Latest in math.ag

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Concentration estimates for algebraic intersectionsJun 13 2019We present an approach over arbitrary fields to bound the degree of intersection of families of varieties in terms of how these concentrate on algebraic sets of smaller codimension.
Elliptic curves in hyper-Kähler varietiesJun 13 2019We show that the moduli space of elliptic curves of minimal degree in a general Fano variety of lines of a cubic fourfold is a non-singular curve of genus $631$. The curve admits a natural involution with connected quotient. We find that the general Fano ... More
Fibred algebraic surfaces and commutators in the Symplectic groupJun 13 2019We describe the minimal number of critical points and the minimal number $s$ of singular fibres for a non isotrivial fibration of a surface $S$ over a curve $B$ of genus $1$, constructing a fibration with $s=1$ and irreducible singular fibre with $4$ ... More
Combinatorially equivalent hyperplane arrangementsJun 13 2019We study the combinatorics of hyperplane arrangements over arbitrary fields. Specifically, we determine in which situation an arrangement and its reduction modulo a prime number have isomorphic lattices via the use of minimal strong $\sigma$-Gr\"obner ... More
Automorphism group of principal bundles, Levi reduction and invariant connectionsJun 12 2019Let $M$ be a compact connected complex manifold and $G$ a connected reductive complex affine algebraic group. Let $E_G$ be a holomorphic principal $G$--bundle over $M$ and $T\, \subset\, G$ a torus containing the connected component of the center of $G$. ... More
Relations in the Cremona group over perfect fieldsJun 12 2019For perfect fields $k$ satisfying $[\bar k:k]>2$, we construct new normal subgroups of the plane Cremona group and provide an elementary proof of its non-simplicity, following the melody of the recent proof by Blanc, Lamy and Zimmermann that the Cremona ... More
Virtual classes of parabolic $\mathrm{SL}_2(\mathbb{C})$-character varietiesJun 12 2019In this paper, we compute the virtual classes in the Grothendieck ring of algebraic varieties of $\mathrm{SL}_2(\mathbb{C})$-character varieties over compact orientable surfaces with parabolic points of semi-simple type. When the parabolic punctures are ... More
On the product of the singular values of a binary tensorJun 12 2019A real binary tensor consists of $2^d$ real entries arranged into hypercube format $2^{\times d}$. For $d=2$, a real binary tensor is a $2\times 2$ matrix with two singular values. Their product is the determinant. We generalize this formula for any $d\ge ... More
Unramified F-divided objects and the étale fundamental pro-groupoid in positive characteristicJun 12 2019Fix a scheme $S$ of characteristic $p$. Let $\mathcal M$ be an $S$-algebraic stack and $Fdiv(\mathcal M)$ the stack of $F$-divided objects, that is sequences of objects $x_i\in\mathcal M$ with isomorphisms $\sigma_i:x_i\to F^*x_{i+1}$. Let $\mathcal X$ ... More
Deformation of Milnor AlgebrasJun 12 2019We investigate deformations of Milnor algebras of smooth homogeneous polynomials, and prove in particular that any smooth degree $d$ homogeneous polynomial in $n+1$ variables that is not of Sebastiani-Thom type is determined by the degree $k$ homogeneous ... More
Algebraic cycles and Verra fourfoldsJun 11 2019This note is about the Chow ring of Verra fourfolds. For a general Verra fourfold, we show that the Chow group of homologically trivial $1$-cycles is generated by conics. We also show that Verra fourfolds admit a multiplicative Chow-K\"unneth decomposition, ... More
Relative quantum cohomologyJun 11 2019We establish a system of PDE, called open WDVV, that constrains the bulk-deformed superpotential and associated open Gromov-Witten invariants of a Lagrangian submanifold $L \subset X$ with a bounding chain. Simultaneously, we define the quantum cohomology ... More
Fibrations in sextic del Pezzo surfaces with mild singularitiesJun 11 2019We study sextic del Pezzo surface fibrations via root stacks.
