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arXiv2000

Krichever correspondence for algebraic varieties

In 70's there was discovered a construction how to attach to some algebraic-geometric data an infinite-dimensional subspace in the space k((z)) of the Laurent power series. Now this construction is called the Krichever map. In e-print math.AG/9911097 A.N. Parshin suggested a generalization of the Krichever map for the case of algebraic surfaces from the point of view of 2-dimensional local fields. In this work we suggest a generalization of the Krichever map to the case of algebraic varieties of arbitrary dimension from the point of view of multidimensional local fields. For surfaces our construction coincides with the Parshin construction. Besides, we obtain new explicit acyclic resolutions of quasicoherent sheaves connected with multidimensional local fields.

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arXiv2002

Geometric Syzygies of Mukai Varieties and General Canonical Curves with Genus at most 8

We describe the spaces of minimal rank last syzygies for the Mukai Varieties of sectional genus 6,7 and 8. Based on this we show: 1. The first geometric syzygies of a general canonical curve of genus 6 form a non degenerate configuration of 5 lines in P^4. 2. The first geometric syzygies of a general canonical curve of genus 7 form a non degenerate, linearly normal, ruled surface of degree 84 on a spinor variety S in P^15. 3. The second geometric syzygies of a general canonical curve of genus 8 form a non degenerate configuration of 14 conics on a 2-uple embedded P^5 in P^20. This proves a natural generalization of Green's conjecture [1984], namely that the geometric syzygies should span the space of all syzygies, in these cases. We have generalized results 1 and 3 to general curves of even genus in math.AG/0108078. Result 2 is the main new result of this paper.

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arXiv2016

Birational motives, II: Triangulated birational motives

We develop birational versions of Voevodsky's triangulated categories of motives over a field, and relate them with the pure birational motives studied in arXiv:0902.4902 [math.AG]. We also get an interpretation of unramified cohomology in this framework, leading to "higher derived functors of unramified cohomology".

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arXiv2020

Airy structures and deformations of curves in surfaces

An embedded curve in a symplectic surface $Σ\subset X$ defines a smooth deformation space $\mathcal{B}$ of nearby embedded curves. A key idea of Kontsevich and Soibelman arXiv:1701.09137 [math.AG], is to equip the symplectic surface $X$ with a foliation in order to study the deformation space $\mathcal{B}$. The foliation, together with a vector space $V_Σ$ of meromorphic differentials on $Σ$, endows an embedded curve $Σ$ with the structure of the initial data of topological recursion, which defines a collection of symmetric tensors on $V_Σ$. Kontsevich and Soibelman define an Airy structure on $V_Σ$ to be a formal quadratic Lagrangian $\mathcal{L}\subset T^*(V_Σ^*)$ which leads to an alternative construction of the tensors of topological recursion. In this paper we produce a formal series $θ$ on $\mathcal{B}$ of meromorphic differentials on $Σ$ which takes it values in $\mathcal{L}$, and use this to produce the Donagi-Markman cubic from a natural cubic tensor on $V_Σ$, giving a generalisation of a result of Baraglia and Huang, arXiv:1707.04975 [math.DG].

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arXiv2008

Positive toric fibrations

A principal toric bundle $M$ is a complex manifold equipped with a free holomorphic action of a compact complex torus $T$. Such a manifold is fibered over $M/T$, with fiber $T$. We discuss the notion of positivity in fiber bundles and define positive toric bundles. Given an irreducible complex subvariety $X\subset M$ of a positive principal toric bundle, we show that either $X$ is $T$-invariant, or it lies in an orbit of $T$-action. For principal elliptic bundles, this theorem is known (math.AG/0403430). As follows from Borel-Remmert-Tits theorem, any compact simply connected homogeneous complex manifold is a principal toric bundle. We show that compact Lie groups with left-invariant complex structure $I$ are positive toric bundles, if $I$ is generic. Other examples of positive toric bundles are discussed.

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arXiv2008

On ground fields of arithmetic hyperbolic reflection groups. III

This paper continues arXiv.org:math.AG/0609256, arXiv:0708.3991 and arXiv:0710.0162 . Using authors's methods of 1980, 1981, some explicit finite sets of number fields containing all ground fields of arithmetic hyperbolic reflection groups in dimension at least 3 are defined, and explicit bounds of their degrees (over Q) are obtained. Thus, now, explicit bound of degree of ground fields of arithmetic hyperbolic reflection groups is known in all dimensions. Thus, now, we can, in principle, obtain effective finite classification of arithmetic hyperbolic reflection groups in all dimensions together.

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arXiv2007

On the boundary motive of a Shimura variety

Applying the main results of math.AG/0408295, we identify the motives occurring as cones in the co-localization and localization filtrations of the boundary motive of a Shimura variety.

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arXiv2007

Quantum cohomology of the Grassmannian and alternate Thom-Sebastiani

We introduce the notion of alternate product of Frobenius manifolds and we give, after [math.AG/0610265], an interpretation of the Frobenius manifold structure canonically attached to the quantum cohomology of G(r,n+1) in terms of alternate products. We also investigate the relationship with the alternate Thom-Sebastiani product of Laurent polynomials.

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arXiv2008

Towards an enumerative geometry of the moduli space of twisted curves and r-th roots

The enumerative geometry of r-th roots of line bundles is the subject of Witten's conjecture and occurs in the calculation of Gromov-Witten invariants of orbifolds. It requires the definition of the suitable compact moduli stack and the generalization of the standard techniques from the theory of moduli of stable curves. In math.AG/0603687, we construct a compact stack by describing the notion of stability in the context of twisted curves. In this paper, by working with stable twisted curves, we extend Mumford's formula for the Chern character of the Hodge bundle to the direct image of the universal r-th root in K-theory.

