bramIn this paper, the authors consider the asymptotic behavior of the distribution of short closed geodesics on random hyperbolic spaces as the genera tend to infinity. The moduli space \(\mathcal{M}_g\) of Riemann surfaces of genus \(g\) admits a natural probability measure \(\mathbb{P}_g\) induced from the Weil-Petersson metric. For \(X\in \mathcal{M}_g\) and an interval \([a,b]\subset \mathbb{R}_+=\{x\in \mathbb{R}\mid x>0\}\), let \(N_{g,[a,b]}(X)\) denote the number of primitive closed geodesics on \(X\) with lengths in the given interval. For \(a_1<b_1\le a_2<b_2\le\) \(\cdots\) \(\le a_k<b_k\), let \((N_{[a_i,b_i]})_{i=1}^k\) be a vector of independent Poisson distributed random variables with means \(\lambda_{[a_i,b_i]}=\int_{a_i}^{b_i}(e^t+e^{-t}-2)/(2t)\, dt\) (\(1\le i\le k\)) on a probability space (with the probabilty measure \(\mathbb{P}\)) that is rich enough to carry such a variable.
The authors first show that for any \(m_i\in \mathbb{N}\) (\(i=1,\dots,k\)), \(\mathbb{P}_g(N_{g,[a_i,b_i]}=m_i, 1\le i\le k)\) tends to \(\mathbb{P}(N_{[a_i,b_i]}=m_i, 1\le i\le k)=\prod_{i=1}^k(\lambda_i^{m_i}e^{-\lambda_i}/m_i!)\) as \(g\to \infty\), where \(\lambda_i=\lambda_{[a_i,b_i]}\).
Let \(\mathrm{sys}(X)\) be the systole of \(X\in \mathcal{M}_g\) and \(\mathbb{E}_g(\mathrm{sys})\) the expectation. Using the relation \(\mathbb{P}_g(\mathrm{sys}\le x)=1-\mathbb{P}_g(N_{g,[0,x]}=0)\), the authors show that \(\mathbb{E}_g(\mathrm{sys})\) tends to \(\int_0^\infty e^{-\lambda_{[0,R]}}dR=1.61498\ldots\) as \(g\to \infty\). In the comparison with Brooks and Makover's result [\textit{R. Brooks} and \textit{E. Makover}, J. Differ. Geom. 68, No. 1, 121--157 (2004; Zbl 1095.30037)], the authors also notice by applying the above result that \(\mathbb{P}_g(\mathrm{sys}\ge b)\to e^{-\lambda_{[0,b]}}=0.339043\ldots\) as \(g\to \infty\) for \(b=2\cdot\cosh^{-1}(3/2)\), while the probability measure from Brooks and Makover's model asymptotically concentrates in the \(b\)-thick part of the moduli space.
In the appendix, the authors also give a sketch of an unpublished result by M. Mirzakhani that there is a universal constant \(A\), \(B>0\) so that for any sequence \(\{c_g\}_g\) of positive numbers with \(c_g<A\log g\), \(\mathbb{P}_g(\mathrm{sys}\ge c_g)<Bc_ge^{-c_g}\).Small-time asymptotics for subelliptic Hermite functions on \(SU(2)\) and the CR spherehttps://zbmath.org/1472.580182021-11-25T18:46:10.358925Z"Campbell, Joshua"https://zbmath.org/authors/?q=ai:campbell.joshua"Melcher, Tai"https://zbmath.org/authors/?q=ai:melcher.tai\textit{J. J. Mitchell} [J. Funct. Anal. 164, No. 2, 209--248 (1999; Zbl 0928.22010)] studied the small-time behavior of Hermite functions on compact Lie groups. In particular, he demonstrated that, when written in exponential coordinates with a natural rescaling, these functions converge to the classical Euclidean Hermite polynomials. In a subsequent work [\textit{J. J. Mitchell}, J. Math. Anal. Appl. 263, No. 1, 165--181 (2001; Zbl 0997.43007)], he proved that Hermite functions on compact Riemannian manifolds, again written in exponential coordinates with appropriate rescaling, admit asymptotic expansions with a classical Hermite polynomial as the leading coefficient.
In the paper under review, the authors investigate heat kernels related to the natural subRiemannian structure on \(\mathrm{SU}(2)\simeq \S^3\) and, more generally, on higher-order odd-dimensional spheres.
More specifically, they prove that under a natural scaling, the small-time behavior of the logarithmic derivatives of the subelliptic heat kernel on \(\mathrm{SU}(2)\) converges to their analogues on the Heisenberg group at time 1. Next, they generalize these results to the CR sphere \(\S^{2d+1} \) equipped with their natural subRiemannian structure, where the limiting spaces are now the higher-dimensional Heisenberg groups. In other words, they obtain similar results as in [Mitchell, loc. cit.] for the Hermite functions on the CR spheres.Note on de Sitter vacua from perturbative and non-perturbative dynamics in type IIB/F-theory compactificationshttps://zbmath.org/1472.811512021-11-25T18:46:10.358925Z"Basiouris, Vasileios"https://zbmath.org/authors/?q=ai:basiouris.vasileios"Leontaris, George K."https://zbmath.org/authors/?q=ai:leontaris.george-kSummary: The properties of the effective scalar potential are studied in the framework of type IIB string theory, taking into account perturbative and non-perturbative corrections. The former modify the Kähler potential and include \(\alpha^\prime\) and logarithmic corrections generated when intersecting \(D7\) branes are part of the internal geometric configuration. The latter add exponentially suppressed Kähler moduli dependent terms to the fluxed superpotential. The possibility of partial elimination of such terms which may happen for particular choices of world volume fluxes is also taken into account. That being the case, a simple set up of three Kähler moduli is considered in the large volume regime, where only one of them is assumed to induce non-perturbative corrections. It is found that the shape of the F-term potential crucially depends on the parametric space associated with the perturbative sector and the volume modulus. De Sitter vacua can be obtained by implementing one of the standard mechanisms, i.e., either relying on D-terms related to \(U(1)\) symmetries associated with the \(D7\) branes, or introducing \(\overline{D 3}\) branes. In general it is observed that the combined effects of non-perturbative dynamics and the recently introduced logarithmic corrections lead to an effective scalar potential displaying interesting cosmological and phenomenological properties.Closed superstring moduli tree-level two-point scattering amplitudes in type IIB orientifold on \(T^6/(Z_2 \times Z_2)\)https://zbmath.org/1472.811892021-11-25T18:46:10.358925Z"Aldi, Alice"https://zbmath.org/authors/?q=ai:aldi.alice"Firrotta, Maurizio"https://zbmath.org/authors/?q=ai:firrotta.maurizioSummary: We reconsider the two-point string scattering amplitudes of massless Neveu-Schwarz-Neveu-Schwarz states of Type IIB orientifold superstring theory on the disk and projective plane in ten dimensions and analyse its \(\alpha^\prime\) expansion. We also discuss the unoriented Type IIB theory on \(T^6 / \mathbb{Z}_2 \times \mathbb{Z}_2\) where two-point string scattering amplitudes of the complex Kähler moduli and complex structures of the untwisted sector are computed on the disk and projective plane. New results are obtained together with known ones. Finally, we compare string scattering amplitudes results at \({\alpha^{\prime}}^2\)-order with the (curvature)\(^2\) terms in the low energy effective action of D-branes and \(\Omega \)-planes in both cases.