Wild Singularities of Kummer VarietiesJun 11 2019In characteristic $p=2$, we compute the singularities of Kummer varieties arising from products of elliptic curves. This result is generalized to Kummer varieties associated to ordinary abelian varieties.
The Deligne-Illusie Theorem and exceptional Enriques surfacesJun 11 2019Building on the results of Deligne and Illusie on liftings to truncated Witt vectors, we give a criterion for non-liftability that involves only the dimension of certain cohomology groups of vector bundles arising from the Frobenius pushforward of the ... More
Solution of all quartic matrix modelsJun 11 2019We consider the quartic analogue of the Kontsevich model, which is defined by a measure $\exp(-N\,\mathrm{Tr}(E\Phi^2+(\lambda/4)\Phi^4)) d\Phi$ on Hermitean $N \times N$-matrices, where $E$ is any positive matrix and $\lambda$ a scalar. We prove that ... More
Spines for amoebas of rational curvesJun 11 2019To every rational complex curve $C \subset (\mathbf{C}^\times)^n$ we associate a rational tropical curve $\Gamma \subset \mathbf{R}^n$ so that the amoeba $\mathcal{A}(C) \subset \mathbf{R}^n$ of $C$ is within a bounded distance from $\Gamma$. In accordance ... More
Parabolic Hitchin Maps and Their Generic FibersJun 11 2019We set up a BNR correspondence for moduli spaces of vector bundles over a curve with a parabolic structure. This leads to a concrete description of generic fibers of the associated parabolic Hitchin map. We also prove that this map (and a weak version ... More
Differential Topology of Gaussian Random Fields: applications to Random Algebraic GeometryJun 11 2019Using the tools that we have developed in arXiv:1902.03805, we study properties of random Kostlan polynomial maps (viewed as random variables in the space of $C^{\infty}$-maps). We apply these tools to the study of problems in random real algebraic geometry, ... More
The logarithmic gauged linear sigma modelJun 11 2019We introduce the notion of log R-maps, and develop a proper moduli stack of stable log R-maps in the case of a hybrid gauged linear sigma model. Two virtual cycles (canonical and reduced) are constructed for these moduli stacks. The main results are two ... More
Abelian tropical coversJun 10 2019The goal of this article is to classify unramified covers of a fixed tropical base curve $\Gamma$ with an action of a finite abelian group G that preserves and acts transitively on the fibers of the cover. We introduce the notion of dilated cohomology ... More
Equivariant Landau--Ginzburg mirror symmetryJun 10 2019We give a new proof of the computation of Hodge integrals we have previously obtained for the quantum singularity (FJRW) theory of chain polynomials. It uses the classical localization formula of Atiyah--Bott and we phrase our proof in a general framework ... More
A geometric Jacquet-Langlands correspondence for paramodular Siegel threefoldsJun 10 2019We study the Picard-Lefschetz formula for the Siegel modular threefold of paramodular level and prove the weight-monodromy conjecture for its middle degree inner cohomology with arbitrary automorphic coefficients. We give some applications to the Langlands ... More
Steady State Invariants and Multistationarity for Families of Toric Reaction NetworksJun 10 2019We study families of chemical reaction networks, with toric steady states. Larger family members are constructed algorithmically from a smallest network and we show that many results about the entire family can be obtained by studying the small family ... More
Smoothing of rational singularities and Hodge structureJun 10 2019We show that the frontier Hodge numbers $h^{p,q}$ (that is, for $pq(n-p)(n-q)=0$) do not change by passing to a desingularization of the singular fiber of a one-parameter degeneration of smooth projective varieties of dimension $n$ if the singular fiber ... More
