skip to main content

Title: Hermitian–Yang–Mills Connections on Collapsing Elliptically Fibered K3 Surfaces

Let$$X\rightarrow {{\mathbb {P}}}^1$$XP1be an elliptically fiberedK3 surface, admitting a sequence$$\omega _{i}$$ωiof Ricci-flat metrics collapsing the fibers. LetVbe a holomorphicSU(n) bundle overX, stable with respect to$$\omega _i$$ωi. Given the corresponding sequence$$\Xi _i$$Ξiof Hermitian–Yang–Mills connections onV, we prove that, ifEis a generic fiber, the restricted sequence$$\Xi _i|_{E}$$Ξi|Econverges to a flat connection$$A_0$$A0. Furthermore, if the restriction$$V|_E$$V|Eis of the form$$\oplus _{j=1}^n{\mathcal {O}}_E(q_j-0)$$j=1nOE(qj-0)forndistinct points$$q_j\in E$$qjE, then these points uniquely determine$$A_0$$A0.

more » « less
Author(s) / Creator(s):
Publisher / Repository:
Springer Science + Business Media
Date Published:
Journal Name:
The Journal of Geometric Analysis
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Abstract

    In the (special) smoothing spline problem one considers a variational problem with a quadratic data fidelity penalty and Laplacian regularization. Higher order regularity can be obtained via replacing the Laplacian regulariser with a poly-Laplacian regulariser. The methodology is readily adapted to graphs and here we consider graph poly-Laplacian regularization in a fully supervised, non-parametric, noise corrupted, regression problem. In particular, given a dataset$$\{x_i\}_{i=1}^n$${xi}i=1nand a set of noisy labels$$\{y_i\}_{i=1}^n\subset \mathbb {R}$${yi}i=1nRwe let$$u_n{:}\{x_i\}_{i=1}^n\rightarrow \mathbb {R}$$un:{xi}i=1nRbe the minimizer of an energy which consists of a data fidelity term and an appropriately scaled graph poly-Laplacian term. When$$y_i = g(x_i)+\xi _i$$yi=g(xi)+ξi, for iid noise$$\xi _i$$ξi, and using the geometric random graph, we identify (with high probability) the rate of convergence of$$u_n$$untogin the large data limit$$n\rightarrow \infty $$n. Furthermore, our rate is close to the known rate of convergence in the usual smoothing spline model.

    more » « less
  2. A<sc>bstract</sc>

    Measurements of the production cross sections of prompt D0, D+, D*+,$$ {\textrm{D}}_{\textrm{s}}^{+} $$Ds+,$$ {\Lambda}_{\textrm{c}}^{+} $$Λc+, and$$ {\Xi}_{\textrm{c}}^{+} $$Ξc+charm hadrons at midrapidity in proton-proton collisions at$$ \sqrt{s} $$s= 13 TeV with the ALICE detector are presented. The D-meson cross sections as a function of transverse momentum (pT) are provided with improved precision and granularity. The ratios ofpT-differential meson production cross sections based on this publication and on measurements at different rapidity and collision energy provide a constraint on gluon parton distribution functions at low values of Bjorken-x(105–104). The measurements of$$ {\Lambda}_{\textrm{c}}^{+} $$Λc+($$ {\Xi}_{\textrm{c}}^{+} $$Ξc+) baryon production extend the measuredpTintervals down topT= 0(3) GeV/c. These measurements are used to determine the charm-quark fragmentation fractions and the$$ \textrm{c}\overline{\textrm{c}} $$cc¯production cross section at midrapidity (|y|<0.5) based on the sum of the cross sections of the weakly-decaying ground-state charm hadrons D0, D+,$$ {\textrm{D}}_{\textrm{s}}^{+} $$Ds+,$$ {\Lambda}_{\textrm{c}}^{+} $$Λc+,$$ {\Xi}_{\textrm{c}}^0 $$Ξc0and, for the first time,$$ {\Xi}_{\textrm{c}}^{+} $$Ξc+, and of the strongly-decaying Jmesons. The first measurements of$$ {\Xi}_{\textrm{c}}^{+} $$Ξc+and$$ {\Sigma}_{\textrm{c}}^{0,++} $$Σc0,++fragmentation fractions at midrapidity are also reported. A significantly larger fraction of charm quarks hadronising to baryons is found compared to e+eand ep collisions. The$$ \textrm{c}\overline{\textrm{c}} $$cc¯production cross section at midrapidity is found to be at the upper bound of state-of-the-art perturbative QCD calculations.

    more » « less
  3. Abstract

    Let us fix a primepand a homogeneous system ofmlinear equations$$a_{j,1}x_1+\dots +a_{j,k}x_k=0$$aj,1x1++aj,kxk=0for$$j=1,\dots ,m$$j=1,,mwith coefficients$$a_{j,i}\in \mathbb {F}_p$$aj,iFp. Suppose that$$k\ge 3m$$k3m, that$$a_{j,1}+\dots +a_{j,k}=0$$aj,1++aj,k=0for$$j=1,\dots ,m$$j=1,,mand that every$$m\times m$$m×mminor of the$$m\times k$$m×kmatrix$$(a_{j,i})_{j,i}$$(aj,i)j,iis non-singular. Then we prove that for any (large)n, any subset$$A\subseteq \mathbb {F}_p^n$$AFpnof size$$|A|> C\cdot \Gamma ^n$$|A|>C·Γncontains a solution$$(x_1,\dots ,x_k)\in A^k$$(x1,,xk)Akto the given system of equations such that the vectors$$x_1,\dots ,x_k\in A$$x1,,xkAare all distinct. Here,Cand$$\Gamma $$Γare constants only depending onp,mandksuch that$$\Gamma Γ<p. The crucial point here is the condition for the vectors$$x_1,\dots ,x_k$$x1,,xkin the solution$$(x_1,\dots ,x_k)\in A^k$$(x1,,xk)Akto be distinct. If we relax this condition and only demand that$$x_1,\dots ,x_k$$x1,,xkare not all equal, then the statement would follow easily from Tao’s slice rank polynomial method. However, handling the distinctness condition is much harder, and requires a new approach. While all previous combinatorial applications of the slice rank polynomial method have relied on the slice rank of diagonal tensors, we use a slice rank argument for a non-diagonal tensor in combination with combinatorial and probabilistic arguments.

