arXiv subjects
Publications and source records attributed to Ziyu Zhang.
Let M be a projective fine moduli space of stable sheaves on a smooth projective variety X with a universal family E. We prove that in four examples, E can be realized as a complete flat family of stable sheaves on M parametrized by X, which identifies X with a smooth connected component of some moduli space of stable sheaves on M.
Given a strict simple degeneration $f \colon X\to C$ the first three authors previously constructed a degeneration $I^n_{X/C} \to C$ of the relative degree $n$ Hilbert scheme of $0$-dimensional subschemes. In this paper we investigate the geometry of this degeneration, in particular when the fibre dimension of $f$ is at most $2$. In this case we show that $I^n_{X/C} \to C$ is a dlt model. This is even a good minimal dlt model if $f \colon X \to C$ has this property. We compute the dual complex of the central fibre $(I^n_{X/C})_0$ and relate this to the essential skeleton of the generic fibre. For a type II degeneration of $K3$ surfaces we show that the stack ${\mathcal I}^n_{X/C} \to C$ carries a nowhere degenerate relative logarithmic $2$-form. Finally we discuss the relationship of our degeneration with the constructions of Nagai.
Point clouds-based Networks have achieved great attention in 3D object classification, segmentation and indoor scene semantic parsing. In terms of face recognition, 3D face recognition method which directly consume point clouds as input is still under study. Two main factors account for this: One is how to get discriminative face representations from 3D point clouds using deep network; the other is the lack of large 3D training dataset. To address these problems, a data-free 3D face recognition method is proposed only using synthesized unreal data from statistical 3D Morphable Model to train a deep point cloud network. To ease the inconsistent distribution between model data and real faces, different point sampling methods are used in train and test phase. In this paper, we propose a curvature-aware point sampling(CPS) strategy replacing the original furthest point sampling(FPS) to hierarchically down-sample feature-sensitive points which are crucial to pass and aggregate features deeply. A PointNet++ like Network is used to extract face features directly from point clouds. The experimental results show that the network trained on generated data generalizes well for real 3D faces. Fine tuning on a small part of FRGCv2.0 and Bosphorus, which include real faces in different poses and expressions, further improves recognition accuracy.
We construct nontrivial L-equivalence between curves of genus one and degree five, and between elliptic surfaces of multisection index five. These results give the first examples of L-equivalence for curves (necessarily over non-algebraically closed fields) and provide a new bit of evidence for the conjectural relationship between L-equivalence and derived equivalence. The proof of the L-equivalence for curves is based on Kuznetsov's Homological Projective Duality for Gr(2,5), and L-equivalence is extended from genus one curves to elliptic surfaces using the Ogg--Shafarevich theory of twisting for elliptic surfaces. Finally, we apply our results to K3 surfaces and investigate when the two elliptic L-equivalent K3 surfaces we construct are isomorphic, using Neron--Severi lattices, moduli spaces of sheaves and derived equivalence. The most interesting case is that of elliptic K3 surfaces of polarization degree ten and multisection index five, where the resulting L-equivalence is new.
We give a proof of the formality conjecture of Kaledin and Lehn: on a complex projective K3 surface, the DG algebra RHom(F,F) is formal for any sheaf F polystable with respect to an ample line bundle. Our main tool is the uniqueness of DG enhancement of the bounded derived category of coherent sheaves. We also extend the formality result to derived objects that are polystable with respect to a generic Bridgeland stability condition.
Generative Adversarial Nets (GANs) are very successful at modeling distributions from given samples, even in the high-dimensional case. However, their formulation is also known to be hard to optimize and often not stable. While this is particularly true for early GAN formulations, there has been significant empirically motivated and theoretically founded progress to improve stability, for instance, by using the Wasserstein distance rather than the Jenson-Shannon divergence. Here, we consider an alternative formulation for generative modeling based on random projections which, in its simplest form, results in a single objective rather than a saddle-point formulation. By augmenting this approach with a discriminator we improve its accuracy. We found our approach to be significantly more stable compared to even the improved Wasserstein GAN. Further, unlike the traditional GAN loss, the loss formulated in our method is a good measure of the actual distance between the distributions and, for the first time for GAN training, we are able to show estimates for the same.
We study duality spectral sequences for Weierstrass fibrations. Using these spectral sequences, we show that on a K-trivial Weierstrass threefold over a K-numerically trivial surface, any line bundle of nonzero fiber degree is taken by a Fourier-Mukai transform to a slope stable locally free sheaf.
Generating diverse questions for given images is an important task for computational education, entertainment and AI assistants. Different from many conventional prediction techniques is the need for algorithms to generate a diverse set of plausible questions, which we refer to as "creativity". In this paper we propose a creative algorithm for visual question generation which combines the advantages of variational autoencoders with long short-term memory networks. We demonstrate that our framework is able to generate a large set of varying questions given a single input image.