Partially complex ranks for real projective varietiesJun 10 2019Let $X(\mathbb {C})\subset \mathbb {P}^r(\mathbb {C})$ be an integral non-degenerate variety defined over $\mathbb {R}$. For any $q\in \mathbb {P}^r(\mathbb {R})$ we study the existence of $S\subset X(\mathbb {C})$ with small cardinality, invariant for ... More
On $2\times 2$ Tropical Commuting MatricesJun 10 2019This paper investigates the geometric properties of the a special case of the two-sided system given by $2 \times 2$ tropical commuting constraints. Given a finite matrix $A \in \mathbb{R}^{2\times 2}$, the paper studies the extreme vertices of the tropical ... More
Monodromy of Rank 2 Parabolic Hitchin SystemsJun 10 2019We study the monodromy of the Hitchin fibration for moduli spaces of parabolic G-Higgs bundles in the cases when G=SL(2,R), GL(2,R) and PGL(2,R) A calculation of the orbits of the monodromy with Z2-coefficients provides an exact count of the components ... More
The stability manifold of local orbifold elliptic quotientsJun 09 2019In this paper, we investigate the stability manifold of local models of orbifold quotients of elliptic curves. In particular, we describe a component of the stability manifold which maps as a covering space onto the universal unfolding space of the mirror ... More
Essential dimension of double covers of symmetric and alternating groupsJun 09 2019I. Schur studied double covers $\widetilde{\Sym}^{\pm}_n$ and $\widetilde{\Alt}_n$ of symmetric groups $\Sym_n$ and alternating groups $\Alt_n$, respectively. Representations of these groups are closely related to projective representations of $\Sym_n$ ... More
The A.B.C.Ds of Schubert calculusJun 09 2019We collect Atiyah-Bott Combinatorial Dreams (A.B.C.Ds) in Schubert calculus. One result relates equivariant structure coefficients for two isotropic flag manifolds, with consequences to the thesis of C. Monical. We contextualize using work of N. Bergeron-F. ... More
Vector bundles on Fano threefolds and K3 surfacesJun 09 2019Let X be a Fano threefold, and let S be a K3 surface in X . Any moduli space M of simple vector bundles on S carries a holomorphic symplectic structure. Following an idea of Tyurin, we show that in some cases, those vector bundles which come from X form ... More
The LLV decomposition of hyper-Kaehler cohomologyJun 08 2019Looijenga-Lunts and Verbitsky showed that the cohomology of a compact hyper-Kaehler manifold $X$ admits a natural action by the Lie algebra $\mathfrak{so} (4, b_2(X)-2)$, generalizing the Hard Lefschetz decomposition for compact Kaehler manifolds. In ... More
Embeddings of maximal tori in groups of type $F_4$Jun 07 2019We classify maximal tori in groups of type $F_4$ over a local or global field of characteristic different from $2$ and $3$. We prove a local-global principle for embeddings of maximal tori in groups of type $F_4$.
The Noether problem for spinor groups of small rankJun 07 2019Building on prior work of Bogomolov, Garibaldi, Guralnick, Igusa, Kordonskii, Merkurjev and others, we show that the Noether Problem for $\operatorname{Spin}_n$ has a positive solution for every $n\leq 14$ over an arbitrary field of characteristic $\neq ... More
An explicit geometric Langlands correspondence for the projective line minus four pointsJun 07 2019This article deals with the tamely ramified geometric Langlands correspondence for GL_2 on $\mathbf{P}_{\mathbf{F}_q}^1$, where $q$ is a prime power, with tame ramification at four distinct points $D = \{\infty, 0,1, t\} \subset \mathbf{P}^1(\mathbf{F}_q)$. ... More
Singularities and radical initial idealsJun 07 2019What kind of reduced monomial schemes can be obtained as a Gr\"obner degeneration of a smooth projective variety? Our conjectured answer is: only Stanley-Reisner schemes associated to acyclic Cohen-Macaulay simplicial complexes. This would imply, in particular, ... More
Concurrent lines on del Pezzo surfaces of degree oneJun 07 2019Let $X$ be a del Pezzo surface of degree one over an algebraically closed field $k$, and let $K_X$ be its canonical divisor. The morphism $\varphi$ induced by the linear system $|-2K_X|$ realizes $X$ as a double cover of a cone in $\mathbb{P}^3$ that ... More