    more » « less
  4. Abstract

    Approximate integer programming is the following: For a given convex body$$K \subseteq {\mathbb {R}}^n$$KRn, either determine whether$$K \cap {\mathbb {Z}}^n$$KZnis empty, or find an integer point in the convex body$$2\cdot (K - c) +c$$2·(K-c)+cwhich isK, scaled by 2 from its center of gravityc. Approximate integer programming can be solved in time$$2^{O(n)}$$2O(n)while the fastest known methods for exact integer programming run in time$$2^{O(n)} \cdot n^n$$2O(n)·nn. So far, there are no efficient methods for integer programming known that are based on approximate integer programming. Our main contribution are two such methods, each yielding novel complexity results. First, we show that an integer point$$x^* \in (K \cap {\mathbb {Z}}^n)$$x(KZn)can be found in time$$2^{O(n)}$$2O(n), provided that theremaindersof each component$$x_i^* \mod \ell $$ximodfor some arbitrarily fixed$$\ell \ge 5(n+1)$$5(n+1)of$$x^*$$xare given. The algorithm is based on acutting-plane technique, iteratively halving the volume of the feasible set. The cutting planes are determined via approximate integer programming. Enumeration of the possible remainders gives a$$2^{O(n)}n^n$$2O(n)nnalgorithm for general integer programming. This matches the current best bound of an algorithm by Dadush (Integer programming, lattice algorithms, and deterministic, vol. Estimation. Georgia Institute of Technology, Atlanta, 2012) that is considerably more involved. Our algorithm also relies on a newasymmetric approximate Carathéodory theoremthat might be of interest on its own. Our second method concerns integer programming problems in equation-standard form$$Ax = b, 0 \le x \le u, \, x \in {\mathbb {Z}}^n$$Ax=b,0xu,xZn. Such a problem can be reduced to the solution of$$\prod _i O(\log u_i +1)$$iO(logui+1)approximate integer programming problems. This implies, for example thatknapsackorsubset-sumproblems withpolynomial variable range$$0 \le x_i \le p(n)$$0xip(n)can be solved in time$$(\log n)^{O(n)}$$(logn)O(n). For these problems, the best running time so far was$$n^n \cdot 2^{O(n)}$$nn·2O(n).

    more » « less
  5. Abstract

    Let$$(h_I)$$(hI)denote the standard Haar system on [0, 1], indexed by$$I\in \mathcal {D}$$ID, the set of dyadic intervals and$$h_I\otimes h_J$$hIhJdenote the tensor product$$(s,t)\mapsto h_I(s) h_J(t)$$(s,t)hI(s)hJ(t),$$I,J\in \mathcal {D}$$I,JD. We consider a class of two-parameter function spaces which are completions of the linear span$$\mathcal {V}(\delta ^2)$$V(δ2)of$$h_I\otimes h_J$$hIhJ,$$I,J\in \mathcal {D}$$I,JD. This class contains all the spaces of the formX(Y), whereXandYare either the Lebesgue spaces$$L^p[0,1]$$Lp[0,1]or the Hardy spaces$$H^p[0,1]$$Hp[0,1],$$1\le p < \infty $$1p<. We say that$$D:X(Y)\rightarrow X(Y)$$D:X(Y)X(Y)is a Haar multiplier if$$D(h_I\otimes h_J) = d_{I,J} h_I\otimes h_J$$D(hIhJ)=dI,JhIhJ, where$$d_{I,J}\in \mathbb {R}$$dI,JR, and ask which more elementary operators factor throughD. A decisive role is played by theCapon projection$$\mathcal {C}:\mathcal {V}(\delta ^2)\rightarrow \mathcal {V}(\delta ^2)$$C:V(δ2)V(δ2)given by$$\mathcal {C} h_I\otimes h_J = h_I\otimes h_J$$ChIhJ=hIhJif$$|I|\le |J|$$|I||J|, and$$\mathcal {C} h_I\otimes h_J = 0$$ChIhJ=0if$$|I| > |J|$$|I|>|J|, as our main result highlights: Given any bounded Haar multiplier$$D:X(Y)\rightarrow X(Y)$$D:X(Y)X(Y), there exist$$\lambda ,\mu \in \mathbb {R}$$λ,μRsuch that$$\begin{aligned} \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}-\mathcal {C})\text { approximately 1-projectionally factors through }D, \end{aligned}$$λC+μ(Id-C)approximately 1-projectionally factors throughD,i.e., for all$$\eta > 0$$η>0, there exist bounded operatorsABso thatABis the identity operator$${{\,\textrm{Id}\,}}$$Id,$$\Vert A\Vert \cdot \Vert B\Vert = 1$$A·B=1and$$\Vert \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}-\mathcal {C}) - ADB\Vert < \eta $$λC+μ(Id-C)-ADB<η. Additionally, if$$\mathcal {C}$$Cis unbounded onX(Y), then$$\lambda = \mu $$λ=μand then$${{\,\textrm{Id}\,}}$$Ideither factors throughDor$${{\,\textrm{Id}\,}}-D$$Id-D.

    more » « less