Papers

We construct, through Hodge-theoretical methods, what we call the $E_r$-Jacobian torus, the $E_r$-Albanese torus and the $E_r$-Albanese map of any compact complex manifold that is either {\it page-(r-1)-$\partial\bar\partial$} or {\it $E_r$-sGG}. These two classes of manifolds, the former of which is contained in the latter, have been introduced recently by both authors jointly with J. Stelzig, respectively by the first-named author. A Hodge theory is also developed for the latter class of manifolds. We then apply our results to give structure theorems and an identity of algebraic dimensions in terms of the $E_r$-Albanese map and torus. Other applications result in cohomological and metrical theorems for the $6$-dimensional sphere when it is equipped either with a hypothetical complex structure or with the complex structures very recently claimed to exist in the literature. For example, we show that in the latter case no strongly Gauduchon metric exists on $S^6$.

We construct and study the Nash manifold of four-point configurations in the real plane modulo the action of a Nash subgroup of the group of similarity transformations. We do so by using the finer notion of a quadrangle in place of that of a four-point configuration, since this retains limiting line data in degenerations. We prove that the space ${\mathcal Q}$ of quadrangles is an 8-dimensional Nash manifold and that, for every Nash subgroup $G$ of the similarity group, the orbit space ${\mathcal Q}/G$ is a Nash manifold. We define a geometric invariant, called aspect, with values in $[-1,1]$, and prove that, over each of the intervals $(-1,0)$ and $(0,1)$, the corresponding part of ${\mathcal Q}/G$ is Nash diffeomorphic to the product of the interval with a fixed fiber. Thus the nonexceptional part of the moduli problem reduces to the analysis of two model fibers, each having a natural geometric interpretation in terms of the quadrangles themselves.

We build a cohesive topological theory of orientations on moduli stacks for non-compact complex-analytic Calabi-Yau $4$-folds by extending and generalising the work of Joyce and Upmeier, and the work of Bojko. As a result, we get, in particular, canonical family orientations for all moduli spaces of Pandharipande-Thomas pairs. Our proofs use the algebraic topology of mapping spaces representing $KU$- and $KO$- theory, as well as some properties of the Lie groups $E_8$ and, surprisingly, $\operatorname{HSpin}(16)$. This is part of a joint project with Felix Thimm dedicated to the proof of the $GW/PT$-correspondence for Calabi-Yau $4$-folds by implementing Pardon's programme in dimension $4$.

We study quasi-modularity of normalized $q$-traces on bosonic Fock spaces associated with finite-dimensional quadratic spaces and superspaces. For a natural class of operators obtained from free-boson fields and their descendants, we prove that their normalized $q$-traces are quasi-modular, with weight bounded by the sum of the weights of the insertions. As applications, we prove Qin's quasi-modularity conjecture for tautological integrals on Hilbert schemes of points of a surface with numerically trivial canonical class, obtain quasi-modularity of arbitrary zero-mode correlation functions in the Heisenberg vertex operator algebra, and give a new proof of the Bloch–Okounkov quasi-modularity theorem as a rank-one specialization of our general result.

Let $G$ be a simply connected semisimple algebraic group. On the (undecorated) braid variety associated with a power of a Coxeter element, there is a finite-order autoequivalence, called the Zamolodchikov transformation, induced by cyclic rotation of words. In this paper, we show that the coarse moduli space of the fixed-point stack of this autoequivalence is isomorphic to the spectrum of the fusion ring of the corresponding affine Lie algebra. The above results also extend to twisted Coxeter elements, and the fusion rings of all affine Lie algebras are obtained in this way.

A net of conics can remain unchanged as the four-point configuration generating it varies. We study this phenomenon over a field ${\mathbb{k}}$ of characteristic different from $2$. We allow degenerate quadrangles, consisting of four not necessarily distinct points and six joining lines, provided that three sides form a nondegenerate triangle. The net of a quadrangle is obtained by adjoining constants to the defining polynomials in the pencil generated by its pairs of opposite sides, viewed as conics. We call two quadrangles entangled when they generate the same net. We prove that entanglement is equivalent to the apparently asymmetric condition that every side of one quadrangle bisects the other quadrangle at that side's midpoint. The line-pair members of the net form a bisector field that determines the entire net, so entanglement is also equivalent to equality of bisector fields. Finally, we prove that the locus of centers of the central conics in the net, together with one nonparallel line-pair member, determines the net.