Reductivity of the automorphism group of K-polystable Fano varietiesJun 07 2019We prove that K-polystable log Fano pairs have reductive automorphism groups. In fact, we deduce this statement by establishing more general results concerning the S-completeness and $\Theta$-reductivity of the moduli of K-semistable log Fano pairs. Assuming ... More
Bernstein-Sato polynomials for general ideals vs. principal idealsJun 07 2019We show that given an ideal I generated by regular functions f_1,...,f_r on the smooth complex variety X, the Bernstein-Sato polynomial of I is equal to the reduced Bernstein-Sato polynomial of the function g=\sum_{i=1}^rf_iy_i on the product of X with ... More
Bernstein-Sato polynomials for general ideals vs. principal idealsJun 07 2019Jun 12 2019We show that given an ideal I generated by regular functions f_1,...,f_r on the smooth complex variety X, the Bernstein-Sato polynomial of I is equal to the reduced Bernstein-Sato polynomial of the function g=\sum_{i=1}^rf_iy_i on the product of X with ... More
Quotients of del Pezzo surfacesJun 07 2019Let $\Bbbk$ be any field of characteristic zero, $X$ be a del Pezzo surface and $G$ be a finite subgroup in $\operatorname{Aut}(X)$. In this paper we study when the quotient surface $X / G$ can be non-rational over $\Bbbk$. Obviously, if there are no ... More
Three real Artin-Tate motivesJun 07 2019We analyze the spectrum of the tensor-triangulated category of Artin-Tate motives over the base field R of real numbers, with integral coefficients. Away from 2, we obtain the same spectrum as for complex Tate motives, previously studied by the second-named ... More
Explicit logarithmic formulas of special values of hypergeometric functions 3F2Jun 07 2019In a joint paper [4] by Otsubo, Terasoma and the first author, we proved that the special value 3F2(a,b,q;a+b,q;1) of the generalized hypergeometric function is a linear combination of log of algebraic numbers if the triplet (a,b,q) of rational numbers ... More
A topological groupoid representing the topos of presheaves on a monoidJun 06 2019Butz and Moerdijk famously showed that every (Grothendieck) topos with enough points is equivalent to the category of sheaves on some topological groupoid. We give an alternative, more algebraic construction in the special case of a topos of presheaves ... More
Symmetries of order eight on K3 surfaces without high genus curves in the fixed locusJun 06 2019In this paper we classify non-symplectic automorphisms of order 8 on complex K3 surfaces in case that the fourth power of the automorphism has only rational curves in its fixed locus. We show that the fixed locus is the disjoint union of a rational curve ... More
Virtual classes and virtual motives of Quot schemes on threefoldsJun 06 2019For a simple, rigid vector bundle $F$ on a Calabi-Yau $3$-fold $Y$, we construct a symmetric obstruction theory on the Quot scheme $\textrm{Quot}_Y(F,n)$, and we solve the associated enumerative theory. We discuss the case of other $3$-folds. Exploiting ... More
Veneroni mapsJun 06 2019Veneroni maps are a class of birational transformations of projective spaces. This class contains the classical Cremona transformation of the plane, the cubo-cubic transformation of the space and the quatro-quartic transformation of $\mathbb{P}^4$. Their ... More
Families of Supermanifolds: Splitting Types and Obstruction MapsJun 06 2019In this article we study the notion of supermanifolds families, starting from Green's general classification of supermanifolds. The topics studied divide this article into two distinct parts, labelled I and II respectively. Part I concerns the splitting ... More
