*-Quantizations of Fourier–Mukai Transforms

We study deformations of Fourier–Mukai transforms in general complex analytic settings. Suppose X and Y are complex manifolds, and let P be a coherent sheaf on X ×  Y. Suppose that the Fourier–Mukai transform ${\Phi}$ Φ given by the kernel P is an equivalence between the coherent derived categories of X and of Y. Suppose also that we are given a formal *-quantization ${\mathbb{X}}$ X of X. Our main result is that ${\mathbb{X}}$ X gives rise to a unique formal *-quantization ${\mathbb{Y}}$ Y of Y. For the statement to hold, *-quantizations must be understood in the framework of stacks of algebroids. The quantization ${\mathbb{Y}}$ Y is uniquely determined by the condition that ${\Phi}$ Φ deforms to an equivalence between the derived categories of ${\mathbb{X}}$ X and ${\mathbb{Y}}$ Y . Equivalently, the condition is that P deforms to a coherent sheaf ${\tilde{P}}$ P ~ on the formal *-quantization ${\mathbb{X} \times\mathbb{Y}^{op}}$ X × Y o p of X × Y; here ${\mathbb{Y}^{op}}$ Y o p is the opposite of the quantization ${\mathbb{Y}}$ Y .     

Preservation of semistability under Fourier–Mukai transforms

For a trivial elliptic fibration \(X=C \times S\) with C an elliptic curve and S a projective K3 surface of Picard rank 1, we study how various notions of stability behave under the Fourier–Mukai autoequivalence \(\Phi \) on \(D^b(X)\), where \(\Phi \) is induced by the classical Fourier–Mukai autoequivalence on \(D^b(C)\). We show that, under some restrictions on Chern classes, Gieseker semistability on coherent sheaves is preserved under \(\Phi \) when the polarisation is ‘fiber-like’. Moreover, for more general choices of Chern classes, Gieseker semistability under a ‘fiber-like’ polarisation corresponds to a notion of \(\mu _*\)-semistability defined by a ‘slope-like’ function \(\mu _*\).     

Fourier–Mukai transform on abelian surfaces

We study moduli spaces of stable sheaves on abelian surfaces whose Mukai vectors are related by a cohomological Fourier-Mukai transform. We show that there is a Fourier-Mukai transform inducing a birational map between them.     

Fourier–Mukai transform in the quantized setting

We prove that a coherent DQ-kernel induces an equivalence between the derived categories of DQ-modules with coherent cohomology if and only if the graded commutative kernel associated to it induces an equivalence between the derived categories of coherent sheaves.     

Moduli of sheaves,Fourier–Mukai transform,and partial desingularization

We study birational maps among (1) the moduli space of semistable sheaves of Hilbert polynomial \(4m+2\) on a smooth quadric surface, (2) the moduli space of semistable sheaves of Hilbert polynomial \(m^{2}+3m+2\) on \(\mathbb {P}^{3}\), (3) Kontsevich’s moduli space of genus-zero stable maps of degree 2 to the Grassmannian Gr(2, 4). A regular birational morphism from (1) to (2) is described in terms of Fourier–Mukai transforms. The map from (3) to (2) is Kirwan’s partial desingularization. We also investigate several geometric properties of 1) by using the variation of moduli spaces of stable pairs.     

Stability and Fourier–Mukai transforms on elliptic fibrations

We systematically develop Bridgeland's [7] and Bridgeland–Maciocia's [10] techniques for studying elliptic fibrations, and identify criteria that ensure 2-term complexes are mapped to torsion-free sheaves under a Fourier–Mukai transform. As an application, we construct an open immersion from a moduli of stable complexes to a moduli of Gieseker stable sheaves on elliptic threefolds. As another application, we give various 1–1 correspondences between fibrewise semistable torsion-free sheaves and codimension-1 sheaves on Weierstrass surfaces.     

A Fourier–Mukai approach to the K-theory of compact Lie groups

Let G be a compact, connected, simply-connected Lie group. We use the Fourier–Mukai transform in twisted K-theory to give a new proof of the ring structure of the K-theory of G.     

Decompositions of Gelfand–Shilov kernels into kernels of similar class

We prove that any linear operator with a kernel in a Gelfand–Shilov space is a composition of two operators with kernels in the same Gelfand–Shilov space. We also give links on numerical approximations for such compositions. We apply these composition rules to establish Schatten–von Neumann properties for such operators.     

Lefschetz fixed point theorems for Fourier–Mukai functors and DG algebras

We propose some variants of Lefschetz fixed point theorem for Fourier–Mukai functors on a smooth projective algebraic variety. Independently we also suggest a similar theorem for endo-functors on the category of perfect modules over a smooth and proper DG algebra.     

DG-resolutions of NC-smooth thickenings and NC-Fourier–Mukai transforms

We give a construction of NC-smooth thickenings [a notion defined by Kapranov (J Reine Angew Math 505:73–118, 1998)] of a smooth variety equipped with a torsion free connection. We show that a twisted version of this construction realizes all NC-smooth thickenings as \(0\) th cohomology of a differential graded sheaf of algebras, similarly to Fedosov’s construction in (J Differ Geom 40:213–238, 1994). We use this dg resolution to construct and study sheaves on NC-smooth thickenings. In particular, we construct an NC version of the Fourier–Mukai transform from coherent sheaves on a (commutative) curve to perfect complexes on the canonical NC-smooth thickening of its Jacobian. We also define and study analytic NC-manifolds. We prove NC-versions of some of GAGA theorems, and give a \(C^\infty \) -construction of analytic NC-thickenings that can be used in particular for Kähler manifolds with constant holomorphic sectional curvature. Finally, we describe an analytic NC-thickening of the Poincaré line bundle for the Jacobian of a curve, and the corresponding Fourier–Mukai functor, in terms of \(A_\infty \) -structures.     