Given a subvariety of the Grassmannian and a data point, we seek to find a point on the subvariety minimizing the Euclidean distance to the data point. The number of complex critical points of this optimization problem is the Euclidean distance (ED) degree. We show that the ED discriminant of the whole Grassmannian, that is, the set of data points with a number of critical points different than the ED degree, is the discriminant of the characteristic polynomial of the data as a projection matrix. Another closely connected algebraic complexity measure for a subvariety in the Grassmannian is the Grassmann distance (GD) degree, which is the number of complex critical points of the distance optimization problem when the data point itself is in the Grassmannian. We give formulae for ED and GD degrees of geometrically meaningful subvarieties of the Grassmannian, namely, products of smaller Grassmannians, matroid realization varieties, and Schubert varieties.

The Mumford–Morita–Miller classes on the moduli space of Riemann surfaces may be defined not only in ordinary cohomology but in any complex-oriented cohomology theory. We begin their investigation, emphasising the new phenomena that arise.

Liftability was first introduced for paving matroids, where it was used to systematically construct polynomials in the ideals of the associated matroid varieties. In this paper, we generalise the notion of liftability to matroids with realisable truncations. We additionally introduce the notion of generic realisability of a matroid; a strong notion of realisability that implies both realisability and liftability of the matroid. We study the behaviour of liftability and generic realisability under the weak order of matroids within the lattice of matroids with fixed truncation. We develop a unified framework for studying liftability of matroids through Oxley–Wang derived matroid of generic matroid realisations. We characterise liftability and generic realisability in terms of Oxley–Wang derived matroids. Moreover, we give an algebraic description of the bases of the generic Oxley–Wang derived matroids in terms of coordinate sections of matroid realisation spaces and relate our constructions to slack realisation spaces of matroids. As an application of our framework, we obtain a combinatorial characterisation of liftable and generically realisable paving matroids and connect it with Whiteley's theory of scene analysis. In particular, our approach gives a new proof, via Oxley–Wang derived matroids, of Whiteley's characterisation of hypergraphs admitting non-trivial (sharp) scenes. Finally, we compute the generic Oxley–Wang derived matroid for a special class of sparse paving matroids.

In an earlier paper of P. Rossi and the first author, a construction of a wide class of quantum hierarchies was presented, with the input data given by an arbitrary cohomological field theory. A crucial role in the construction is played by the double ramification (DR) cycles on the moduli spaces of stable curves. As a special case, one gets a quantization of the hierarchy of the Intermediate Long Wave (ILW) equation. The ILW equation has two famous limits: the Korteweg–de Vries equation and the Benjamin–Ono (BO) equation, and for the latter one a quantum hierarchy was constructed by Nazarov and Sklyanin. In our paper, we propose a conjecture describing a relation between a limit of the quantum ILW hierarchy and the Nazarov–Sklyanin hierarchy and present various checks to support it.

We study the Lie algebra $L_{\ge2}$ of divergence-free polynomial vector fields on $k^n$, $n\ge3$, with coefficients of degree at least two, graded by coefficient degree. Over every field its derived algebra in degree $d\ge3$ is the space of exact fields, those whose contraction with the volume form is an exact form, and it is already spanned by brackets with quadratic fields. In characteristic zero this is the whole degree-$d$ component. In characteristic $p>0$ the abelianization is nonzero above degree two exactly in the degrees $d\ge(p-1)(n-1)$ with $d\equiv1-n\pmod p$, and Cartier descent identifies it with a Frobenius twist of a rational $\mathrm{GL}_n$-module, tensored with a power of the determinant. For $p\ge5$ each exact component is obtained from the previous one by bracketing with quadratic fields.

We compute the rational cohomology of the fine compactified universal Jacobians over the moduli spaces of stable curves of genus $2$ with $0$ and $1$ marked points.

We show that all closed geodesics in any arithmetic complex hyperbolic surface of second type are simple. These include the fake projective planes studied by Mumford, Klingler, Prasad-Yeung, and Cartwright-Steger.