From hyperelliptic to superelliptic curvesJun 06 2019In this long survey article we show that the theory of elliptic and hyperelliptic curves can be extended naturally to all superelliptic curves. We focus on automorphism groups, stratification of the moduli space $\mathcal{M}_g$, binary forms, invariants ... More
Modèles géométriques attachés aux paires réductivesJun 05 2019L'objet de cet article d'exposition est de pr\'esenter une introduction \`a l'article "The ext algebra of a quantized cycle" \'ecrit en collaboration avec Damien Calaque, et d'expliquer plus g\'en\'eralement comment le dictionnaire entre th\'eorie de ... More
Low degree points on curvesJun 05 2019In this paper we investigate an arithmetic analogue of the gonality of a smooth projective curve $C$ over a number field $k$: the minimal $e$ such there are infinitely many points $P \in C(\bar{k})$ with $[k(P):k] \leq e$. Developing techniques that make ... More
Estimates for the number of rational points on simple abelian varieties over finite fieldsJun 05 2019Let $A$ be a simple abelian variety of dimension $g$ over the field $\mathbb{F}_q$. The paper provides improvements on the Weil estimates for the size of $A(\mathbb{F}_q)$. For an arbitrary value of $q$ we prove $(\lfloor(\sqrt{q}-1)^2 \rfloor + 1)^g ... More
Rational points and derived equivalenceJun 05 2019We give the first examples of derived equivalences between varieties defined over non-closed fields where one has a rational point and the other does not. We begin with torsors over Jacobians of curves over Q and F_q(t), and conclude with a pair of hyperkaehler ... More
ECH capacities, Ehrhart theory, and toric varietiesJun 05 2019ECH capacities were developed by Hutchings to study embedding problems for symplectic $4$-manifolds with boundary. They have found especial success in the case of certain toric symplectic manifolds where many of the computations resemble calculations ... More
Aspects of Scattering Amplitudes and Moduli Space LocalizationJun 05 2019We elaborate on the recent proposal that intersection numbers of certain cohomology classes on the moduli space of genus-zero Riemann surfaces with punctures compute tree-level scattering amplitudes in quantum field theories. The relevant cohomology classes ... More
Degree 3 unramified cohomology of classifying spaces for exceptional groupsJun 05 2019Let $G$ be a reductive group defined over an algebraically closed field of characteristic $0$ such that the Dynkin diagram of $G$ is the disjoint union of diagrams of types $G_{2}, F_{4}, E_{6}, E_{7}, E_{8}$. We show that the degree $3$ unramified cohomology ... More
Locally Heavy Hyperplanes in MultiarrangementsJun 05 2019Hyperplane Arrangements of rank $3$ admitting an unbalanced Ziegler restriction are known to fulfil Terao's conjecture. This long-standing conjecture asks whether the freeness of an arrangement is determined by its combinatorics. In this note we prove ... More
Critical values of coamoebas of codimension two affine planesJun 05 2019We describe the topology of critical loci of coamoeba of generic affine planes in four-space.
On the Quot scheme $\mathrm{Quot}_{\mathcal O_{\mathbb P^1}^r/\mathbb P^1/k}^d$Jun 05 2019We consider the quot scheme $\mathrm{Quot}^d_{\mathcal F^r/ \mathbb P^1/ k}$ of locally free quotients of $\mathcal F^r:= \bigoplus ^{ r} \mathcal O_{\mathbb P^1 }$ with Hilbert polynomial $p(t)=d$. We prove that it is a smooth variety of dimension $dr$, ... More
Generalized orbifold Euler characteristics on the Grothendieck ring of varieties with actions of finite groupsJun 05 2019The notion of the orbifold Euler characteristic came from physics at the end of 80's. There were defined higher order versions of the orbifold Euler characteristic and generalized ("motivic") versions of them. In a previous paper the authors defined a ... More
Simpson's geometric P=W conjecture in the Painlevé VI case via abelianizationJun 05 2019We use abelianization of Higgs bundles near infinity to prove the homotopy commutativity assertion of Simpson's geometric P=W conjecture in the Painlev\'e VI case.