Semistability of Lazarsfeld–Mukai bundles via parabolic structures

Our aim in this article is to produce new examples of semistable Lazarsfeld–Mukai bundles on smooth projective surfaces X using the notion of parabolic vector bundles. In particular, we associate natural parabolic structures to any rank two (dual) Lazarsfeld–Mukai bundle and study the parabolic stability of these parabolic bundles. We also show that the orbifold bundles on Kawamata coverings of X corresponding to the above parabolic bundles are themselves certain (dual) Lazarsfeld–Mukai bundles. This gives semistable Lazarsfeld–Mukai bundles on Kawamata covers of the projective plane and of certain K3 surfaces.     

On the divergence of double Fourier–Walsh and Fourier–Walsh–Kaczmarz series of continuous functions

Acta Mathematica Hungarica - We prove that there exists a continuous function on $$[0,1]^2$$ , with a certain smoothness, whose double Fourier–Walsh series diverges on a set of positive...     

Oscillation of holomorphic Bergman–Besov kernels on the ball

We estimate the oscillation of holomorphic Bergman–Besov reproducing kernels on the unit ball of \(\mathbb {C}^n\). As an application of this estimate we characterize holomorphic Bergman–Besov spaces \(A_\alpha ^p\,(\alpha \in \mathbb {R})\) in terms of double integrals of the fractions \(|f(z)-f(w)|/|z-w|\) and \(|f(z)-f(w)|/|1-\langle z,w \rangle |\) and complete the earlier works done on this subject. Our results provide, when \(\alpha \le -1\), a derivative-free characterization of \(A_\alpha ^p\).     

Universality properties of Walsh–Fourier series

It is shown that Walsh–Fourier series of \(W\) -continuous functions can have maximal sets of limit functions on small subsets of the unit interval.     

Unitarity of Generalized Fourier–Gauss Transforms

Abstract

A generalized Fourier–Gauss transform is an operator acting in a Boson Fock space and is formulated as a continuous linear operator acting on the space of test white noise functions. It does not admit, in general, a unitary extension with respect to the norm of the Boson Fock space induced from the Gaussian measure with variance 1 but is extended to a unitary isomorphism if the Gaussian measure is replaced with the ones with different covariance operators. As an application, unitarity of a generalized dilation is discussed.     

Non-trivial elements in the Abel–Jacobi kernels of higher-dimensional varieties

The purpose of this paper is to construct non-trivial elements in the Abel–Jacobi kernels in any codimension by specializing correspondences with non-trivial Hodge-theoretical invariants at points with different transcendence degrees over a subfield in C$\mathbb{C}$.     

Fourier–Jacobi periods of classical Saito–Kurokawa lifts

Kohnen–Skoruppa (Invent Math 95(3): 541–558, 1989) proved a formula for the ratio of the Petersson inner products of the half integral weight modular form and its Saito–Kurokawa lifting. We give an interpretation of this formula in the framework of the refined Gan–Gross–Prasad conjecture. This provides us with an example of the refined Gan–Gross–Prasad conjecture for the nontempered representations.     

Volatility forecasting via SVR–GARCH with mixture of Gaussian kernels

The support vector regression (SVR) is a supervised machine learning technique that has been successfully employed to forecast financial volatility. As the SVR is a kernel-based technique, the choice of the kernel has a great impact on its forecasting accuracy. Empirical results show that SVRs with hybrid kernels tend to beat single-kernel models in terms of forecasting accuracy. Nevertheless, no application of hybrid kernel SVR to financial volatility forecasting has been performed in previous researches. Given that the empirical evidence shows that the stock market oscillates between several possible regimes, in which the overall distribution of returns it is a mixture of normals, we attempt to find the optimal number of mixture of Gaussian kernels that improve the one-period-ahead volatility forecasting of SVR based on GARCH(1,1). The forecast performance of a mixture of one, two, three and four Gaussian kernels are evaluated on the daily returns of Nikkei and Ibovespa indexes and compared with SVR–GARCH with Morlet wavelet kernel, standard GARCH, Glosten–Jagannathan–Runkle (GJR) and nonlinear EGARCH models with normal, student-t, skew-student-t and generalized error distribution (GED) innovations by using mean absolute error (MAE), root mean squared error (RMSE) and robust Diebold–Mariano test. The results of the out-of-sample forecasts suggest that the SVR–GARCH with a mixture of Gaussian kernels can improve the volatility forecasts and capture the regime-switching behavior.     

On the semistability of certain Lazarsfeld–Mukai bundles on abelian surfaces

Let \(X=\mathscr {J}(\widetilde{\mathscr {C}})\), the Jacobian of a genus 2 curve \(\widetilde{\mathscr {C}}\) over \({\mathbb {C}}\), and let Y be the associated Kummer surface. Consider an ample line bundle \(L=\mathscr {O}(m\widetilde{\mathscr {C}})\) on X for an even number m, and its descent to Y, say \(L'\). We show that any dominating component of \({\mathscr {W}}^1_{d}(|L'|)\) corresponds to \(\mu _{L'}\)-stable Lazarsfeld–Mukai bundles on Y. Further, for a smooth curve \(C\in |L|\) and a base-point free \(g^1_d\) on C, say (A, V), we study the \(\mu _L\)-semistability of the rank-2 Lazarsfeld–Mukai bundle associated to (C, (A, V)) on X. Under certain assumptions on C and the \(g^1_d\), we show that the above Lazarsfeld–Mukai bundles are \(\mu _L\)-semistable.