Laface and Ugaglia recently constructed an irrational one-point Seshadri constant on the blow-up of $\mathbb{P}^2$ at nine very general points by combining a dihedral orbit on $\mathbb{P}^1\times\mathbb{P}^1$, a sequence of de Jonquières transformations, and a reflection argument along a $(-4)$-curve with balanced normal bundle. We show that the same mechanism extends uniformly to every odd integer $n\geq 5$. For $n=2k+1$ we prove $$ \varepsilon\bigl(\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}(n-4,1);p_1,\ldots,p_{2n}\bigr)=\sqrt{\frac{n-4}{n}} $$ at $2n$ very general points, and already at a very general free orbit of a fixed dihedral group of order $2n$. For every odd $n\geq 7$ this produces an explicit ample line bundle on the blow-up of $\mathbb{P}^2$ at $k+7=(n+13)/2$ very general points whose one-point Seshadri constant at a very general point equals $$ 2\sqrt{n(n-4)}. $$ More precisely, after one quadratic transformation we obtain the ample divisor $$ L_n=(3n-4)H-nE_1-(n-2)(E_2+\cdots+E_5)-4(E_6+\cdots+E_{k+5})-2(E_{k+6}+E_{k+7}), $$ with $L_n^2=4n(n-4)$ and $\varepsilon(L_n;x)=\sqrt{L_n^2}$. We also isolate an abstract balanced-reflection principle underlying the construction: a nef class on a special fiber can be reflected across a rational curve of square $-2a$ whenever the curve has normal bundle $\mathcal{O}_{\mathbb{P}^1}(-a)^{\oplus 2}$ in the total space, and the reflected class is nef on very general fibers.

In this paper, we study the number of points of $\mathbb{Z}$-schemes $X$ of finite type over the finite rings $\mathbb{Z}/p^m\mathbb{Z}$ for positive integers $m$ and prime numbers $p$. Motivated by Igusa's conjecture for exponential sums and the Lang-Weil estimate, we establish an estimate of $X(\mathbb{Z}/p^m\mathbb{Z})$ in terms of $m$, $X(\mathbb{Z}/p^{m-1}\mathbb{Z})$ and poles of the $p$-adic Igusa local zeta function of $X$ uniformly in $m\geq 2$ and $p$. Our estimate might be viewed as an inductive version for counting points of schemes over finite rings of the Lang-Weil estimate. Our estimate agrees with the expected bound of Igusa's conjecture for exponential sums if $X_{\mathbb{Q}}$ is not a locally complete intersection having only rational singularities. This can be regarded as an analogous version of the estimate of exponential sums modulo powers of primes of polynomials with non-rational singularities given by Cluckers, Mustaţǎ and the author in 2019. In the case that $X_{\mathbb{Q}}$ is a locally complete intersection having only rational singularities, our estimate is very close to the expected bound of Igusa's conjecture for exponential sums. To prove our estimate, beside using the usual tools such as resolutions of singularities and Denef's formula for Igusa local zeta functions, we also use the Hermit interpolation formula to study the numerators of Igusa local zeta functions. By the same method, we are able to approach Igusa's conjecture for exponential sums in the case of rational singularities. Lastly, to have further applications of our result in the future, we study heavily on $p$-adic analysis to give an upper bound of the real parts of non-trivial poles of Igusa local zeta functions in terms of suitable generalized log canonical thresholds.

Let $X$ be a nonsingular Hermitian variety of even dimension over a finite field, and let $\Lambda$ be the lattice spanned by the classes of its maximal totally isotropic linear subspaces defined over the base field. By work of Dummigan, Dummigan–Tiep and Shimada, these classes span the middle cohomology, and the intersection form on the primitive part is definite, as predicted by the standard conjecture of Hodge type. We give an elementary proof, which avoids the representation theory of finite unitary groups and uses only the dual polar graph attached to the Hermitian form. It shows that the intersection form on the primitive part is an explicit positive multiple of a Euclidean inner product. The advantage of the new proof is that it also controls $\Lambda$ $p$-adically. If $X$ is defined over $\F_{q^2}$ and has dimension $2m$, every $p$-adic elementary divisor of the intersection matrix of these subspaces divides $q^m$. Consequently $q^m$ times every algebraic cycle of codimension $m$ is numerically equivalent to an integral combination of them. As a by-product, the standard conjecture of Hodge type holds for varieties finitely covered by Hermitian varieties, and the $p$-adic bound descends along finite morphisms of degree prime to $p$.