Lagrangian tens of planes, Enriques surfaces and holomorphic symplectic fourfoldsJun 04 2019Fano models of Enriques surfaces in $\mathbb P^5$. The first one parametrizes the varieties of lines on smooth cubic hypersurfaces containing 10 mutually intersecting planes. The second one is a family of double EPW sextics introduced by K. O'Grady. The ... More
Good Reduction of Unitary Groups of Quaternionic Skew-Hermitian FormsJun 04 2019Given a field $K$ equipped with a set of discrete valuations $V$, we develop a general theory to relate ramification of skew-hermitian forms over a quaternion $K$-algebra $Q$ to ramification of quadratic forms over the function field $K(Q)$ obtained via ... More
Chern numbers of uniruled threefoldsJun 04 2019In this paper we show that the Chern numbers of a smooth Mori fibre space in dimension three are bounded in terms of the underlying topological manifold. We also generalise a theorem of Cascini and the second named author on the boundedness of Chern numbers ... More
Classification of rationally elliptic toric orbifoldsJun 04 2019In this note we classify rationally elliptic compact toric orbifolds up to algebraic isomorphism.
Nilpotence in normed MGL-modulesJun 04 2019We establish a motivic version of the May Nilpotence Conjecture: if E is a normed motivic spectrum that satisfies $E \wedge HZ \simeq 0$, then also $E \wedge MGL \simeq 0$. In words, motivic homology detects vanishing of normed modules over the algebraic ... More
Vertical Vafa-Witten invariantsJun 04 2019We show that \emph{vertical} contributions to (possibly semistable) Tanaka-Thomas-Vafa-Witten invariants are well defined for surfaces with $p_g(S)>0$, partially proving conjectures of \cite{TT2} and \cite{T}. Moreover, we show that such contributions ... More
The hyperplane property for the genus one stable quasimap invariants of hypersurfaceJun 04 2019We analyze the local structure of moduli space of genus one stable quasimaps. Combining it with the p-fields theory developed in \cite{L}, we prove the hyperplane property for genus one invariants of stable quasimaps to hypersurface in $\mathbb{P}^n$
Deligne-Lusztig duality on the stack of local systemsJun 03 2019In the setting of the geometric Langlands conjecture, we argue that the phenomenon of divergence at infinity on Bun_G (that is, the difference between $!$-extensions and $*$-extensions) is controlled, Langlands-dually, by the locus of semisimple $\check{G}$-local ... More
Automorphisms of categories of schemesJun 03 2019Given two schemes $S$ and $S'$, we prove that every equivalence between $\mathbf{Sch}_S$ and $\mathbf{Sch}_{S'}$ comes from a unique isomorphism between $S$ and $S'$. This eliminates all Noetherian and finite type hypotheses from a result of Mochizuki ... More
K-theory formulas for orthogonal and symplectic orbit closuresJun 03 2019The complex orthogonal and symplectic groups both act on the complete flag variety with finitely many orbits. We study two families of polynomials introduced by Wyser and Yong representing the $K$-theory classes of the closures of these orbits. Our polynomials ... More
A Tannakian framework for $G$-displays and Rapoport-Zink spacesJun 03 2019We develop a Tannakian framework for group-theoretic analogs of displays, originally introduced by B\"ultel and Pappas, and further studied by Lau. We use this framework to define Rapoport-Zink functors associated to triples $(G,\{\mu\},[b])$, where $G$ ... More
Global mirrors and discrepant transformations for toric Deligne-Mumford stacksJun 03 2019We introduce a global Landau-Ginzburg model which is mirror to several toric Deligne-Mumford stacks and describe the change of the Gromov-Witten theories under discrepant transformations. We prove a formal decomposition of the quantum cohomology D-modules ... More
On minimal model theory for log abundant lc pairsJun 03 2019Jun 04 2019Under the assumption of the minimal model theory for projective klt pairs of dimension $n$, we establish the minimal model theory for projective lc pairs such that the log canonical divisor is log abundant and its restriction to any lc center has numerical ... More
