arXiv subjects

Igor Reider

Publications and source records attributed to Igor Reider.

9 recordsLinked to original sources

arXiv2023

Refinement of the Infinitesimal Variation of Hodge Structure: the case of canonical curves

Let $C$ be a smooth complex projective curve with canonical divisor $K_C$ very ample. We explore the relation between the cup-product $$ H^1 (\Theta_C ) \longrightarrow (H^0({\cal O}_C (K_C))^{\ast} \otimes H^1 ({\cal O}_C) $$ where $\Theta_C ={\cal O}_C (-K_C)$ is the holomorphic tangent bundle of $C$, and the geometry of the canonical embedding of $C$. The cup-product, following Griffiths, stratifies ${\mathbb P}(H^1 (\Theta_C ))$ by the subvarieties $\Sigma_r$, according to the rank $r$ of $\xi \in H^1 (\Theta_C )$ viewed as the linear map $$ \xi:H^0({\cal O}_C (K_C)) \longrightarrow H^1 ({\cal O}_C) $$ or, equivalently, by the dimension of the kernel of $\xi$ $$ W_{\xi}=ker(\xi). $$ The refinement consists of the filtration of $W^{\bullet}_{\xi} ([\phi])$ of $W_{\xi}$, varying with $[\phi] \in {\mathbb P}(W_{\xi})$. This filtration has geometric meaning: 1) it is related to special divisors on $C$, 2) it `counts' certain rational normal curves in the canonical embedding of $C$. As an illustration, the results about the strata $\Sigma_0$ and $\Sigma_1$ are recovered and as corollaries one obtains the classical theorems of Max Noether on projective normality of the canonical embedding and Babbage-Enriques-Petri about the canonical curve being cut out by quadrics. The refinement brings out new aspects: quiver representations, Fano toric varieties with a distinguished anti-canonical divisor, dimer models. The quiver emerges from the construction and properties of the refinement; the Fano variety arises from the graph underlying the quiver and related to the Higgs structures. The graph underlying the refinement becomes an important part of the theory: it connects to topics such as the Topological Quantum field theory, moduli of elliptic curves with marked points, modular curves, higher categorical structures.

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arXiv2021

Bridgeland stability conditions and the tangent bundle of surfaces of general type

Let $X$ be a smooth compact complex surface with the canonical divisor $K_X$ ample and let $\Theta_X$ be its holomorphic tangent bundle. Bridgeland stability conditions are used to study the space $H^1 (\Theta_X)$ of infinitesimal deformations of complex structures of $X$ and its relation to the geometry/topology of $X$. The main observation is that for $X$ with $H^1 (\Theta_X)$ nonzero and the Chern numbers $(c_2 (X), K^2_X)$ subject to $$ \tau_X :=2ch_2 (\Theta_X)=K^2_X -2c_2(X) >0 $$ the object $\Theta_X [1]$ of the derived category of bounded complexes of coherent sheaves on $X$ is Bridgeland unstable in a certain part of the space of Bridgeland stability conditions. The Harder-Narasimhan filtrations of $\Theta_X [1]$ for those stability conditions are expected to provide new insights into geometry of surfaces of general type and the study of their moduli. The paper provides a certain body of evidence that this is indeed the case.

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arXiv2018

On the Infinitesimal Torelli theorem for regular surfaces with very ample canonical divisor

Let $X$ be a smooth compact complex surface subject to the following conditions: (i) the canonical line bundle $\mathcal{O}_X(K_X) $ is very ample, (ii) the irregularity $q(X): = h^1(\mathcal{O}_X) =0$, (iii) $X$ contains no rational normal curves of degree $\leq (p_g-1)$, (iv) the multiplication map $m_2: Sym^2(H^0(\mathcal{O}_X(K_X))) \longrightarrow H^0 (\mathcal{O}_X (2K_X))$ is surjective. It is shown that the Infinitesimal Torelli holds for such $X$. Our proof is based on the study of the cup-product $$ H^1 (\Theta_X) \longrightarrow (\mathcal{O}_X(K_X))^{\ast} \otimes H^1 (\Omega_X) $$ where $\Theta_X$ (resp. $\Omega_X$) is the holomorphic tangent (resp. cotangent) bundle of $X$. Conceptually, the approach consists of lifting the data of the cohomological cup-product above to the category of complexes of coherent sheaves of $X$. This establishes connections between the geometry of the canonical map and the above cup-product by exhibiting geometrically meaningful objects in the category of (short) exact complexes of coherent sheaves on $X$.

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arXiv2016

Surfaces in $\mathbb{P}^4$ lying on small degree hypersurfaces

Since the work of Ellingsrud and Peskine at the end of 1980s, it has been known that, with the exception of a finite number of families, smooth compact complex surfaces in $\mathbb{P}^4$ with prescribed Chern classes must lie on hypersurfaces of degree $m\leq 5$. The study of surfaces lying on a small degree hypersurface in $\mathbb{P}^4$---small meaning $\leq5$---seems to be a way of obtaining empirical data leading to a better conceptual understanding of surfaces in $\mathbb{P}^4$. From this perspective, two main issues are considered in the paper: - an analogue of the Hartshorne-Lichtenbaum finiteness results for smooth surfaces of general type contained in a small degree hypersurface in $\mathbb{P}^4$, - a study of the irregularity of smooth surfaces contained in a small degree hypersurface in $\mathbb{P}^4$.