Following Bayer and Macrì, we study the birational geometry of singular moduli spaces $M$ of sheaves on a K3 surface $X$ which admit symplectic resolutions. More precisely, we use the Bayer-Macrì map from the space of Bridgeland stability conditions $\mathrm{Stab}(X)$ to the cone of movable divisors on $M$ to relate wall-crossing in $\mathrm{Stab}(X)$ to birational transformations of $M$. We give a complete classification of walls in $\mathrm{Stab}(X)$ and show that every birational model of $M$ obtained by performing a finite sequence of flops from $M$ appears as a moduli space of Bridgeland semistable objects on $X$. An essential ingredient of our proof is an isometry between the orthogonal complement of a Mukai vector inside the algebraic Mukai lattice of $X$ and the Néron-Severi lattice of $M$ which generalises results of Yoshioka, as well as Perego and Rapagnetta. Moreover, this allows us to conclude that the symplectic resolution of $M$ is deformation equivalent to the 10-dimensional irreducible holomorphic symplectic manifold found by O'Grady.
In [BM14b], the first author and Macrì constructed a family of nef divisors on any moduli space of Bridgeland-stable objects on a smooth projective variety X. In this article, we extend this construction to the setting of any separated scheme Y of finite type over a field, where we consider moduli spaces of Bridgeland-stable objects on Y with compact support. We also show that the nef divisor is compatible with the polarising ample line bundle coming from the GIT construction of the moduli space in the special case when Y admits a tilting bundle and the stability condition arises from a θ-stability condition for the endomorphism algebra. Our main tool generalises the work of Abramovich--Polishchuk [AP06] and Polishchuk [Pol07]: given a t-structure on the derived category D_c(Y) on Y of objects with compact support and a base scheme S, we construct a constant family of t-structures on a category of objects on YxS with compact support relative to S.
Our aim is to provide a pixel-wise instance-level labeling of a monocular image in the context of autonomous driving. We build on recent work [Zhang et al., ICCV15] that trained a convolutional neural net to predict instance labeling in local image patches, extracted exhaustively in a stride from an image. A simple Markov random field model using several heuristics was then proposed in [Zhang et al., ICCV15] to derive a globally consistent instance labeling of the image. In this paper, we formulate the global labeling problem with a novel densely connected Markov random field and show how to encode various intuitive potentials in a way that is amenable to efficient mean field inference [Krähenbühl et al., NIPS11]. Our potentials encode the compatibility between the global labeling and the patch-level predictions, contrast-sensitive smoothness as well as the fact that separate regions form different instances. Our experiments on the challenging KITTI benchmark [Geiger et al., CVPR12] demonstrate that our method achieves a significant performance boost over the baseline [Zhang et al., ICCV15].
In this paper we tackle the problem of instance-level segmentation and depth ordering from a single monocular image. Towards this goal, we take advantage of convolutional neural nets and train them to directly predict instance-level segmentations where the instance ID encodes the depth ordering within image patches. To provide a coherent single explanation of an image we develop a Markov random field which takes as input the predictions of convolutional neural nets applied at overlapping patches of different resolutions, as well as the output of a connected component algorithm. It aims to predict accurate instance-level segmentation and depth ordering. We demonstrate the effectiveness of our approach on the challenging KITTI benchmark and show good performance on both tasks.
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 $Φ$ on $D^b(X)$, where $Φ$ 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 $Φ$ 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 $μ_\ast$-semistability defined by a `slope-like' function $μ_\ast$.
We give an explicit proof of a Bogomolov-type inequality for $c_3$ of reflexive sheaves on $\mathbb{P}^3$. Then, using resolutions of rank-two reflexive sheaves on $\mathbb{P}^3$, we prove that some strata of the moduli of rank-two complexes that are both PT-stable and dual-PT-stable are quotient stacks. Using monads, we apply the same techniques to $\mathbb{P}^2$ and show that some strata of the moduli of Bridgeland-stable complexes are quotient stacks.
We view the moduli space of semistable sheaves on a K3 surface as a global quotient stack, and compute its cotangent complex in terms of the universal sheaf on the Quot scheme. Relevant facts on the classical and reduced Atiyah classes are reviewed. We also define the notion of a symplectic stack, and show that it includes all moduli stacks of semistable sheaves on K3 surfaces.
This paper studies deformations and birational maps between singular moduli spaces of semistable sheaves with 2-divisible Mukai vectors on K3 surfaces. It is showed that under certain conditions, two such moduli spaces of the same dimension can be connected by deformations and birational maps.
This paper studies formality of the differential graded algebra $RHom(E,E)$, where $E$ is a semistable sheaf on a K3 surface. The main tool is Kaledin's theorem on formality in families. For a large class of sheaves $E$, this DG algebra is formal, therefore we have an explicit description of the singularity type of the moduli space of semistable sheaves at the point represented by $E$. This paper also explains why Kaledin's theorem fails to apply in the remaining case.