The Hard Lefschetz Theorem for PL spheresJun 03 2019Jun 10 2019We provide a simpler proof of the hard Lefschetz Theorem for face rings of PL spheres: While the algebraic theory remains the same, we replace the geometric constructions by Pachner's Theorem. This simplifies the reasoning for an important special case ... More
The Hard Lefschetz Theorem for PL spheresJun 03 2019Jun 04 2019We provide a simpler proof of the hard Lefschetz Theorem for face rings of PL spheres: While the algebraic theory remains the same, we replace the geometric constructions by Pachner's Theorem. This simplifies the reasoning for an important special case ... More
The module of Valabrega-Valla of the Jacobian ideal of points in projective planeJun 03 2019The module of Valabrega-Valla of the Jacobian ideal of a reduced projective variety $V$ is the torsion of the Aluffi algebra. One considers the problem of its vanishing in the case of where $V$ is a reduced set of points in the projective plane. It is ... More
Virtual Resolutions of Monomial Ideals on Toric VarietiesJun 03 2019We use cellular resolutions of monomial ideals to prove an analog of Hilbert's syzygy theorem for virtual resolutions of monomial ideals on smooth toric varieties.
Logarithmic concavity for morphisms of matroidsJun 02 2019Morphisms of matroids are combinatorial abstractions of linear maps and graph homomorphisms. We introduce the notion of basis for morphisms of matroids, and show that its generating function is strongly log-concave. As a consequence, we obtain a generalization ... More
Dévisser, découper, éclater et aplatir les espaces de BerkovichJun 01 2019We develop in this article flattening techniques for coherent sheaves in the realm of Berkovich spaces; we are inspired by the general strategy that Raynaud and Gruson have used for dealing with the analogous problem in scheme theory. As an application, ... More
Moduli of hypersurfaces in toric orbifoldsJun 01 2019We construct and study the moduli of hypersurfaces in toric orbifolds. Let $X$ be a projective toric orbifold and $\alpha \in Cl(X)$ an ample class. The moduli space is constructed as a quotient of the linear system $|\alpha|$ by $G = Aut(X)$. Since the ... More
On smooth projective D-affine varietiesJun 01 2019We show various properties of smooth projective D-affine varieties. In particular, any smooth projective D-affine variety is algebraically simply connected and its image under a fibration is D-affine. In characteristic zero such D-affine varieties are ... More
Equivariant Grothendieck-Riemann-Roch theorem via formal deformation theoryJun 01 2019We use the formalism of traces in higher categories to prove a common generalization of the holomorphic Atiyah-Bott fixed point formula and the Grothendieck-Riemann-Roch theorem. The proof is quite different from the original one proposed by Grothendieck ... More
Cohomology ring of the flag variety vs Chow cohomology ring of the Gelfand-Zetlin toric varietyJun 01 2019We compare the cohomology ring of the flag variety $FL_n$ and the Chow cohomology ring of the Gelfand-Zetlin toric variety $X_{GZ}$. We show that $H^*(FL_n, \mathbb{Q})$ is the Gorenstein quotient of the subalgebra $L$ of $A^*(X_{GZ}, \mathbb{Q})$ generated ... More
Three dimensional mirror self-symmetry of the contangent bundle of the full flag varietyJun 01 2019Let X be a holomorphic symplectic variety with a torus T action and a finite fixed point set of cardinality k. Let $A_{I,J}=\left. Stab(J)|_{I}$ be the k by k matrix of restrictions of the elliptic stable envelopes of X to the fixed points. The entries ... More
A Universal HKR TheoremMay 31 2019In this work we study the failure of the HKR theorem over rings of positive and mixed characteristic. For this we construct a \emph{filtered circle} interpolating between the usual topological circle and a formal version of it. By mapping to schemes we ... More
Frieze varieties are invariant under mutationMay 31 2019We define a generalized version of the frieze variety introduced by Lee, Li, Mills, Seceleanu and the second author. The generalized frieze variety is an algebraic variety determined by an acyclic quiver and a generic specialization of cluster variables ... More
A counter-example to the equivariance structure on semi-universal deformationMay 31 2019We provide a counter-example to the the $G$-equivariant structure on semi-universal deformation in the case that $G$ is nonreductive.