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arXiv2014

Infinitesimal Torelli Theorem for regular surfaces with very ample canonical divisor

The article proves the Infinitesimal Torelli theorem for surfaces subject to the following conditions: 1) the canonical bundle of a surface is ample and generated by its global sections, 2)the geometric genus $p_g \geq 4$, 3) the irregularity $q=0$ . The main novelty is a realization of the Kodaira-Spencer classes lying in the kernel of the cohomology cup-product controlling the derivative of the period map of weight 2 in the category of the coherent sheaves of a surface.

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arXiv2011

Twisted Kodaira-Spencer classes and the geometry of surfaces of general type

We study the cohomology groups $H^1(X,Θ_X(-mK_X))$, for $m\geq1$, where $X$ is a smooth minimal complex surface of general type, $Θ_X$ its holomorphic tangent bundle, and $K_X$ its canonical divisor. One of the main results is a precise vanishing criterion for $H^1(X,Θ_X (-K_X))$. The proof is based on the geometric interpretation of non-zero cohomology classes of $H^1(X,Θ_X (-K_X))$. This interpretation in turn uses higher rank vector bundles on $X$. We apply our methods to the long standing conjecture saying that the irregularity of surfaces in $\PP^4$ is at most 2. We show that if $X$ has prescribed Chern numbers, no irrational pencil, and is embedded in $\PP^4$ with a sufficiently large degree, then the irregularity of $X$ is at most 3.

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arXiv2011

Nonabelian Jacobian of Smooth Projective Surfaces - A Survey

The nonabelian Jacobian $\JA$ of a smooth projective surface $X$ is inspired by the classical theory of Jacobian of curves. It is built as a natural scheme interpolating between the Hilbert scheme $\XD$ of subschemes of length $d$ of $X$ and the stack ${\bf M}_X (2,L,d)$ of torsion free sheaves of rank 2 on $X$ having the determinant $\OO_X (L)$ and the second Chern class (= number) $d$. It relates to such influential ideas as variations of Hodge structures, period maps, nonabelian Hodge theory, Homological mirror symmetry, perverse sheave, geometric Langlands program. These relations manifest themselves by the appearance of the following structures on $\JA$: 1) a sheaf of reductive Lie algebras, 2) (singular) Fano toric varieties whose hyperplane sections are (singular) Calabi-Yau varieties, 3) trivalent graphs. This is an expository paper giving an account of most of the main properties of $\JA$ uncovered in [R1] and [R2].

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arXiv2011

Configuration of points and strings

Let $X$ be a smooth projective variety of dimension $n\geq 2$. It is shown that a finite configuration of points on $X$ subject to certain geometric conditions possesses rich inner structure. On the mathematical level this inner structure is a variation of Hodge-like structure. As a consequence one can attach to such point configurations: (i) Lie algebras and their representations (ii) Fano toric variety whose hyperplane sections are Calabi-Yau varieties. These features lead to a picture which is very suggestive of quantum gravity according to string theory.

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arXiv2011

Nonabelian Jacobian of smooth projective surfaces and representation theory

The paper studies representation theoretic aspects of a nonabelian version of the Jacobian for a smooth complex projective surface $X$ introduced in [R1]. The sheaf of reductive Lie algebras $\bf\calG$ associated to the nonabelian Jacobian is determined and its Lie algebraic properties are explicitly related to the geometry of configurations of points on $X$. In particular, it is shown that the subsheaf of centers of $\bf\calG$ determines a distinguished decomposition of configurations into the disjoint union of subconfigurations. Furthermore, it is shown how to use $sl_2$-subalgebras associated to certain nilpotent elements of $\bf\calG$ to write equations defining configurations of $X$ in appropriate projective spaces. The same nilpotent elements are used to establish a relation of the nonabelian Jacobian with such fundamental objects in the representation theory as nilpotent orbits, Springer resolution and Springer fibres of simple Lie algebras of type $A_n$, for appropriate values of $n$. This leads to a construction of distinguished collections of objects in the category of representations of symmetric groups as well as in the category of perverse sheaves on the appropriate Hilbert schemes of points of $X$. Hence two ways of categorifying the second Chern class of vector bundles of rank 2 on smooth projective surfaces. We also give a `loop' version of the above construction by relating the nonabelian Jacobian to the Infinite Grassmannians of simple Lie groups of type $SL_n(\bf C)$, for appropriate values of n. This gives, via the geometric version of the Satake isomorphism, a distinguished collection of irreducible representations of the Langlands dual groups thus indicating a relation of the nonabelian Jacobian to the Langlands duality for smooth projective surfaces.

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