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 Compositio MathematicaJournal Prestige (SJR): 3.139 Citation Impact (citeScore): 1Number of Followers: 0      Subscription journal ISSN (Print) 0010-437X - ISSN (Online) 1570-5846 Published by Cambridge University Press  [387 journals]
• COM volume 156 Issue 7 Cover and Front matter
• PubDate: 2020-07-01T00:00:00.000Z
DOI: 10.1112/S0010437X19007917
Issue No: Vol. 156, No. 7 (2020)

• COM volume 156 Issue 7 Cover and Back matter
• PubDate: 2020-07-01T00:00:00.000Z
DOI: 10.1112/S0010437X19008030
Issue No: Vol. 156, No. 7 (2020)

• Computing a categorical Gromov–Witten invariant
• Authors: Andrei Căldăraru; Junwu Tu
Pages: 1275 - 1309
Abstract: We compute the $g=1$ , $n=1$ B-model Gromov–Witten invariant of an elliptic curve $E$ directly from the derived category $\mathsf{D}_{\mathsf{coh}}^{b}(E)$ . More precisely, we carry out the computation of the categorical Gromov–Witten invariant defined by Costello using as target a cyclic $\mathscr{A}_{\infty }$ model of $\mathsf{D}_{\mathsf{coh}}^{b}(E)$ described by Polishchuk. This is the first non-trivial computation of a positive-genus categorical Gromov–Witten invariant, and the result agrees with the prediction of mirror symmetry: it matches the classical (non-categorical) Gromov–Witten invariants of a symplectic 2-torus computed by Dijkgraaf.
PubDate: 2020-07-01T00:00:00.000Z
DOI: 10.1112/S0010437X20007174
Issue No: Vol. 156, No. 7 (2020)

• Homological mirror symmetry for higher-dimensional pairs of pants
• Authors: Yankı Lekili; Alexander Polishchuk
Pages: 1310 - 1347
Abstract: Using Auroux’s description of Fukaya categories of symmetric products of punctured surfaces, we compute the partially wrapped Fukaya category of the complement of $k+1$ generic hyperplanes in $\mathbb{CP}^{n}$ , for $k\geqslant n$ , with respect to certain stops in terms of the endomorphism algebra of a generating set of objects. The stops are chosen so that the resulting algebra is formal. In the case of the complement of $n+2$ generic hyperplanes in $\mathbb{C}P^{n}$ ( $n$ -dimensional pair of pants), we show that our partial wrapped Fukaya category is equivalent to a certain categorical resolution of the derived category of the singular affine variety $x_{1}x_{2}\ldots x_{n+1}=0$ . By localizing, we deduce that the (fully) wrapped Fukaya category of the $n$ -dimensional pair of pants is equivalent to the derived category of $x_{1}x_{2}\ldots x_{n+1}=0$ . We also prove similar equivalences for finite abelian covers of the $n$ -dimensional pair of pants.
PubDate: 2020-07-01T00:00:00.000Z
DOI: 10.1112/S0010437X20007150
Issue No: Vol. 156, No. 7 (2020)

• The test function conjecture for local models of Weil-restricted groups
• Authors: Thomas J. Haines; Timo Richarz
Pages: 1348 - 1404
Abstract: We prove the test function conjecture of Kottwitz and the first named author for local models of Shimura varieties with parahoric level structure attached to Weil-restricted groups, as defined by B. Levin. Our result covers the (modified) local models attached to all connected reductive groups over $p$ -adic local fields with $p\geqslant 5$ . In addition, we give a self-contained study of relative affine Grassmannians and loop groups formed using general relative effective Cartier divisors in a relative curve over an arbitrary Noetherian affine scheme.
PubDate: 2020-07-01T00:00:00.000Z
DOI: 10.1112/S0010437X20007162
Issue No: Vol. 156, No. 7 (2020)

• The bounded height conjecture for semiabelian varieties
• Authors: Lars Kühne
Pages: 1405 - 1456
Abstract: The bounded height conjecture of Bombieri, Masser, and Zannier states that for any sufficiently generic algebraic subvariety of a semiabelian $\overline{\mathbb{Q}}$ -variety $G$ there is an upper bound on the Weil height of the points contained in its intersection with the union of all algebraic subgroups having (at most) complementary dimension in  $G$ . This conjecture has been shown by Habegger in the case where $G$ is either a multiplicative torus or an abelian variety. However, there are new obstructions to his approach if $G$ is a general semiabelian variety. In particular, the lack of Poincaré reducibility means that quotients of a given semiabelian variety are intricate to describe. To overcome this, we study directly certain families of line bundles on  $G$ . This allows us to demonstrate the conjecture for general semiabelian varieties.
PubDate: 2020-07-01T00:00:00.000Z
DOI: 10.1112/S0010437X20007198
Issue No: Vol. 156, No. 7 (2020)

• Summands of theta divisors on Jacobians
• Authors: Thomas Krämer
Pages: 1457 - 1475
Abstract: We show that the only summands of the theta divisor on Jacobians of curves and on intermediate Jacobians of cubic threefolds are the powers of the curve and the Fano surface of lines on the threefold. The proof only uses the decomposition theorem for perverse sheaves, some representation theory and the notion of characteristic cycles.
PubDate: 2020-07-01T00:00:00.000Z
DOI: 10.1112/S0010437X20007204
Issue No: Vol. 156, No. 7 (2020)

• Étale Steenrod operations and the Artin–Tate pairing
• Authors: Tony Feng
Pages: 1476 - 1515
Abstract: We prove a 1966 conjecture of Tate concerning the Artin–Tate pairing on the Brauer group of a surface over a finite field, which is the analog of the Cassels–Tate pairing. Tate asked if this pairing is always alternating and we find an affirmative answer, which is somewhat surprising in view of the work of Poonen–Stoll on the Cassels–Tate pairing. Our method is based on studying a connection between the Artin–Tate pairing and (generalizations of) Steenrod operations in étale cohomology. Inspired by an analogy to the algebraic topology of manifolds, we develop tools allowing us to calculate the relevant étale Steenrod operations in terms of characteristic classes.
PubDate: 2020-07-01T00:00:00.000Z
DOI: 10.1112/S0010437X20007216
Issue No: Vol. 156, No. 7 (2020)

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