Let $K$ be any complete non-Archimedean field, not necessarily spherically complete. We prove that every polystable representation of a finite quiver over $K$ admits a split harmonic norm in the sense of Haiden–Katzarkov–Kontsevich–Pandit (HKKP), answering a question from their preprint Towards Categorical Kähler Geometry. Moret-Bailly's closed-image theorem gives a uniform separation from representations with destabilizing subobjects. A finite estimate on norm weights then yields a linear coercivity bound for stable representations and hence a minimum of the extended HKKP potential on the complete CAT(0) space of all norms. A direct first-variation computation identifies split minimizers with HKKP harmonic norms. Strict supporting hinges reduce split attainment to semistable reduction for quivers with arbitrary real radius labels. Small changes of the lifted splitting make each reversed extension nonsplit; a finite slope–rank descent proves termination. Consequently, every minimum in the completed norm space is attained by a split norm with the same arrow singular profiles, without restrictions on the value group or the quiver. Nonsplit minimizers illustrate split attainment in one and two radius modifications, and a two-parameter Berkovich analytic family separates polystability, semistability, and harmonicity of a chosen split norm.

For a third-order symmetric tensor, the Comon rank gap is the difference between its CP rank and its symmetric rank. We study direct sums of Lovitz's rational $27$-dimensional tensor, whose two ranks are $55$ and $56$. Two copies have ranks $110$ and $112$, and three copies have ranks $165$ and $168$, over both the real and the complex fields. Thus direct sums yield concise cubic tensors on spaces of dimensions $54$ and $81$ with Comon rank gaps two and three.

We prove a factorization theorem for ordinary topological recursion near a regular nodal degeneration of a spectral curve, allowing logarithmic spectral coordinates. The partition function for closed genera $g\ge2$ factors into the partition function of the normalization, a universal Gaussian vacuum, and the exponential of a connected graph sum. This graph sum has strictly positive order in the vanishing period, and thus the factorization recovers the conifold gap, identifies the term of degree zero in the vanishing period with the free energy of the normalization, and gives finite graph formulas for the coefficients of positive powers of the period. At fixed genus, the regular series converges jointly in the period and the parameters along the nodal locus. The Gaussian neck tensors can be expressed in terms of relative Gromov–Witten invariants of a parametrized $\mathbb P^1$. As examples, we use remodeling to obtain Gromov–Witten factorizations for local $\mathbb P^2$ and local $\mathbb F_0$ in the conifold frame, with backgrounds $\mathbb C^3$ and the resolved conifold, respectively. The analogous local $\mathbb F_1$ factorization has background $\mathcal O(1)\oplus\mathcal O(-3)$ over $\mathbb P^1$.

We prove that the tangent sheaf of $\SU_C(n,L)$ is slope stable for every smooth complex projective curve $C$ of genus $g\geq2$, every $n\geq2$, and every determinant $L$. For $g\geq3$, we prove a more general stability theorem on moduli spaces of principal bundles for connected reductive groups, using cohomology sheaves associated with representations in place of tangent sheaves. The adjoint representation gives tangent sheaf stability for the case of almost simple groups. The moduli spaces may be singular, and stability holds with respect to every ample Cartier polarization.

We study abeloid varieties over the adic space $\operatorname{Spa}(K, K^+)$, where $K$ is a complete non-Archimedean field and $K^+$ is a valuation ring of arbitrary rank. We extend the well-known Raynaud uniformization to this setting. We also show the existence of suitable Néron models in algebraic and formal situations over higher rank valuation rings: We prove the existence of algebraic (lft-)Néron models for semi-abelian varieties over valuation rings with algebraically closed fraction fields, as well as, up to finite separable base change, over complete rank one valuation rings with algebraically closed residue field. Building on this result, we show the existence of formal marked Néron models for abeloid varieties over $\operatorname{Spa}(K, K^+)$ after finite separable base change. We also investigate the connection between the higher rank Raynaud extension and the formal Néron model.

We construct a smooth projective Calabi–Yau threefold with a flat elliptic fibration of Mordell–Weil rank twelve and Hodge numbers $(h^{1,1},h^{2,1})=(25,1)$. The construction starts from the Kummer surfaces associated with the products of a fixed elliptic curve and the level-four modular family. The threefold is birational to an anticanonical quartic in $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}(3)^{\oplus2}\oplus\mathcal{O}_{\mathbb{P}^1}(4)\oplus\mathcal{O}_{\mathbb{P}^1})$. We give twelve independent sections and construct a projective crepant resolution of the associated Weierstrass model.

We develop a geometric way to uniformly carry out Hörmander's $L^2$-estimates for Hermitian-Yang-Mills connections with bounded energy on Kähler surfaces. We then show the Uhlenbeck convergence of smooth HYM connections on Kähler surfaces can be realized as algebraic convergence inside an analytic flat family. As applications, we prove a continuity theorem for smoothable families and a boundedness result for $\mu$-semistable sheaves with fixed topology. We also discuss potential applications for the moduli problem.

