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We study the non-klt locus of singularities of pairs. We show that given a pair (X, B) and a projective morphism X → Z with connected fibres such that −(KX +B) is nef over Z, the non-klt locus of (X, B) has at most two connected components near each fibre of X → Z. This was conjectured by Hacon and Han. In a different direction we answer a question of Mark Gross on connectedness of the non-klt loci of certain pairs. This is motivated by constructions in Mirror Symmetry.

Finite Quot schemes were used by Bertram, Johnson, and the first author to study Le Potier’s strange duality conjecture on del Pezzo surfaces when one of the moduli spaces is the Hilbert scheme of points. In order to rigorously enumerate the finite Quot scheme, we study the moduli space of limit stable pairs in which the target has rank one on a smooth complex projective surface. We obtain an embedding of this moduli space into a smooth space that induces a perfect obstruction theory. This obstruction theory yields a virtual fundamental class that can be computed explicitly. Because the moduli space coincides with the Quot scheme when they have dimension 0, this makes the desired count of the finite Quot scheme in Bertram et al. rigorous. As another application, we obtain a universality result for tautological integrals on the moduli space of stable pairs.

In this paper, we prove that the degree of regularity of square systems, a subfamily of the HFE systems, over a prime finite field of odd characteristic q is exactly q and, therefore, prove that inverting square systems algebraically using Gröbner basis algorithm is exponential, when q = \Omega(n), where n is the number of variables of the system.

This paper studies a notion of enumerative invariants for stable A-branes, and discusses its relation to invariants defined by spectral and exponential networks. A natural definition of stable A-branes and their counts is provided by the string theoretic origin of the topological A-model. This is the Witten index of the supersymmetric quantum mechanics of a single D3 brane supported on a special Lagrangian in a Calabi-Yau threefold. Geometrically, this is closely related to the Euler characteristic of the A-brane moduli space. Using the natural torus action on this moduli space, we reduce the computation of its Euler characteristic to a count of fixed points via equivariant localization. Studying the A-branes that correspond to fixed points, we make contact with definitions of spectral and exponential networks. We find agreement between the counts defined via the Witten index, and the BPS invariants defined by networks. By extension, our definition also matches with Donaldson-Thomas invariants of B-branes related by homological mirror symmetry.

We study hybrid models arising as homological projective duals (HPD) of certain projective embeddings f:X→P(V) of Fano manifolds X. More precisely, the category of B-branes of such hybrid models corresponds to the HPD category of the embedding f. B-branes on these hybrid models can be seen as global matrix factorizations over some compact space B or, equivalently, as the derived category of the sheaf of A-modules on B, where A is an A_∞ algebra. This latter interpretation corresponds to a noncommutative resolution of B. We compute explicitly the algebra A by several methods, for some specific class of hybrid models, and find that in general it takes the form of a smash product of an A_∞ algebra with a cyclic group. Then we apply our results to the HPD of f corresponding to a Veronese embedding of projective space and the projective embedding of Fano complete intersections in P^n.

We study BPS states of 5d N=1 SU(2) Yang-Mills theory on S^1×R^4. Geometric engineering relates these to enumerative invariants for the local Hirzebruch surface F_0. We illustrate computations of Vafa-Witten invariants via exponential networks, verifying fiber-base symmetry of the spectrum at certain points in moduli space, and matching with mirror descriptions based on quivers and exceptional collections. Albeit infinite, parts of the spectrum organize in families described by simple algebraic equations. Varying the radius of the M-theory circle interpolates smoothly with the spectrum of 4d N=2 Seiberg-Witten theory, recovering spectral networks in the limit.

The authors describe the relationships between categories of B-branes in different phases of the non-Abelian gauged linear sigma model. The relationship is described explicitly for the model proposed by Hori and Tong with non-Abelian gauge group that connects two non-birational Calabi-Yau varieties studied by Rødland. A grade restriction rule for this model is derived using the hemisphere partition function and it is used to map B-type D-branes between the two Calabi-Yau varieties.

In this paper, we study smooth complex projective varieties X such that some exterior power ⋀^rT_X of the tangent bundle is strictly nef. We prove that such varieties are rationally connected. We also classify the following two cases. If T_X is strictly nef, then X isomorphic to the projective space P^n. If ⋀^2T_X is strictly nef and if X has dimension at least 3, then X is either isomorphic to P^n or a quadric Q^n.

Let (X, g) be a compact Riemannian manifold with quasi-positive Riemannian scalar curvature. If there exists a complex structure J compatible with g, then the Kodaira dimension of (X, J) is equal to −∞ and the canonical bundle KX is not pseudo-effective. We also introduce the complex Yamabe number λc(X) for compact complex manifold X, and show that if λ_c(X) is greater than 0, then κ(X) is equal to −∞; moreover, if X is also spin, then the Hirzebruch A-hat genus \hat{A}(X) is zero.