Dimensional interpolation and the Selberg integralMay 31 2019Jun 04 2019We show that a version of dimensional interpolation for the Riemann--Roch--Hirzebruch formalism in the case of a grassmannian leads to an expression for the Euler characteristic of line bundles in terms of a Selberg integral. We propose a way to interpolate ... More
Hodge-Riemann bilinear relations for Schur classes of ample vector bundlesMay 31 2019Let $X$ be a $d$ dimensional projective manifold, $E$ be an ample vector bundle on $X$ and $0\le \lambda_N\le \lambda_{N-1} \le \cdots \le \lambda_1 \le \operatorname{rank}(E)$ be a partition of $d-2$. We prove that the Schur class $s_{\lambda}(E)\in ... More
Three theorems on the $\partial\bar{\partial}$-lemmaMay 31 2019We study the $\partial\bar{\partial}$-lemma on projective bundles, blowups and product complex manifolds. We also discuss projective bundle formulas and blowup formulas of de Rham, Dolbeault, Bott-Chern and Aeppli cohomologies on (not necessarily compact) ... More
Stable models of plane quartics with hyperelliptic reductionMay 31 2019Let C/K: F = 0 be a smooth plane quartic over a complete discrete valuation field K. In a previous paper the authors togetehr with Q. Liu give various characterizations of the reduction (i.e. non-hyperelliptic genus 3 curve, hyperelliptic genus 3 curve ... More
Simply connected Sasaki-Einstein rational homology 5-spheresMay 30 2019We completely determine which simply connected rational homology 5-spheres admit Sasaki-Einstein metrics.
The Fourier-Mukai transform made easyMay 30 2019We propose a slightly modified definition for the Fourier-Mukai transform (on abelian varieties) that makes it much easier to remember various formulas. As an application, we give relatively short proofs for two important theorems: the characterization ... More
Two remarks on sums of squares with rational coefficientsMay 30 2019There exist homogeneous polynomials $f$ with $\mathbb Q$-coefficients that are sums of squares over $\mathbb R$ but not over $\mathbb Q$. The only systematic construction of such polynomials that is known so far uses as its key ingredient totally imaginary ... More
K-Theoretic $I$-function of $V//_θ \mathbf{G}$ and ApplicationMay 30 2019In this paper, we compute K-theoretic $I$-function with level structure (defined by quasi-map theory) of GIT-quotient of a vector space via abelian and non-abelian correspondence. As a consequence, we generalize Givental-Lee's result to find the analogous ... More
K-Theoretic $I$-function of $V//_θ \mathbf{G}$ and ApplicationMay 30 2019Jun 07 2019In this paper, we compute K-theoretic $I$-function with level structure (defined by quasi-map theory) of GIT-quotient of a vector space via abelian and non-abelian correspondence. As a consequence, we generalize Givental-Lee's result to find the analogous ... More
Twisted Schubert polynomialsMay 30 2019We prove that twisted versions of Schubert polynomials defined by $\widetilde{\mathfrak S}_{w_0} = x_1^{n-1}x_2^{n-2} \cdots x_{n-1}$ and $\widetilde{\mathfrak S}_{ws_i} = (s_i+\partial_i)\widetilde{\mathfrak S}_w$ are monomial positive and give a combinatorial ... More
Regularity of the superstring supermeasure and the superperiod mapMay 30 2019The supermeasure whose integral is the genus $g$ vacuum amplitude of superstring theory is potentially singular on the locus in the moduli space of supercurves where the corresponding even theta-characteristic has nontrivial sections. We show that the ... More