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We develop an analogue of Eisenbud-Floystad-Schreyer's Tate resolutions for toric varieties. Our construction, which is given by a noncommutative analogue of a Fourier- Mukai transform, works quite generally and provides a new perspective…
We show a precise proof of Steenbrink's formula for the spectrum of convenient Newton non-degenerate functions, and prove the symmetry of combinatorial polynomials in the simplicial case. Combined with the modified Steenbrink conjecture for…
We introduce the D=4 twistorial tensionfull bosonic string by considering the canonical twistorial 2--form in two--twistor space. We demonstrate its equivalence to two bosonic string models: due to Siegel (with covariant worldsheet…
Based on recent advances on the relation between geometry and representation theory, we propose a new approach to elliptic Schubert calculus. We study the equivariant elliptic characteristic classes of Schubert varieties of the generalized…
We exhaustively analyze the toric symmetries of CP^3 and its toric blowups. Our motivation is to study toric symmetry as a computational technique in Gromov-Witten theory and Donaldson-Thomas theory. We identify all nontrivial toric…
We prove that the Okounkov body of a big divisor with respect to a general flag on a smooth projective surface whose pseudo-effective cone is rational polyhedral decomposes as the Minkowski sum of finitely many simplices and line segments…
We continue to explore the numerical nature of the Okounkov bodies focusing on the local behaviors near given points. More precisely, we show that the set of Okounkov bodies of a pseudoeffective divisor with respect to admissible flags…
Let $T$ be a maximal torus of a semisimple complex algebraic group, $\mathrm{BS}(s)$ be the Bott-Samelson variety for a sequence of simple reflections $s$ and $\mathrm{BS}(s)^T$ be the set of $T$-fixed points of $\mathrm{BS}(s)$. We prove…
The global Okounkov body of a projective variety is a closed convex cone that encodes asymptotic information about every big line bundle on the variety. In the case of a rank two toric vector bundle E on a smooth projective toric variety,…
We study solutions for boson stars in the multi-scalar field theory with global symmetry $[U(1)]^N$. The properties of the boson stars are investigated by the Newtonian approximation with the large coupling limit. Our purpose is to study…
S.~Elnitsky (1997) gave an elegant bijection between rhombic tilings of $2n$-gons and commutation classes of reduced words in the symmetric group on $n$ letters. P.~Magyar (1998) found an important construction of the Bott-Samelson…
We give a new construction of the Bott-Samelson variety $Z$ as the closure of a $B$-orbit in a product of flag varieties $(G/B)^l$. This also gives an embedding of the projective coordinate ring of the variety into the function ring of a…
It is shown that new leading ($\al'$) as well as all-order solutions of String theory can be obtained by taking appropriate singular limits of the known solutions. We give several leading order solutions for the bosonic as well as the…
Let $k$ be a perfect field and let $C_0:f=0$ be a smooth curve in the torus $\mathbb{G}_{m,k}^2$. Let $\mathbb{T}_\Delta$ be the toric variety associated to the Newton polygon of $f$. Extending the toric resolution of $C_0$ on…
For every rational homology 3-sphere with 2-torsion only we construct a unified invariant (which takes values in a certain cyclotomic completion of a polynomial ring), such that the evaluation of this invariant at any odd root of unity…
Sinha and Vafa \cite {sinha} had conjectured that the $SO$ Chern-Simons gauge theory on $S^3$ must be dual to the closed $A$-model topological string on the orientifold of a resolved conifold. Though the Chern-Simons free energy could be…
A Newton-Okounkov polytope of a complete flag variety can be turned into a convex geometric model for Schubert calculus. Namely, we can represent Schubert cycles by linear combinations of faces of the polytope so that the intersection…
Grothendieck and Harder proved that every principal bundle over the projective line with split reductive structure group (and trivial over the generic point) can be reduced to a maximal torus. Furthermore, this reduction is unique modulo…
We describe a class of isolated nondegenerate hypersurface singularities that give a polynomial contribution to Batyrev's stringy E-function. These singularities are obtained by imposing a natural condition on the facets of the Newton…
Tree tensor networks such as the tensor train format are a common tool for high dimensional problems. The associated multivariate rank and accordant tuples of singular values are based on different matricizations of the same tensor. While…