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Of all publications in the section: 12
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Article
Kaledin D. Pure and Applied Mathematics Quarterly. 2014. Vol. 10. No. 2. P. 348-354.

In this Appendix, we will try to explain the constructions of the paper in a slightly more general context of D-modules and “formal geometry” of Gelfand and Kazhdan. To save space, we only sketch the proofs, and wework in the algebraic setting (generalization to complex-analytic varieties is immediate, exactly the same arguments work).

Article
Fedor Bogomolov, De Oliveira B. Pure and Applied Mathematics Quarterly. 2013. Vol. 9. No. 4. P. 613-642.
A closed symmetric differential of the 1st kind is a differential that locally is the product of closed holomorphic 1-forms. We show that closed symmetric 2-differentials of the 1st kind on a projective manifold X come from maps of X to cyclic or dihedral quotients of Abelian varieties and that their presence implies that the fundamental group of X (case of rank 2) or of the complement X∖E of a divisor E with negative properties (case of rank 1) contains subgroup of finite index with infinite abelianization. Other results include: i) the identification of the differential operator characterizing closed symmetric 2-differentials on surfaces (using a natural connection to flat Riemannian metrics) and ii) projective manifolds X having symmetric 2-differentials w that are the product of two closed meromorphic 1-forms are irregular, in fact if w is not of the 1st kind (which can happen), then X has a fibration f:X→C over a curve of genus ≥1.
Article
Verbitsky M. Pure and Applied Mathematics Quarterly. 2008. Vol. 4. No. 3/2. P. 651-714.
Article
Campana F., Amerik E. Pure and Applied Mathematics Quarterly. 2008. No. 4(2). P. 1-37.
Article
Verbitsky M. Pure and Applied Mathematics Quarterly. 2014. Vol. 10. No. 2. P. 325-354.

Let S be a smooth rational curve on a complex manifold M. It is called ample if its normal bundle is positive: NS=⨁O(i_k),i_k<0. We assume that M is covered by smooth holomorphic deformations of S. The basic example of such a manifold is a twistor space of a hyperkähler or a 4–dimensional anti-selfdual Riemannian manifold X (not necessarily compact). We prove “a holography principle” for such a manifold: any meromorphic function defined in a neighbourhood U of S can be extended to M, and any section of a holomorphic line bundle can be extended from U to M. This is used to define the notion of a Moishezon twistor space: this is a twistor space admitting a holomorphic embedding to a Moishezon variety M′. We show that this property is local on X, and the variety M′ is unique up to birational transform. We prove that the twistor spaces of hyperkähler manifolds obtained by hyperkähler reduction of flat quaternionic-Hermitian spaces by the action of reductive Lie groups (such as Nakajima’s quiver varieties) are always Moishezon.

Article
Penkov I., Tikhomirov A. S. Pure and Applied Mathematics Quarterly. 2014. Vol. 10. No. 2. P. 289-323.

We consider ind-varieties obtained as direct limits of chains of embeddings $X_1\stackrel{\phi_1}{\hookrightarrow}\dots\stackrel{\phi_{m-1}}{\hookrightarrow} X_m\stackrel{\phi_m}{\hookrightarrow}X_{m+1}\stackrel{\phi_{m+1}}{\hookrightarrow}\dots$, where each $X_m$ is a grassmannian or an isotropic grassmannian (possibly mixing grassmannians and isotropic grassmannians), and the embeddings $\phi_m$ are linear in the sense that they induce isomorphisms of Picard groups. We prove that any such ind-variety is isomorphic to one of certain standard ind-grassmannians and that the latter are pairwise non-isomorphic ind-varieties.

Article
Kaledin D. Pure and Applied Mathematics Quarterly. 2008. No. 4:3. P. 785-875.
Article
Ebeling W., Gusein-Zade S. Pure and Applied Mathematics Quarterly. 2020. Vol. 16. No. 4. P. 1099-1113.

In the framework of constructing mirror symmetric pairs of Calabi-Yau manifolds, P.Berglund, T.Hubsch and M.Henningson considered a pair (f,G) consisting of an invertible polynomial f and a finite abelian group G of its diagonal symmetries and associated to this pair a dual pair (f~, G~). A.Takahashi suggested a generalization of this construction to pairs (f, G) where G is a non-abelian group generated by some diagonal symmetries and some permutations of variables. In a previous paper, the authors showed that some mirror symmetry phenomena appear only under a special condition on the action of the group G: a parity condition. Here we consider the orbifold Euler characteristic of the Milnor fibre of a pair (f,G). We show that, for an abelian group G, the mirror symmetry of the orbifold Euler characteristics can be derived from the corresponding result about the equivariant Euler characteristics. For non-abelian symmetry groups we show that the orbifold Euler characteristics of certain extremal orbit spaces of the group G and the dual group G~ coincide. From this we derive that the orbifold Euler characteristics of the Milnor fibres of dual periodic loop polynomials coincide up to sign.

Article
Oliveira B., Bogomolov F. A. Pure and Applied Mathematics Quarterly. 2011. Vol. 7. No. 4. P. 1085-1104.
Article
Buryak A., Janda F., Pandharipande R. Pure and Applied Mathematics Quarterly. 2015. Vol. 11. No. 4. P. 591-631.

The relations in the tautological ring of the moduli space $M_g$ of nonsingular curves conjectured by Faber-Zagier in 2000 and extended to the moduli space $\overline{M}_{g,n}$ of stable curves by Pixton in 2012 are based upon two hypergeometric series $A$ and $B$. The question of the geometric origins of these series has been solved in at least two ways (via the Frobenius structures associated to 3-spin curves and to $P^1$). The series $A$ and $B$ also appear in the study of descendent integration on the moduli spaces of open and closed curves. We survey here the various occurrences of $A$ and $B$ starting from their appearance in the asymptotic expansion of the Airy function (calculated by Stokes in the $19^{th}$ century). Several open questions are proposed.