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arXiv1999

Plane curves with small linear orbits II

The `linear orbit' of a plane curve of degree d is its orbit in P^{d(d+3)/2} under the natural action of PGL(3). We classify curves with positive dimensional stabilizer, and we compute the degree of the closure of the linear orbits of curves supported on unions of lines. Together with the results of math.AG/9805020, this encompasses the enumerative geometry of all plane curves with small linear orbit. This information will serve elsewhere as an ingredient in the computation of the degree of the orbit closure of an arbitrary plane curve.

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arXiv2006

Tropical constructive Pappus' theorem

In this paper, we state the validity of Cramer's rule in a tropical context and its correspondence with classical Cramer's rule. This correspondence will allow us to prove a constructive version of Pappus' theorem conjectured in the paper First Steps in Tropical Geometry, math.AG/0306366

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arXiv2004

Green-Lazarsfeld's Gonality Conjecture for a Generic Curve of Odd Genus

The gonality conjecture predicts that the gonality of a curve can be read off Koszul cohomology of line bundles of sufficiently large degree. We verify this conjecture for generic curves of odd genus. The even-genus case was previously solved in the joint work arXiv:math.AG/0301261.

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arXiv2009

On the Hall algebra of an elliptic curve, II

This paper is a sequel to math.AG/0505148, where the Hall algebra U^+_E of the category of coherent sheaves on an elliptic curve E defined over a finite field was explicitly described, and shown to be a two-parameter deformation of the ring of diagonal invariants R^+=C[x_1^{\pm 1}, ..., y_1,...]^{S_{\infty}} (in infinitely many variables). In the present work, we study a geometric version of this Hall algebra, by considering a convolution algebra of perverse sheaves on the moduli spaces of coherent sheaves on E. This allows us to define a "canonical basis" B of U^+_E in terms of intersection cohomology complexes. We also give a characterization of this basis in terms of an involution, a lattice and a certain PBW-type basis. Finally we provide some (loose) interpretation of the above construction in terms of Schubert varieties in (nonexistent!) "double affine grassmanian", and in terms of character sheaves of loop groups (both of type A).

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arXiv2008

Zero loci of admissible normal functions with torsion singularities

We show that the zero locus of a normal function on a smooth complex algebraic variety S is algebraic provided that the normal function extends to a admissible normal function on a smooth compactification of S with torsion singularity. This result generalizes our previous result for admissible normal functions on curves [arxiv:math/0604345 [math.AG]]. It has also been obtained by M. Saito using a different method in a recent preprint [arXiv:0803.2771v2].

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arXiv2010

Poisson deformations of affine symplectic varieties II

This is a continuation of math.AG/0609741. Let Y be an affine symplectic variety with a C^*-action with positive weights, and let π: X -> Y be its crepant resolution. Then πinduces a natural map PDef(X) -> PDef(Y) of Kuranishi spaces for the Poisson deformations of X and Y. In the Part I, we proved that PDef(X) and PDef(Y) are both non-singular, and this map is a finite surjective map. In this paper (Part II), we prove that it is a Galois covering. Markman already obtained a similar result in the compact case, which was a motivation of this paper. As an application, we shall construct explicitly the universal Poisson deformation of the normalization \tilde{O} of a nilpotent orbit closure \bar{O} in a complex simple Lie algebra when \tilde{O} has a crepant resolution.

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arXiv2013

Phases of Lagrangian-invariant objects in the derived category of an abelian variety

We continue the study of Lagrangian-invariant objects (LI-objects for short) in the derived category $D^b(A)$ of coherent sheaves on an abelian variety, initiated in arXiv:1109.0527. For every element of the complexified ample cone $D_A$ we construct a natural phase function on the set of LI-objects, which in the case $\dim A=2$ gives the phases with respect to the corresponding Bridgeland stability (see math.AG/0307164). The construction is based on the relation between endofunctors of $D^b(A)$ and a certain natural central extension of groups, associated with $D_A$ viewed as a hermitian symmetric space. In the case when $A$ is a power of an elliptic curve, we show that our phase function has a natural interpretation in terms of the Fukaya category of the mirror dual abelian variety. As a byproduct of our study of LI-objects we show that the Bridgeland's component of the stability space of an abelian surface contains all full stabilities.

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arXiv2025

An $l$-adic norm residue epimorphism theorem

We show that the continuous étale cohomology groups $H^n_{\mathrm{cont}}(X,\mathbf{Z}_l(n))$ of smooth varieties $X$ over a finite field $k$ are spanned as $\mathbf{Z}_l$-modules by the $n$-th Milnor $K$-sheaf locally for the Zariski topology, for all $n\ge 0$. Here $l$ is a prime invertible in $k$. This is the first general unconditional result towards the conjectures of arXiv:math/9801017 (math.AG) which put together the Tate and the Beilinson conjectures relative to algebraic cycles on smooth projective $k$-varieties.

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arXiv2020

Generalized B-Opers

Opers were introduced by Beilinson-Drinfeld [arXiv:math.AG/0501398]. In [J. Math. Pures Appl. 82 (2003), 1-42] a higher rank analog was considered, where the successive quotients of the oper filtration are allowed to have higher rank. We dedicate this paper to introducing and studying generalized $B$-opers (where "$B$" stands for "bilinear"), obtained by endowing the underlying vector bundle with a bilinear form which is compatible with both the filtration and the connection. In particular, we study the structure of these $B$-opers, by considering the relationship of these structures with jet bundles and with geometric structures on a Riemann surface.

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