We generalize Serre's conditions for coherent sheaves to triangulated categories equipped with a $t$-structure and duality, which we call Verdier-type categories. We present examples of such categories and their associated Serre conditions, then focus on constructible sheaves, where we refer to these conditions as geometric Serre conditions. We investigate their relationship with perversity and illustrate the theory through a range of examples. Finally, we prove a geometric version of the Auslander-Reiten conjecture.

Given a $\mathbb{Q}$-Gorenstein normal projective Calabi-Yau variety $Y$ with ordinary double points as singularities and a small crepant resolution $\pi\colon X\to Y$ with $X$ in the Fujiki class $\mathcal{C}$, we establish uniform estimates away from the exceptional locus where the given metrics may degenerate. When $Y$ is a threefold, we prove that these metrics also converge in a Gromov-Hausdorff sense to the metric completion of $Y$ on the regular locus. In particular, our results extend the work of Song on a conjecture of Candelas and de la Ossa.

This paper studies flasque resolutions for linear reductive group schemes over arbitrary bases, extending Colliot-Thelene's theory. A flasque resolution exists for any linear reductive group scheme over a connected base scheme (resp. a quasi-compact, quasi-separated base scheme). The construction relies on Borovoi's fundamental group and descent theory involving automorphisms of Chevalley groups. Applications include R-equivalence for reductive group schemes over rings and over a field of cohomological dimension at most 2.

Let $F$ be a projective variety, let $H$ be a normally generated line bundle on $F$, and let $X\in |H^{\otimes d}|$ be a general projective variety. We establish a uniform criterion, formulated in terms of the graded Betti numbers of the section ring $R(F,H)$, that provides sufficient conditions for every automorphism of $X$ to extend to an automorphism of $F$. This extends the classical results of Matsumura-Monsky on automorphisms of hypersurfaces in projective spaces to a broader class of polarized varieties $(F,H)$, including rational homogeneous varieties of Picard number one, smooth Fano threefolds with their anticanonical polarization, and examples of hyperkähler and Calabi-Yau varieties.

We prove that, for every genus $g\geq 3$, the primary obstruction of each parity component of the moduli superstack of smooth unpunctured super Riemann surfaces is nonzero, both algebraically and holomorphically. Consequently, both components are non-projected and, in particular, non-split: neither admits an algebraic or holomorphic retraction onto its reduced moduli stack of spin curves.

Infinite-distance limits play a central role in the Swampland program, but understanding their behaviour at intersections of boundary divisors is subtle, as it requires control over simultaneous limits of several moduli. To address this, we establish a necessary and sufficient criterion for a boundary stratum of arbitrary codimension in the complex-structure moduli space of Calabi-Yau manifolds to lie at infinite Weil-Petersson distance. In particular, intersections of infinite-distance boundary divisors necessarily remain at infinite distance. The result holds for the full Weil-Petersson metric and arbitrary trajectories, including those with unrestricted hierarchies in the saxions and axion winding. In compactifications of Type IIB string theory on Calabi-Yau threefolds, our proof is based on an inequality relating the Weil-Petersson length of a trajectory to the BPS mass associated with an auxiliary charge vector. If this mass tends to zero while the charge-to-mass ratio remains uniformly bounded, every trajectory approaching the limit has infinite length. Using asymptotic Hodge theory, we construct an auxiliary charge vector satisfying these conditions and derive the criterion directly from the limiting mixed Hodge structure associated with the degeneration. More generally, our result applies to polarized variations of Hodge structure of Calabi-Yau type. In particular, we prove Wang's conjecture on finite Weil-Petersson distance for degenerations over higher-dimensional bases.

We study the marked analytic moduli space of polystable degree-zero $\mathrm{SL}_r(\C)$-Higgs bundles over Teichmüller space. The connected components of the orbit-type loci defined by the full determinant-one stabilizers form a complex Whitney stratification. The projection to Teichmüller space is a holomorphic submersion on every stratum. Each stratum carries a canonical relative holomorphic symplectic form. These forms determine a vertical holomorphic Poisson bracket on the reduced structure sheaf, whose symplectic leaves are the connected orbit-type strata in the fibers. On each fiber, the Hitchin–Kobayashi correspondence is biholomorphic on every stratum and identifies its hyperkähler structure. The harmonic representatives and vertical hyperkähler tensors vary smoothly along the strata of the total space.